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., 
 
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 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 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 
 
 A 
 
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 
 
 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 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 /a 2 
 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 - fir 2 ) 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 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. 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 
 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 H 2 0. 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 
 
 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. 
 (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 
 
 153A [July 29, 1892.} The Rotational Ether, Compressible or In- 
 compressible ... ... ... ... ... ... ... 243 
 
 154 First Rotational Analogy : Magnetic Force compared with 
 
 Velocity 245 
 
 155 Circuital Indeterminateness of the Flux of Energy in general 247 
 
 156 Second Rotational Analogy : Induction compared with 
 
 Velocity 249 
 
 157 [Aug. 5, 1892.} Probability of the Kinetic Nature of Magnetic 
 
 Energy 250 
 
 158 Unintelligibility of the Rotational Analogue for a Conducting 
 
 Dielectric when Magnetic Energy is Kinetic ... ... ... 252 
 
 159 The Rotational Analogy, with Electric Energy Kinetic, ex- 
 
 tended to a Conducting Dielectric by means of Translational 
 Friction 253 
 
 160 [Sept. 2, 1892.} Symmetrical Linear Operators, direct and in- 
 
 verse, referred to the Principal Axes ... ... ... ... 256 
 
 161 Geometrical Illustrations. The Sphere and Ellipsoid. Inverse 
 
 Perpendiculars and Maccullagh's Theorem ... ... ... 259 
 
 162 Internal Structure of Linear Operators. Manipulation of 
 
 several when Principal Axes are Parallel ... ... ... 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 283 
 
 169 The Dyadical Structure of Linear Operators ... 285 
 
 170 Hamilton's Theorem ... 287 
 
 171 [Nov. 18, 1892.] Hamilton's Cubic and the Invariants con- 
 
 cerned 289 
 
 172 The Inversion of Linear Operators ... ... ... ... 293 
 
 173 Vector Product of a Vector and a Dyadic. The Differentiation 
 
 of Linear Operators ... ... ... ... ... ... 295 
 
 174 [Dec. 9, 1892.] Summary of Method of Vector Analysis ... 297 
 
 175 Uosuitability of Quaternions for Physical Needs. Axiom : 
 
 Once a Vector, always a Vector ... ... ... ... 301 
 
 CHAPTER IV. 
 
 THEORY OF PLANE ELECTROMAGNETIC WAVES. 
 (Pages 306 to 466.) 
 
 SECTION. 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 
 
 180 Relations between E and H in a Pure Wave. Effect of Self- 
 
 Induction. Fatuity of Mr. Preece's " KR 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- Surf aces. Electromagnetic 
 
 versus Elastic Solid Theories ... . . ... ... ... 325 
 
 184 A Perfect Conductor is a Perfect Obstructor, but does not 
 
 absorb the Energy of Electromagnetic Waves ... ... 328 
 
XX. CONTENTS. 
 
 SECTION. PAGE. 
 
 185 [Feb. 24, 1893.] Conductors at Low Temperatures 330 
 
 186 Equilibrium of Radiation. The Mean Flux of Energy ... 331 
 
 187 The Mean Pressure of Radiation 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. 
 Distinct Effects of Electric and Magnetic Conductance ... 369 
 
 199 Application to Waves along Straight Wires 374 
 
 200 [May 26, 1893.] Transformation of Variables from Electric 
 
 and Magnetic Force to Voltage and Gaussage ... ... 378 
 
 201 Transformation of the Circuital Equations to the Forms in- 
 
 volving Voltage and Gaussage ... ... ... ... ... 381 
 
 202 [June 9, 1893.] The Second Circuital Equation for Wires in 
 
 Terms of V and C when Penetration is Instantaneous ... 386 
 
 203 The Second Circuital Equation when Penetration is Not In- 
 
 stantaneous. Resistance Operators, and their Definite 
 Meaning ... ... ... ... ... ... ... ... 390 
 
 204 Simply Periodic Waves Easily Treated in Case of Imperfect 
 
 Penetration ... ... ... ... ... ... ... 393 
 
 205 [July 7, 1893.} Long Waves and Short Waves. Identity 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 Con- 
 ductance and Permittance when regarded as a Condenser ... 446 
 
 220 Inductive Leaks applied to Submarine Cables 447 
 
 221 General Theory of Transmission of Waves along a Circuit with 
 
 or without Auxiliary Devices ... ... ... ... ... 449 
 
 222 Application of above Theory to Inductive Leakance ... ... 453 
 
 APPENDIX B. 
 A GRAVITATIONAL AND ELECTROMAGNETIC ANALOGY. 
 
 Parti. [July 14, 1893.} 455 
 
 Part II. [Aug. 4,1893.} 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 en- 
 titled *' Electromagnetic Induction and its Propagation," com- 
 menced 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 
 those mentioned for the discontinuance. We 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. Per- 
 haps 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 physi- 
 cal 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 con- 
 dense 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 investiga- 
 tions to himself. But that does not answer, in a general way, 
 though it may do so sometimes. It is like putting away seed 
 in a mummy 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 dis- 
 coveries, and to hoard them up as Cavendish did, without any 
 
 B 2 
 
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 crimi- 
 nality 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 
 themselves organists? We 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 there- 
 fore 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 in- 
 imitable 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 connec- 
 tion 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 electro- 
 dynamic 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 experi- 
 mental basis. The interest excited has been immense, and the 
 theorist can now write about electromagnetic waves without in- 
 curring 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 pheno- 
 mena 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 accu- 
 rately 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 investi- 
 gations 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. 
 
 by their prejudice from giving it fair development. We 
 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 in- 
 tellect. 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 men- 
 tioned, 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 in- 
 cludes, 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. 1 1 
 
 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 
 or at any time, is merely a confession of ignorance. We have 
 no evidence of any such discontinuities. We 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 appro- 
 priately 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 impor- 
 tant 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 ap- 
 plicable 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- 
 pressing the real essence. Unfortunately, in my 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 collec- 
 tion 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 hypo- 
 theses 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 attrac- 
 tions or repulsions of electricity, or of magnetism, or of elements 
 of electric currents upon one another, abolished, and electro- 
 magnetic 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 heaven- 
 sent 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 dis- 
 covery 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- 
 cerned. We 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, 
 within certain limits. We 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 magnetisa- 
 tion (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 en- 
 largement 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 ^produce visible effects. The conception of fields 
 of force nr J irally follows, with the mapping out of space 
 by means o. -ines or tubes of force definitely distributed. A 
 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. 
 
 If E be the electric force at any point and D the displace- 
 ment, we have 
 
 D = cE: (1) 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 21 
 
 and similarly, if H 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, indi- 
 cating 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 fir 1 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, with- 
 out 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 con- 
 cerned 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. 
 
 When the fluxes vary, their rates of increase B and D are the 
 velocities corresponding to the forces E and H, provided no 
 other effects are produced. The activity of E is ED, and 
 that of H is HB. The work spent in producing the fluxes 
 (not counting what may be done simultaneously in other ways) 
 is, therefore 
 
 U = TE dD, T =| B H rfB, ... (5) 
 
22 ELECTROMAGNETIC THEORY. CH. IT. 
 
 where U 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 de- 
 finite 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 consider- 
 able 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 eolo- 
 tropic. 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 prin- 
 cipal 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. If E 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 
 flux components obtain D 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 
 a pair be equal, then E and D 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//A O 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. 
 
 If, for example, H be imagined to be the velocity of a sub- 
 stance, then JfiH 2 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 JcE 2 is the stored energy of the strain. 
 
 Dissipation of Energy. The Conduction-current ; Conduc- 
 tivity and Resistivity. The Electric Current. 
 
 24. Besides influencing the values of .the ether constants w 
 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 electro- 
 magnetic 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) = Q 1 + 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) 
 
 then K is the density of the magnetic conduction current, and 
 the rate of waste of energy is 
 
 Q 2 = HK = <7H 2 (12) 
 
 The activity of the magnetic force is now 
 
 H(K + B) = Q 2 + T (13) 
 
 Compare this equation with (10). The magnetic current is 
 K + B. 
 
 As there is (I believe) no evidence that the property sym- 
 bolised 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 ; 
 it is the part K 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 
 state of things at a point is concerned. We 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- 
 ticular unit. We 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 
 the component perpendicular to the area. We 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 prac- 
 tical 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 
 shape be perfectly insulated, except at two places, A 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) 
 
 where R and K 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 = RC 2 = 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) 
 
 if D 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 J2cE 2 . 
 
 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 cur- 
 rent (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 arc r 
 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 + SV 2 
 
 \ .... (6) 
 
 representing the waste in Joulean heating and the rate of in- 
 crease 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 mag- 
 netic 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. Thomp- 
 son 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 
 as the force. If H 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 
 displacement by treating the ratio B/H as a function of H 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-induc- 
 tion, 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 C r 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 C r Let also B A be the 
 induction through the circuit of C r Then 
 
 BX-LA, ...... (8) 
 
 where, by what has already been explained, LH is the perme- 
 ance 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) 
 
 by using the first expression in (7), remembering that H 
 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 = (L 1 N)C . . . . (8a) 
 to express the flux of induction ; and by (9) and (Sa), 
 
 ...... (9o) 
 
 if L = N 2 L 1} to express the energy. This L is the inductance 
 of the coil. It is N 2 times the permeance of the magnetic 
 circuit. 
 
 Again, regarding the coil as a single circuit, B : N 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 1 N 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-induc- 
 tion. 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 fact r 
 consistent with the original meaning, I believe, as used by 
 Sir W. Thomson in connection with his " electromagnetic defi- 
 nition " 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 inde- 
 pendently 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) 
 
 where E is the circuital voltage, C the current, and R the re- 
 sistance. And in the magnetic circuit we have a formally 
 similar relation, 
 
 H=L-IB, (2) 
 
 where H is the circuital gaussage, B the induction, and Lr 1 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 
 the assertion that H 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, mean- 
 ing 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 circuita- 
 tion of the force is then called its " curl." Thus, if J be the 
 electric current and G the magnetic current, the two laws are 
 
 curlH^J, (4) 
 
 -cur!E 1 = G, (5) 
 
 where E x and H x are the electric and magnetic force of the 
 field. We 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- 
 rent J. Its edge is then the circuit to which H a belongs in (4). 
 The gaussage in this circuit measures the current-density. 
 Similarly, regarding (5), the voltage in a unit circuit perpen- 
 dicular to the magnetic current measures its density (nega- 
 tively). 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 mag- 
 netic 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 pre- 
 sent, 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 
 what results, viz., waste of energy, Q per second, and increase per 
 second of the electric energy U 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. Or, if Q be the 
 total waste, and similarly U 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) 
 
 and A 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 
 and magnetic currents. We 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) 
 
 Instead of (7). This is the integral equation of activity. We 
 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. 
 
 38. The distinction between H x and H and between E x and 
 E is often a matter of considerable importance. We have 
 
 (10) 
 (11) 
 
38 ELECTROMAGNETIC THEORY. CH. II. 
 
 Now it is E and H that are effective in producing fluxes. Thus 
 E is the force of the flux D and also of ; and H 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, (12) 
 
 -curl (E-e) = G, (13) 
 
 equivalent to (4) and (5). To distinguish from the forces of 
 the fluxes, I sometimes call E l and H 3 the forces " of the field." 
 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 mag- 
 netic 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 defi- 
 nition 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 
 uniform in its properties. We 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 x be 
 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, 
 it is clear that we have not got to the root of the matter. We 
 want another Newton, the Newton of molecular physics. Facts 
 there are in plenty to work upon, and perhaps another heaven- 
 born genius may come to make their meaning plain. Pro- 
 perties 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) 
 
 the prefix V 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. 
 
 Example. A Stationary Electromagnetic Sheet. 
 
 45. It should be remembered that we regard the dis- 
 placement 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 
 very simple case of an infinite plane lamina of E and H travel- 
 ling at the speed of light v perpendicularly to itself through a 
 homogeneous dielectric. This is possible, as will appear later, 
 when E and H 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." 
 
 46. A 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. 
 
 47. A third method of arriving at (13) is by considering 
 the rate of change of the amount of induction through a 
 moving circuit. We 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 
 assume that the circulation of the electric force E (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 
 magnetic conductivity, when G 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 cir- 
 cuit, 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, 
 is the element of unit length. Now, suppose the induction at AB 
 is perpendicular to the plane of the paper, directed downwards, 
 and that it moves from right to left perpendicularly across 
 AB. Let also from A to B 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 
 of its increase, so far as AB 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 
 direction is from A 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 
 impressed electric force per unit length in AB 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 support- 
 ing 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 indepen- 
 dent 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 indif- 
 ferent 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 
 
 D = />, B = o-, ..... (5) 
 
 at any distance. 
 
 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 displace- 
 ment, and how it is to be measured, consider a point- 
 source 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 displace- 
 ment 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 to- 
 fixed 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 
 
 if w 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 
 at any point A 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 
 displacement from the charge at A 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) com- 
 bined 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 dis- 
 placement 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 
 form, it changes the law of circuitation (curl H = J) referred to 
 fixed space, so as to refer it in the same form to the moving 
 medium. 
 
 The result is H = 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. 
 
 (2.) Source and Medium in Relative Motion. A 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 dis- 
 placement 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." 
 Let A (in Fig. 1) be the initial position of the charge when it 
 first begins to move, and let AC be the direction of its sub- 
 sequent motion. Describe a sphere of radius AB = 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 
 charge at every moment of its passage from A to B 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 con- 
 cerned. But the spherical surface constitutes an electro- 
 magnetic 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 
 paper above A (at E, for example), and down through the 
 paper below A (at F, for example). The tangential displace- 
 ment and induction surface-densities (or fluxes per unit area of 
 the sheet), say, D and B , are connected by the equation 
 
 or, = cv . 
 
 Or, if E and H be the equivalent forces got by dividing by 
 c and by ft respectively, then, since fj.cv 2 = 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 
 that on the D 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), ac- 
 companied 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 
 of stoppage and after. Describe a sphere of radius vt, with B 
 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 is- 
 described 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 dis- 
 placement 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 
 speed v, so that the relative motion is the same as before. We 
 must now suppose B to be at rest, the charge being there origi- 
 nally, and remaining there, whilst it is A that is travelling 
 from right to left, and the spherical surface has a motion com- 
 pounded of expansion from the centre A 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, with- 
 out 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 understand- 
 ing 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 stop- 
 page 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 corre- 
 sponding 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 propaga- 
 tion 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 sepa- 
 rated 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 the- 
 induction 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. 
 
 In Fig. 4, A 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 con- 
 tinuously 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 electrifi- 
 cation 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 elec- 
 trification 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, q l and q^ for example, on a spherical sheet. One 
 way is to put - q t and q 2 at the centre, and superpose the 
 two solutions. Or the charges - (q l and q 2 ) may be put exter- 
 nally, 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 
 formula, or B = /*VvD , where D 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 disturb- 
 ances. 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- 
 cognised and explicitly formularised. We 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 l be a unit vector 
 along r from the element to the point where H is reckoned. 
 The intensity of H 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 
 the current-element, or the lines of H are circles about this axis. 
 But from the Maxwellian point of view this field of H 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 
 sense. We 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 informa- 
 tion) 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 con- 
 sequences. 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 cur- 
 rents. 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 im- 
 provement in electrical theory conceivable. The electric 
 current in a nonconductor was the very thing wanted to co- 
 ordinate 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 recom- 
 mendations (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 pre- 
 judiced 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 obstruc- 
 tion evidenced by a carping and unreceptive spirit. The 
 labours of many may be required before a satisfactory elemen- 
 tary 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. 
 
 F 2 
 
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 con- 
 vection-current or electrification in motion as a part of the true 
 current. Although Maxwell did not himself explicitly repre- 
 sent 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 experi- 
 mental origin, of course, and, like the first, was long in approxi- 
 mating 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 
 they operated in his " general equations of propagation." We 
 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 properties, the} 7 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 conduc- 
 tion 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) 
 
 - curl (E - e - e) = G = K + B + wcr. . . (2) 
 Now transfer the e and h terms to the right side, producing 
 
 curl (H-h ) = J =0 + D + u/o + j,. . . (3) 
 
 - curl (E - e c ) = G = K + 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 cur- 
 rent 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 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 73 
 
 standpoint of theoretical dynamics. But only (roughly speak- 
 ing) about the middle of the present century did it become 
 recognised by scientific men as a universal truth, extending to 
 all the phenomena of Nature. This arose from the experi- 
 mental demonstrations given (principally those of Joule), that 
 in various cases in which energy disappeared from one form, 
 it was not lost, but made its appearance in other forms, and 
 to the same amount. 
 
 The principle is now-a-days a sure article of scientific faith. 
 Its ultimate basis is probably the conviction (a thoroughly 
 reasonable one) that the laws of motion hold good in the in- 
 visible world as well as in the visible, confirmed by the repeated 
 experimental verification. But the principle is now believed in 
 quite generally ; by the ordinary unscientific man, for example, 
 who could not tell you correctly what the principle meant, or 
 energy either, to save his life, although he might tell you that 
 he could not conceive the possibility of energy being destroyed. 
 He goes by faith, having taken it in when young. 
 
 So it will be with the modern view of the ether as the 
 medium through which energy is being sent when a wire sup- 
 ports a current. Only train up the young to believe this, and 
 they will afterwards look upon the notion of its going through 
 the conductor as perfectly absurd, and will wonder how anyone 
 ever could have believed it. But before this inevitable state of 
 things comes to pass, an intermediate period must be passed 
 through, in which scientific men themselves are learning to 
 believe the modern view thoroughly from the evidence of its 
 truth, and to teach it to others. This period is certainly not 
 yet come to its end ; for almost weekly we may read about 
 " the velocity of electricity in wires," such language emanating 
 from scientific men who are engaged in repeating and extend 
 ing Hertz's experiments. 
 
 The principle of the continuity of energy is a special form of 
 that of its conservation. In the ordinary understanding of the 
 conservation principle it is the integral amount of energy that 
 is conserved, and nothing is said about its distribution or its 
 motion. This involves continuity of existence in time, but not 
 necessarily in space also. 
 
 But if we can localise energy definitely in space, then we are 
 bound to ask how energy gets from place to place. If it 
 
74 ELECTROMAGNETIC THEORY. CH. II. 
 
 possessed continuity in time only, it might go out of existence 
 at one place and come into existence simultaneously at 
 another. This is sufficient for its conservation. This view, 
 however, does not recommend itself. The alternative is to 
 assert continuity of existence in space also, and to enunciate 
 the principle thus : 
 
 When energy goes from place to place, it traverses the 
 intermediate space. 
 
 This is so intelligible and practical a form of the principle, 
 that we should do our utmost to carry it out. 
 
 The idea that energy has position, therefore, naturally 
 involves the idea of a fiux of energy. Let A be the vector flux 
 of energy, or the amount transferred per unit of time per unit 
 of area normal to the direction of its transfer ; and let T be 
 the density of the energy. We may then write 
 
 conv A = f, (5) 
 
 or divA=-T (6) 
 
 That is, the convergence of the flux of energy is accounted for 
 by increased energy in the unit volume where it converges ; 
 or, the divergence of the flux of energy, or the rate at which 
 it leaves the unit volume, is accompanied by simultaneous and 
 corresponding decrease in the density of the energy. 
 
 Examples. Convection of Energy and Flux of Energy due 
 to an active Stress. Gravitational difficulty. 
 
 68. Now, there are numerous cases in which the flux of 
 energy is perfectly plain, and only needs to be pointed out to- 
 be recognised. The simplest of all is the mere convection of 
 energy by motion of the matter with which it is associated. A 
 body in translational motion, for instance, carries its energy 
 with it; not merely the kinetic energy of its translational 
 motion, but also its rotational energy if it- rotates, and like- 
 wise the energy of any internal vibratory or rotatory motions 
 it may possess, and other energy not known to be kinetic, 
 and therefore included under "potential energy," which we 
 may have good reason to localise in the body. Even 
 if part of the energy be outside the body, yet, if it be 
 associated with the body, it will travel with it, on the whole, 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 75 
 
 and so (at least sometimes) may be considered to be rigidly 
 attached to it. This case may be regarded as the convection 
 of energy, if we do not go too closely into detail. 
 
 When energy is thus conveyed, the energy flux is qT, where 
 q is the velocity and T the density of the energy conveyed (of 
 any kind). If, then, all energy were conveyed, equation (5) 
 would become 
 
 convaT = t (7) 
 
 Now this equation, if we regard T as the density of matter, is 
 the well known equation of continuity of matter used in hydro- 
 dynamics and elsewhere. 
 
 This brings us to Prof. Lodge's theory of the identity of 
 energy. (Phil. Mag., 1885.) Has energy personal identity, 
 like matter ? I cannot see it, for one ; and think it is pushing 
 the principle of continuity of energy, which Prof. Lodge was 
 writing about, too far. It is difficult to endow energy with 
 objectivity, or thinginess, or personal identity, like matter. The 
 relativity of motion seems to be entirely against the idea. Nor 
 are we able to write the equation (7) in all cases. Energy may 
 be transferred in other ways than by convection of associated 
 matter. If we atomise the energy we can then imagine the q. 
 above to be the velocity of the energy, not of the matter. But 
 as the science of dynamics is at present understood, we cannot 
 make use of this idea profitably, I think. 
 
 The chief reason why the founders of the modern science of 
 energy did not explicitly make use of the idea of the continuity 
 of energy was probably the very obscure nature of gravitational 
 energy. Where was it before it became localised as the kinetic 
 energy of a mass ? It is of no use to call it potential energy 
 if that is to explain anything, which it does not. It would, seem 
 that the energy must have been in the ether, somewhere, and 
 was transferred into the body, somehow. This makes the ether 
 the great store-house of all gravitational energy. But we are 
 entirely ignorant of its distribution in the ether, and of its 
 mode of transference. 
 
 Observe here that ether must be regarded as a form of 
 matter, because it is the recipient of energy, and that is the 
 characteristic of ordinary matter. There is an unceasing- 
 enormous flux of energy through the ether from the Sun, for 
 
76 ELECTROMAGNETIC THEORY. OH. R 
 
 instance, and we know that it takes several minutes to come : 
 it comes through the ether, without bringing the ether with it; 
 it is not a convection of energy, therefore. Of course, to avoid 
 confusion, it is well to distinguish ether from ordinary matter 
 by its separate name ; but it is important to note that it has 
 some of the characteristics of ordinary matter. It need not be 
 gravitating matter ; it is, perhaps, more likely to be the medium 
 of gravitational action than to gravitate itself. 
 
 In default of ability to represent the flux of gravitational 
 energy its entry into a body must be otherwise represented than 
 by the convergence of a flux. Turn (7) into 
 
 f<l + convA = f (8) 
 
 Here fq. is the activity of the force f, and indicates energy com- 
 municated to the unit volume in an unstated manner, generated 
 within the unit volume, so to speak. The force f is thus an 
 impressed force. By this device we can allow for unknown 
 fluxes of energy. We should also explicitly represent the 
 convective flux. Thus, 
 
 fq + conv(qT + A) = f, .... (9) 
 
 where A is the flux of energy other than convective, not asso- 
 ciated with f. 
 
 When a stress works, for example, we shall have a flux of 
 energy, A, which can hardly be considered to be convective, in 
 the sense explained, a flux of energy through matter or ether, 
 as in wave propagation through an elastic solid, or radiation 
 through the ether. We can, however, sometimes reduce it to 
 the form qT 1? and then it would appear to be convection of 
 energy. 
 
 This occurs in a moving frictionless fluid, for example, when 
 the stress is an isotropic pressure p. The flux of energy other 
 than convective is jpq. But when we pass to a less ideal case, 
 as an elastic solid, in which the stress is of a more general 
 character, the energy flux expressive of the activity of the 
 stress takes the form 
 
 A=-P fl (z, ...... (10) 
 
 where P 7 is the vector expressing the pull per unit area on the 
 plane perpendicular to the velocity q. In this equation q is 
 the tensor of q. 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 77 
 
 Specialised form of expression of the Continuity of Energy. 
 
 69. Passing to more practical and specialised forms of 
 expression of the continuity of energy, it is to be observed 
 first that we are not usually concerned with a resultant flux 
 of energy from all causes, but only with the particular forms- 
 relating to the dynamical question that may be under con- 
 sideration. Secondly, that it is convenient to divide the 
 energy above denoted by T into different parts, denoting 
 potential energy, and kinetic energy, and wasted energy. 
 Thus we have the following as a practical form, 
 
 fq + conv[A + q(U + T)] = Q + U + T, . (11) 
 
 where the terms in the square brackets indicate the energy 
 flux, partly convective, with the factor q, U being the density 
 of potential energy, and T that of kinetic energy, and partly 
 non-convective, viz., A, which may be due to a working stress. 
 Also fq is the activity of impressed force (here merely a trans- 
 lational force). Thus on the left side of equation (11) we- 
 have a statement of the supply of energy to the unit volume- 
 fixed in space. On the right side we account for it by the rates- 
 of increase of the stored potential and kinetic energy, and by 
 Q, which means the rate of waste of energy in the unit volume. 
 The wasted energy is also stored, at least temporarily ; but not 
 recoverably, so that we may ignore energy altogether after it is 
 once wasted. 
 
 In place of the term fq, in which the impressed force is 
 translational, we may have other terms possessing a similar 
 meaning, indicating a supply of energy from certain sources. 
 These sources, too, may be not external, but internal or intrin- 
 sic, as for instance when there is a thermo-electric or voltaic- 
 source of energy within the unit volume considered. 
 
 Also, the terms A, U, T, and Q may have to be split up 
 into different parts, according to the nature of the dynamical 
 connections. The important thing to be grasped is, that when- 
 ever we definitely localise energy we can obtain an equation 
 showing its continuity in space and time, and that when we 
 can only partially localise it, we can still, by proper devices, 
 allow for the absence of definiteness. The equation is simply 
 the equation of activity of the dynamical system, suitably 
 
78 
 
 ELECTROMAGNETIC THEORY. 
 
 CH. II. 
 
 arranged to show the flux of energy, and can always be obtained 
 when the equations of motion are known and also the nature of 
 the stored and wasted energies. 
 
 Electromagnetic Application. Medium at Eest. The 
 Poynting Flux. 
 
 70. Now, in the electromagnetic case, the "equations of 
 motion " are the two circuital laws, and to form the equation 
 of activity, we may multiply (1) by (H - h - h) and (2) by 
 (E - e - e) and add the results ; or else multiply (3) by (H - h ) 
 and (4) by (E - e ) and add the results. The equation of activity 
 thus obtained has then to be dynamically interpreted in accord- 
 ance with the principle of continuity of energy. 
 
 When the medium is stationary, there is no difficulty with 
 the interpretation. We obtain 
 
 e J + h G = Q + U + T + divW, . . (12) 
 where W is a new vector given by 
 
 W = V(E-e )(H-h ), . . . (13) 
 
 and U is the electric energy, T the magnetic energy, and Q the 
 waste per unit volume. 
 
 On the left side of (12) is exhibited the rate of supply of 
 energy from intrinsic sources. On the right side it is accounted 
 for partly by the waste Q, and by increase of the stored energy 
 U and T. The rest is exhibited as the divergence of the flux W 
 given by (13). We conclude, therefore, that W expresses the 
 flux of energy in the electromagnetic field when it is stationary. 
 
 This remarkable formula was first discovered and interpreted 
 by Prof. Poynting [Phil. Trans., 1884, Pt. 2], and independently 
 by myself a little later. It was this discovery that brought 
 the principle of continuity of energy into prominence. But it 
 should be remembered that there is nothing peculiarly electro- 
 magnetic about a flux of energy. It is here made distinct, 
 because the energy is distinctly localised in Maxwell's theory. 
 
 The flux of energy takes place in the direction perpendicular 
 to the plane containing the electric and magnetic forces (of the 
 field), say 
 
 W = VE 1 H 1 , (14) 
 
 if E = e + E 1 ; H = h + H r 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 79 
 
 The formula is not a working formula, in general, for to 
 know W we must first know the distribution of the electric and 
 magnetic forces. But it is a valuable and instructive formula 
 for all that. Its discovery furnished the first proof that Max- 
 well's theory implied that the insulating medium outside a 
 conducting wire supporting an electric current was the medium 
 through which energy is transferred from its source at a dis- 
 tance, provided that we admit the postulated storage of energy. 
 For inside a wire the electric force is axial, and the magnetic 
 force circular about the axis ; the flux of energy is therefore 
 radial. When the current is steady, it comes from the boundary 
 of the wire, and ceases at its axis. It delivers up energy on the 
 way, which is wasted in the Joule-heating. But when the 
 current is not steady, the magnetic and electric energy will be 
 also varying. 
 
 If we work out the distributions of electric and magnetic 
 force outside the wire, according to the conditions to which it 
 is subjected and its environment, we can similarly fully trace 
 how the energy is supplied to the wire from its source. It is 
 passed out from the source into the dielectric medium, and 
 then converges upon the wire where it is wasted. The flux of 
 energy usually takes place nearly parallel to the wire (because 
 the electric force is nearly perpendicular to its boundary) ; 
 its slight slope towards the wire indicates its convergence 
 thereupon. If the wire had no resistance, there would be* 
 no convergence of energy upon it, the flux of energy 
 would be quite parallel to it. Details are best studied 
 in the concfete application, and it is only by the con- 
 sideration of variable states, and the propagation of electro- 
 magnetic waves, that we can obtain a full understanding of 
 the meaning of W considered as a flux of energy. 
 
 In the case of a simple progressive plane wave disturbance, 
 in which a distribution of E and H (mutually perpendicular) in 
 the plane of the wave is propagated unchanged through a 
 medium at constant speed, it is a self-evident result that the 
 energy of the disturbance travels with it. The flux of energy 
 is, therefore (since e = 0, h = 0), 
 
 W = v(U + T) = VEH, ..... (15) 
 where v is the velocity of the wave, and U, T are the densities 
 
80 ELECTROMAGNETIC THEORY. CIL II, 
 
 (which are here equal) of the electric and magnetic energies. 
 In this example, which is approximately that of radiant energy 
 from the sun, the idea of a flux of energy, and the conclusion 
 that its proper measure is the density of the energy multiplied 
 by the wave velocity, are perfectly plain and reasonable. Now, 
 many cases of the propagation of waves along wires can be 
 reduced to this simple case, with a correction for the resistance 
 of the wire, and other cases can be represented by two or more 
 oppositely travelling plane disturbances. In a very complex 
 electromagnetic field, the flux of energy is necessarily also very 
 complex, and hard to follow ; but the fundamental principles 
 concerned are the same throughout. 
 
 The flux of energy W arises from the internal structure of 
 the ether. It is somewhat analogous to the activity of a stress. 
 But the only dynamical analogy that is satisfactory in this 
 respect is that furnished by Sir W. Thomson's rotational ether, 
 when interpreted in a certain manner, so that 2E shall repre- 
 sent a torque, and H the velocity of the medium, with the 
 result that (on this understanding) VEH is the flux of energy, 
 whilst U is the potential energy of the rotation, and T the 
 kinetic translational energy. But it is very difficult to extend 
 this analogy to include electromagnetic phenomena more com- 
 prehensively. [See Appendix at the end of this chapter.] 
 
 Extension to a Moving Medium. Full interpretation of 
 the Equation of Activity and derivation of the Flux 
 of Energy. 
 
 71. Passing now to the case of a moving medium, we shall 
 obtain, from equations (1), (2), 66, the equation of activity, 
 
 (e + e) J + (h + h)G = E J + HG 
 
 + div V(B - 0o - e)(H - ho - h). (16) 
 
 Or from equations (3), (4), 66, we may get the form 
 
 e J + h G = E J + HG + div V(E - e )(H - h ), (1 7) 
 
 and from these a variety of other forms may be derived. 
 The dynamical interpretation in accordance with the prin- 
 ciple of continuity of energy is not so easy as in the former 
 case. I have recently given a full discussion of these 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 81 
 
 equations in another place. The following gives an outline 
 of the results : 
 
 Since there is a flux of energy when the medium is stationary, 
 there will still be a flux of this kind (i.e., independent of the 
 motion) when it is moving, whatever other flux of energy there 
 may then be. I conclude that the Poynting flux W still 
 preserves the form 
 
 W = V(E-e )(H-h ) 5 .... (18) 
 
 from the fact that disturbances are propagated through a 
 medium endowed with a uniform translational motion in the 
 same manner as when it is at rest. Otherwise we could only 
 know that it must reduce to this form when at rest. 
 Next, there is the convective flux of energy 
 
 ...... (19) 
 
 where q. is the velocity of the medium. 
 
 Thirdly, there is a flux of energy representing the activity, 
 A, of the electromagnetic stress. It is given by 
 
 . . . (20) 
 
 Fourthly, there is, in association with this stress, a convective 
 flux of other energy, say, 
 
 1(U + T ) ....... (21) 
 
 The complete flux of energy is the sum X, of these four 
 vectors, i.e. : 
 
 X = W + q(U + T) + A + q(U + T ), . . (22) 
 
 W and A being given by (18) and (20). 
 
 Finally, the activity of the intrinsic forces is 
 
 OoJo + VJo, ..... (23) 
 
 so that J is the true electric current. 
 
 We, therefore, have the equation of activity brought to the 
 standard form 
 
 e J + h 0- = (Q + U + T) + (Q + U + T ) + div. X, (24) 
 
 which is a special form of equation (11), with the convergence 
 of the energy flux in it replaced by its divergence (the negative 
 of convergence) on the other side of the equation. 
 
82 ELECTROMAGNETIC THEORY. CH. II. 
 
 But the unknown terms, on the right side of (24), with the 
 zero suffix, may be entirely eliminated, including those in X, by 
 making use of the secondary equation of translational activity 
 
 T ), . . (25) 
 
 where F is the translational force due to the stress. * 
 When this is done, equation (24) takes the form 
 
 e J + h G = Q + U + T + Fq + div.[W + A + q (U + T)] (26) 
 
 where the energy flux represented includes the Poynting flux, 
 the stress flux, and the convective flux of electric and magnetic 
 energy. 
 
 We may observe that in the expression (20) for A occurs the 
 term - q (U + T), so that this term may be eliminated from (26), 
 making the energy flux in it become 
 
 V(B - e )(H - ho) - VeH -, VEh. . . (27) 
 
 But these changes, with a view to the simplification of expres- 
 sion, cause us to altogether lose sight of the dynamical signi- 
 ficance of the equation of activity, and of the stress function. 
 Equation (26) is, therefore, the best form. 
 
 It should be understood that it is an identity, subject to the 
 two laws of circuitation and the distribution of energy accord- 
 ing to that of and H in the field. But it should be also 
 mentioned that in the establishment of (26) it has been assumed 
 that the medium, as it moves, carries its intrinsic properties of 
 permittivity and inductivity with it unchanged. That is, these 
 properties do not alter for the same portion of the medium, 
 irrespective of its position, although within the same unit 
 volume these properties may be changing by the exit of one 
 and entry of another part of the medium of different permit- 
 tivity and inductivity ; understanding by " medium" whatever 
 is supporting the fluxes, whether matter and ether together, or 
 ether alone ; and it is also to be understood that the three 
 velocities q, u, and w are identical, or that electrification, 
 which is always found associated with matter, moves with the 
 medium, which is then the matter and ether, moving together. 
 In other respects equation (26) is unrestricted as regards either 
 homogeneity and isotropy in respect to permittivity, inductivity, 
 and conductivity (electric, and fictitious magnetic). 
 
 [* I regret to have misrepresented Dr. Burton's notion of a moving strain 
 figure. He does it entirely by conservative elastic forces, and my objection 
 does not apply.] 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 83 
 
 Whether, when matter moves, it carries the immediately 
 surrounding ether with it, or moves through the ether, or only 
 partially carries it forward, and what is the nature of the 
 motion produced in the ether by moving matter, are questions 
 which cannot be answered at present. Optical evidence is 
 difficult of interpretation, and conclusions therefrom are con- 
 flicting. 
 
 But in ordinary large scale electromagnetic phenomena, it 
 can "make very little difference whether the ether moves or 
 stands stock-still in space. For the speed with which it pro- 
 pagates disturbances through itself is so enormous that if the 
 ether round a magnet were stirred up, artificially, like water 
 in a basin, with any not excessive velocity, the distortion in 
 the magnetic field produced by the stirring would be next to 
 nothing. 
 
 Strictly speaking, when matter is strained its elastic and 
 other constants must be somewhat altered by the distortion of 
 the matter. The assumption, therefore, that the permittivity 
 and inductivity of the same part of the medium remain the 
 same as it moves is not strictly correct. The dependence of 
 the permittivity and inductivity on the strain can be allowed for 
 in the reckoning of the stress function. This matter has been 
 lately considered by Prof. Hertz. But the constants may also 
 vary in other ways. It is unnecessary to consider here these 
 small corrections. As usual in such cases, the magnitude of 
 the expressions for the corrections is out of all proportion to 
 their importance, in relation to the primary formula to which 
 they are added. 
 
 Derivation of the Electromagnetic Stress from the Flux 
 of Energy. Division into an Electric and a Magnetic 
 Stress. 
 
 72. From the form of the expression for A in equation (20), 
 viz., the flux- of energy due to the stress, we may derive the 
 expression for the electromagnetic stress itself. If the stress 
 were of the irrotational type considered in works on Elasticity, 
 we could do this by means of the formula (10), 68. ' But for a 
 stress of the most general type the corresponding formula is 
 
 (28) 
 
 UNIVERSITY 
 
84 ELECTROMAGNETIC THEORY. CH. II. 
 
 where Q, is the stress vector conjugate to P q ; these are identi- 
 cally the same when the stress is irrotational. This gives 
 
 Q N = D.EN + B.HN-N(U + T), . . (29) 
 
 from which we obtain P N by merely exchanging E and D, and 
 H and B ; thus 
 
 P N = E.DN + H.BN-N(U + T). . . . (30) 
 
 This is the stress vector for any plane denned by N, a unit 
 vector normal to the plane. 
 
 But it is only in eolotropic bodies that we have to distinguish 
 between the directions of a force and of the corresponding 
 flux. Putting^ these on one side, and considering only ordi- 
 nary isotropy, the interpretation is simple enough. It will be 
 observed that the electromagnetic stress (30) divides into an 
 electric stress 
 
 E.DN-N.JED, (31) 
 
 and a magnetic stress 
 
 H.BN-N.JHB (32) 
 
 To find their meaning, take N in turns parallel to and perpen- 
 dicular to the force E (or to D, since its direction is the same). 
 In the first case, the stress (31) becomes 
 
 N(ED - JED) = N.JED = NU, 
 
 indicating a tension parallel to the electric force of amount U 
 per unit area. 
 
 In the second case, when N is perpendicular to E, we have 
 DN = 0, so that the stress is 
 
 -NU, 
 
 that is, a pressure of amount U. This applies to any direction 
 perpendicular to E, so that the electric stress consists of a 
 tension U parallel to the electric force, combined with an equal 
 lateral pressure. 
 
 Similarly the magnetic stress consists of a tension T parallel 
 to the magnetic force, combined with an equal lateral pressure. 
 
 It will also be found that the tensor of the electric stress 
 vector is always U, and that of the magnetic stress vector is 
 always T. The following construction (Fig. 5) is also useful : 
 Let ABC be the plane on which the electric stress is required, 
 BN the unit normal, BE the electric force, BP the stress. 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 
 
 85 
 
 Then N, E, and P are in the same plane, and the angle be- 
 tween N and E equals that between E and P. Or, the same 
 operation which turns N to E also turns E to P, except as 
 regards the tensor of P. 
 
 To show the transition from a tension to aniequal pressure, 
 imagine the plane ABC to be turned round, and with it the 
 normal N. Of course E remains fixed, being the electric force 
 at the point B. Start with coincidence of N and E. Then P 
 also coincides with them, and represents a normal pull on the 
 surface ABC. As N and E separate, so do E and P equally, 
 so that when E makes an angle of 45 with N the normal pull 
 is turned into a tangential pull, or a shearing stress, P being 
 
 >N 
 
 FIG. 5. 
 
 now at right angles to N. Further increase in the angle E 
 makes with N brings BP to the other side of BC j and when E 
 is at right angles to N, we have P and N in the same line, but 
 oppositely directed. That is, the tension has become converted 
 into an equal pressure. 
 
 Uncertainty regarding the General Application of the 
 Electromagnetic Stress. 
 
 73. We may now consider the practical meaning of the 
 stress whose relation to the electric and magnetic forces has, 
 unaer certain suppositions, been formularised. Go back to the 
 foundation of electromagnetic theory, viz., the mechanical 
 forces experienced by electrically charged bodies, by conductors 
 supporting currents, and by magnets, intrinsic or induced. It 
 
86 ELECTROMAGNETIC THEORY. CH. II. 
 
 is by observation of these forces in the first place, followed by 
 the induction of the laws they obey, and then by deductive 
 work, that the carving out Of space into tubes of force follows; 
 and now, further, we see that the localisation of the stored 
 energies, according to the square of the electric and magnetic 
 force respectively, combined with the two circuital laws, leads 
 definitely to a stress existing in the electromagnetic field, 
 which is the natural concomitant of the stored energy, and 
 which is the immediate cause of the mechanical forces observed 
 in certain cases. But the theory of the stress goes so far be- 
 yond experimental knowledge in some respects, although agree- 
 ing with it in others, that we could only expect it to be true if 
 the theoretical foundations were also rigidly true in all respects. 
 Such is not the case, however. To begin with, the way of ex- 
 pressing the action of ordinary matter merely by altering the 
 values of the two ether constants, and by a fresh property, 
 that of conductivity, is extremely bald. It is, indeed, surpris- 
 ing what a variety of phenomena is explained by so crude a 
 method. 
 
 The objection is sometimes made against some modern theo- 
 retical developments that they are complicated. Considered as 
 an argument, the objection is valueless, and only worthy of 
 superficial minds. Whatever do they expect ? Do they not 
 know that experimental knowledge, even as at present existent, 
 shows that the theory of electromagnetism, when matter is 
 present, must, to be comprehensive, be something far more (in- 
 stead of less) complicated than theory as now developed ? The 
 latter is, as it were, merely a rough sketch of a most elabo- 
 rate subject, only small parts of which can be seen at one 
 time. 
 
 In the next place, even if we take the stated influence of 
 matter on the ether as sufficient for the purposes of a rough 
 sketch, the theory of the stress should, except in certain rela- 
 tively simple cases, be received with much caution. Why this 
 should be so will be apparent on examining the manner in which 
 the stress has been obtained from the circuital laws. If we inves- 
 tigate the subject statically, and, starting from certain mechanical 
 forces regarded as known, endeavour to arrive at a stress which 
 shall explain those forces, we shall find that the problem is essen- 
 tially an indeterminate one. All sorts of stress functions may 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 87 
 
 be made up which are precisely equivalent in their effects in the 
 gross, that is, as regards translating or rotating solid bodies 
 placed in an electric or magnetic field. To remove this indeter- 
 minateness a dynamical method must be adopted, wherein what 
 goes on in the unit volume whilst its electric and magnetic 
 states are changing, and the matter concerned is itself in motion, 
 are considered. If our system of connections is dynamically 
 complete and consistent, and is such that the flux of energy can 
 be traced, then a determinate stress comes out, as we have 
 found. The method is, at any rate, a correct one, however the 
 results may require to be modified by alterations in the data. 
 Besides that, the distribution of energy (electric and magnetic) in 
 bodies is in some cases open to question ; and a really speculative 
 datum is that concerning the motion of the ether as controlled 
 by the motion of matter. Now, this datum appears to be one 
 which is essential to the dynamical method ; the only alternative 
 is the statical and quite indeterminate method. Our attitude 
 towards the general application of the special form of the 
 stress theory obtained should, therefore, be one of scientific 
 scepticism. This should, however, be carefully distinguished 
 from an obstinate prejudice founded upon ignorance, such as 
 is displayed by some anti-Maxwellians, even towards parts of 
 Maxwell's theory which have received experimental demon- 
 stration. 
 
 The stress theory can, nevertheless, sometimes be received 
 with considerable confidence, if not absolute certainty. The 
 simplest case is that of ordinary electrostatics. 
 
 The Electrostatic Stress in Air. 
 
 74. Let there be no magnetic force at all, and the electric 
 force be quite steady, and the medium be at rest, and there be 
 no impressed forces. These limitations bring the circuital laws 
 down to 
 
 = *E, (1) 
 
 -curlE = 0; ....'. (2) 
 
 that is, there must be no conduction current anywhere, and 
 the voltage in any circuit must be zero. The first condition 
 (1) implies that there is no electric force in conductors. We 
 may, therefore, divide space into conducting and non-conducting 
 
88 ELECTROMAGNETIC THEORY. C1I. II. 
 
 regions, and our electric field is entirely confined to the latter. 
 The second condition (2) implies tangential continuity of E 
 at the boundary of the non-conductor, so that as there is no E 
 in the other or conducting side, there must be no tangential 
 E on the non-conducting side. The lines of force, therefore, 
 terminate perpendicularly on the conducting matter. Whether 
 the conductors are also dielectrics or not is quite immaterial. 
 The displacement also terminates normally on the conducting 
 surface in the usual case of isotropy. Thus D, the tensor of D, 
 measures the surface density of electrification, when the positive 
 direction of D is from the conductor to the insulator. But in 
 general it is the normal component of D, that is DN, where 
 N is the unit normal vector, that measures the density of the 
 electrification. Besides this, there may be interior electrification 
 of the non-conductor, its volume density being measured by the 
 divergence of the displacement. The arrangement of the 
 electric force, so that the circuital voltage shall be zero 
 throughout the non-conductor, and give the proper internal 
 electrification, and the charges on the conductors, is uniquely 
 determinate. 
 
 We have thus the ordinary case of a number of charged con- 
 ductors in air, with the difference that the air, or parts thereof, 
 may be replaced by matter of different permittivity. It is also 
 to be noted that one non-conducting region which is entirely 
 separated from another by conducting matter may be taken by 
 itself, and all the rest ignored. 
 
 Now, first without replacing the air by matter of different 
 permittivity, we see that there are two entirely different ways 
 of considering the mutual actions of the conductors. The old 
 way is analogous to Newton's way of expressing the fact of 
 gravitation. We may say that any element of electrification p 
 repels any other p' with a force 
 
 pp'l 47rcr 2 , 
 
 if r be the distance between the two charges, and that the 
 resultant of all such forces makes up the real forcive. 
 
 In the other way, appropriate to the philosophy of Faraday, 
 as developed by Maxwell, these forces acting at a distance are 
 mathematical abstractions only, and have no real existence. 
 What is real is a stress in the electric field, of a peculiar nature, 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 89 
 
 being a tension of amount U parallel to E, combined with an equal 
 lateral pressure, and it is the action of this stress that causes 
 the electrified conductors to move, or strains them, according to 
 the way they are supported, when by constraints they cannot 
 appreciably move. 
 
 Since the electric force if normal to the conducting surfaces, 
 the stress vector is entirely a normal pull of amount U per 
 unit area, and the motions or tendencies to move of the con- 
 ductors are perfectly accounted for by this pull. They do not 
 move because of electrical forces acting at a distance across the 
 air, but because they are subjected to moving force on the spot 
 by the stress terminating upon them. 
 
 Thus a charged soap-bubble is subjected to an external radial 
 tension, and therefore expands; and so, no doubt, does a charged 
 metal sphere to some small extent. The parallel plates of a 
 condenser are pulled together. When they are very large 
 compared with their distance apart, the force on either is 
 
 J ED x area, 
 = J E x charge. 
 
 Here E is the transverse voltage divided by the distance be- 
 tween the plates, so that, if the plates be connected to a constant 
 .-source of voltage, the attraction varies inversely as the square 
 -of the distance between them; whereas, if the plates be in- 
 .sulated and their charges constant, the attraction is the same 
 .at any distance sufficiently small compared with the size of the 
 plates. 
 
 But by sufficiently separating the plates, or by using smaller 
 ^plates, the displacement, which was formerly almost entirely 
 'between them, will spread out, and will terminate in appreci- 
 able amount upon the sides remote from one another. By the 
 pull on the remote sides thus produced the attraction will be 
 lessened, and the further the plates are separated the more 
 displacement goes to their backs, and the less is the attraction. 
 When the distance is great enough it tends to be simply the 
 -attraction between two point charges. Thus the attraction 
 between two distant oppositely charged conducting spheres, 
 which varies closely as the inverse square of the distance, 
 depends entirely upon the slight departure from uniformity 
 of distribution of the electrification over their surfaces, whereby 
 
90 ELECTROMAGNETIC THEORY. CH. II. 
 
 the normal pull on either is made a little greater on the side 
 next the other than on the remote side. Also, the inverse 
 square law itself, which is exactly true for point charges, is 
 merely the ultimate limit of this operation. 
 
 Some attacks have been made on the law of inverse squares, 
 especially in its magnetic aspect. But these attacks appear to 
 have been founded upon misapprehension. The law is true, 
 and always will be. 
 
 The moving Force on Electrification, bodily and superficial. 
 Harmonisation. 
 
 75. In the above electrostatic application of the stress, it 
 will be observed that the tension alone comes into play, at 
 least explicitly, owing to the tubes of displacement terminating 
 perpendicularly on the conductors. Thus each tube may be 
 compared with a rope in a state of tension, pulling whatever 
 its ends may be attached to. But the lateral pressure is 
 needed to keep the medium itself in equilibrium, so that the 
 only places where translational force arises from the stress is 
 where there is electrification. The mechanical force is 
 
 F = EdivD = E/> ..... (3) 
 
 per unit volume. This is the force on volume electrification,. 
 and is the result of the differential action of the stress round 
 about the electrification, as in the case of the inverse square 
 law between point charges, lately mentioned. The correspond- 
 ing surface force is 
 
 F = iE.DN = NU ..... (4) 
 
 per unit area. Now, here DN is the surface equivalent of 
 div D, so there is at first sight a discrepancy between the ex- 
 pressions for the force per unit volume (3), and per unit area 
 (4), on bodily and surface electrification. How the coefficient 
 J comes in may be seen by taking the limiting form of the 
 previous expression. Let there be a thin skin of electrification, 
 of amount o- per unit area ; E falling off from E outside to 
 inside the skin. Evidently the mean E is JE, so that the 
 total force on unit area of the skin obtained by summation 
 of the forces on the volume electrification in the skin, is not. 
 Eo-, but JEo-. This is merely a mathematical harmonisation^ 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 91 
 
 From the point of view of the stress the difficulty does not pre- 
 sent itself. 
 
 The harmonisation is simply evident when the layer is of 
 uniform density, for the electric force will then fall off in in- 
 tensity uniformly. It might, however, be suspected that, per- 
 haps, the result would not come out quite the same if we 
 assumed any other law of distribution, and kept to it in pro- 
 ceeding to the limit by making the skin infinitely thin. But a 
 cursory examination will show that it is all right ; for if E is 
 the electric force within the layer, the electrification density 
 will be c (dE/dx) and the translational force will be cE (dE/dx) 
 per unit volume, if x is measured perpendicularly to the skin's 
 surfaces. Integrate through the skin, and the result is 
 
 where the suffixes refer to the value just outside the skin, on 
 its two sides. In the present case the second term U 2 is zero 
 (within the conductor), so that the result is the single normal 
 pull of the tension on the non-conducting side. 
 
 Depth of Electrified Layer on a Conductor. 
 
 76. In this connection the old question of the depth of the 
 layer of electrification on a conducting surface crops up. Has 
 it any depth at all, and, if so, how much ? The question is not 
 so superficial as it looks, arid the answer thereto lies in the 
 application thereof. If a powerful mental microscope be applied 
 to magnify the molecules and produce evident heterogeneity, 
 the surface of a conductor would become indefinite ; and unless 
 the molecules were found to be very closely packed, it is evident 
 that the displacement in the ether outside the conductor would 
 not terminate entirely upon those molecules which happen to 
 be most superficially situated, but that a portion of the dis- 
 placement would go deeper and in sensible amount reach 
 molecules beneath the first set, and an insensible amount might 
 penetrate through many layers. Thus in a molecular theory 
 the depth of the layer of electrification has meaning, and could 
 be roughly estimated. 
 
 But the case is entirely different in a theory which delibe- 
 rately ignores molecules, and assumes continuity of structure. 
 
92 ELECTROMAGNETIC THEORY. CH. II. 
 
 A conductor is then a conductor all through, and not a 
 heterogeneous mixture ; and the surface of a conductor is an 
 unbroken surface. The electrification on it is therefore surface 
 electrification, and has no depth. For it to be otherwise is 
 simply to make nonsense. It is desirable to be consistent in 
 working out a theory, for the sake of distinctness of ideas ; if, 
 then, we wish to give depth to surface electrification, and still 
 keep in harmony with Maxwell's theory, we must change our 
 way of regarding a conductor, and bring in heterogeneity. 
 Each view is true, in its own way ; but as in the mathematical 
 theory continuity of structure is tacitly assumed, we have a 
 simultaneous evanescence of one dimension in the distribution 
 of electrification. 
 
 The same question occurs in another form in the estimation 
 of -bodily electrification, when the meaning of volume density 
 of electrification is considered. When air is electrified, it is 
 probable that the electrification is carried upon the foreign 
 particles suspended in the air, and it may be partly upon the 
 air molecules themselves. In either case it is ultimately surface 
 electrification, and quite discontinuous. But, merely for the 
 sake of facility of working, it is desirable to ignore all the dis- 
 continuity, and assume a continuous and practically equivalent 
 distribution of bodily electrification. Thus, as previously we 
 saw surface density to be a kind of volume density, so now we 
 see that volume density is a kind of surface density. When, 
 therefore, we say that the translational force per unit volume 
 is E/>, where E is the electric force and p the volume density of 
 electrification, we really mean that E/> is the average, obtained 
 by summation, of the translational forces on the multitudinous 
 electrified particles, every one of these forces being itself a 
 differential effect, as before seen, viz., the resultant of the 
 unequal pulls on different parts of a particle exerted by the 
 electric stress. 
 
 As ether has some of the properties of matter, and as electri- 
 fication is found in association with matter, it is possible, 
 however improbable, that ether itself may become electrified. 
 But of this nothing is known. Nor, more importantly, is it 
 understood why the electric stress appears to act differently on 
 positively and on negatively electrified matter. But, if we 
 begin to talk about what is not understood, we enter illimit- 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 93 
 
 able regions. Men who are engaged in expounding practical 
 problems sometimes make the boast that they take and discuss 
 things as they are, not as they might be. There is a sound of 
 specious plausibility here, which is grateful and comforting ; 
 but, as a matter of plain fact, questions of physics never are 
 theorised upon as they are, but always as they might be. It is 
 a necessity to limit the field of inquiry, for to take things as 
 they are, or as they seem to be, would lead at once to a com- 
 plete tangle. For the problem, as it presents itself in reality, 
 there is always substituted a far simpler one, containing certain 
 features of the real one emphasised, as it were, and othera 
 altogether omitted. The juveniles, who take things as they 
 are, do not do it ; they only think so. They may strain out a 
 few gnats successfully, but swallow, quite unawares, all the 
 camels in Arabia. But the principle and practice of limitation 
 and substitution is the same all over; in politics, for instance, 
 where a fictitious British Constitution does brave duty, as a 
 scarecrow, and in other useful ways. 
 
 Electric Field disturbed by Foreign Body. Effect of a. 
 Spherical Non-conductor. 
 
 77. To further exemplify the significance of the electric 
 stress, let us introduce a foreign body into a stationary electric 
 field. The field will be disturbed by its introduction, and will 
 settle down to a new state ; the change depending upon the 
 nature of the foreign body, whether conducting or non-conduct- 
 ing, in substance or superficially, and upon whether it has a 
 charge itself, or contains any other source of displacement. If 
 it be a good conductor, either charged or uncharged, the final, 
 state, reached very quickly, will be such that the displacement 
 will terminate normally upon its surface, thus reproducing the 
 previous case ( 74 to 76). But if it be a non-conductor, the 
 result is somewhat different. If superficially conducting, we 
 may indeed have an ultimate electrification of the surface, so as 
 to come wholly or partly under the same case : but if there be 
 no superficial or internal conduction, or only so little that a 
 long time must elapse for it to become fully effective, what we 
 do is simply to replace the dielectric air in a certain region by 
 another dielectric of different permittivity, usually greater.. 
 
94 ELECTROMAGNETIC THEORY. CH. II. 
 
 Then, supposing the external field to be due to charges upon 
 insulated conductors or to internal electrification in the air 
 (kept at rest for the purposes of the argument), no change will 
 occur in their amounts ; but there will be merely an alteration 
 in the distribution of the electrification on the conductors, 
 caused by the displacement becoming denser within the foreign 
 body than before (or less dense, if its permittivity can be less 
 than that of the air). The intrinsic electrification of the body 
 itself, if any, must also be allowed for; but should it have 
 none previously, it will remain unelectrified when introduced 
 into the field, and the displacement will pass freely through it, 
 and out again in a solenoidal manner. This is expressed by 
 the surface condition 
 
 where D : and D 2 are the displacements in the air and foreign 
 body respectively at their interface, and N 1? N 2 are the unit 
 normals from the interface to the two media. 
 
 The only quite simple case (excepting that of infinite plane 
 sheets) is that of a sphere of uniform permittivity brought into 
 a previously uniform field. The ultimate displacement in the 
 sphere is then parallel to the original displacement in the air. 
 It may vary between zero and three times the original displace- 
 ment, as the permittivity of the sphere varies from zero up to 
 infinity. It is certainly a little surprising that the ultimate 
 displacement with infinite permittivity should be only three 
 times the original (and it is not much less when the permittivity 
 is only 10 times that of the air) ; whilst, on the other hand, the 
 zero displacement when the permittivity is zero (a quite ideal 
 case) is obvious enough, because the displacement never enters 
 the sphere at all, but goes round it. The original uniform 
 field may be conveniently that between the parallel plates of a 
 very large air condenser. There is, however, a double action 
 taking place. When the transverse voltage of the condenser is 
 maintained constant by connection with a suitable source, the 
 insertion of the foreign body, which increases or reduces the 
 permittance of the condenser, will increase or reduce its charge 
 under the action of the constant source, besides concentrating 
 the displacement within the body, or the reverse. With constant 
 charges, however, when the plates are insulated, the insertion 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 95 
 
 of the foreign body will reduce the transverse voltage when its 
 permittivity exceeds that of the air ; and conversely. Increase 
 of permittivity also increases the stored energy when the trans- 
 verse voltage is constant, but reduces it when the charges are 
 constant. 
 
 Now, since the electric force does not terminate on the boun- 
 dary of the foreign body, but extends all through it, so does the 
 electric stress. So far, however, as the resultant force and 
 torque on the body, when solid, is concerned, we may ignore 
 the internal stress altogether, and consider only the external, or 
 stress in the air. This is a particular case of a somewhat impor- 
 tant and wide prope^ in abstract dynamics, which we may 
 state separately thus. 
 
 Dynamical Principle. Any Stress Self-equilibrating. 
 
 78. The resultant force and torque due to any stress in any 
 region is zero. Or, any stress in any region forms a self-equili- 
 brating system. 
 
 Imagine any distribution of stress to exist in a region A, and 
 to terminate abruptly on its boundary. Or, equivalently, 
 imagine a piece of a stressed solid to be removed from its place 
 without altering the stress. The stress-variation, when esti- 
 mated in a certain way, constitutes mechanical force tending to 
 move the body. This will be, in the case of an ordinary irrota- 
 tional stress, entirely translational force. But there are two kinds. 
 First, there is internal force, reckoned per unit volume, due to 
 :he continuous variation of the stress in the body. Next, 
 ;here is superficial force, reckoned per unit area, due to the 
 abrupt cessation of the stress. This surface traction is repre- 
 sented simply by the stress vector itself, acting on the inner 
 side of the surface of the body. Now, the resultant effect of 
 these two forcives, over the surface and throughout the volume 
 of the solid respectively, in tending to translate and rotate it, 
 is zero. Or, in other words, the force and torque equivalent to the 
 surface forcive are the negatives of those due to the internal 
 forcive. If it were otherwise, the differential action would 
 cause indefinite increase in the translational and rotational 
 energy to arise out of the internal mutual forces only of a 
 body. 
 
96 ELECTROMAGNETIC THEORY. CH. II. 
 
 If the stress be of the rotational type, there will be an in- 
 ternal torque (per unit volume) as well as a translational 
 force. Still, however, the resultant force and torque due to 
 the surface tractions will cancel those due to the internal forces 
 and torques. 
 
 In case there be any difficulty in conceiving the traction exerted 
 on the surface of a body by the stress within itself, we may 
 replace the sudden cessation of the stress by a gradual cessation 
 through a thin skin. The solid is then under the influence of 
 continuous bodily force only (and torque also, if the stress be 
 rotational) conveniently divisible into the internal force all 
 over, and the force in the skin. Otherwise it is the same. 
 
 Now put the solid piece back into its place again. Since its 
 own stress balances itself, we see that whatever the forcive on 
 the piece may be it must be statically equivalent to the action 
 upon its boundary of the external stress only, constituting an 
 external surface traction. There need be no connection between 
 the external and the internal stress. The latter may be any- 
 thing we like, so far as the resultant force and torque on the 
 piece are concerned. The difference will arise in the strains 
 produced, or in the relative internal motions, when for one stress 
 another is substituted*- 
 
 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. 
 
 79. Returning to the electric field, we see that whatever 
 be the nature of the reaction of the foreign body on the original 
 state of the field, the resultant mechanical action on the body 
 as a whole is fully represented by the stress in the air just 
 outside it, in its actual state, as modified by the presence of 
 the body, and that we need not concern ourselves with the 
 internal state of stress. Nor are we limited, in this respect, by 
 any assumed proportionality of electric force to displacement 
 in the body, or assumed absence of absorption, or other irregu- 
 larities and complications. That is, we need not have any 
 theory to explicitly account for the change made in the electric 
 field by the body. Nor do we gain any information regarding 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 97 
 
 the internal stress from merely a knowledge of the external 
 stress, although that involves the reaction of the body on the 
 electric field. This is the meaning of the statical indeter- 
 minateness of the stress before referred to, and the principle 
 applies generally. 
 
 The air stress vector P will usually have both a normal and 
 a tangential component at the surface of a body, viz.: 
 
 U cos 26 normal, 
 U sin 28 tangential, 
 
 if be the angle between the normal N to the surface (drawn 
 from the body to the air) and the electric force E, and U the 
 density of the electric energy, or the tensor of the stress vector, 
 In only one case, however, will this external surface traction 
 represent the real forcive in detail (as well as in the lump), 
 viz., when there is no stress at all on the other side of the 
 boundary, that is, in the case of a conductor in static equilibrium. 
 In general, the real surface force is represented by the vector 
 P 1 -P 2 , the stress difference at the boundary, P : being the 
 external and P 2 the internal stress vector, which two stresses 
 may, if we please, be imagined to be united continuously by a 
 gradually changing intermediate stress existing in a thin skin, 
 an idea appropriate to molecular theories. (It may be remem- 
 bered that P when positive means a pull.) Each of these may 
 be split into a normal and a tangential component. Now, the 
 tangential components are 
 
 Uj sin 20j and U 2 sin 20 2 , 
 
 where the suffix x relates to the air, and 2 to the other medium. 
 Or E x Dj sin l cos 6 l and E 2 D 2 sin 2 cos 6%, 
 
 But here we have normal continuity of the flux D, and tangential 
 continuity of the force E; (otherwise the surface would be 
 electrified and covered with a magnetic current sheet); that is 
 
 D 1 cos^ 1 = D 2 cos #2, ) 
 and E 1 sin^=E 2 sm0 2 . / 
 
 These relations make the tangential tractions equal and 
 opposite, so that there is no resultant tangential traction, and 
 
 H 
 
98 ELECTROMAGNETIC THEORY. CH. II. 
 
 the actual traction is entirely normal, being the difference of the 
 normal components of P l and P 2 , or, 
 
 .... (6) 
 which is the same as (being subject to (5),) 
 
 ^D x cos O l (E x cos 1 - E 2 cos 2 ) 
 - P! sin ^ (D x sin 6^-0,3 sin 2 ) . . . (7) 
 The coefficient J comes in for a similar reason to before, 75. 
 
 This formula (6) or (7) being the real surface traction when 
 E varies as D in the body, and the stress is of the same type as 
 in air, furnishes a second way of calculating the resultant force 
 and torque on the body, when its permittivity is uniform ; and 
 it is noteworthy that the surface traction is, as in the case of an 
 electrified conductor in equilibrium, entirely normal, although 
 it may now be either a pull or a push. 
 
 Translational Force due to Variation of Permittivity. 
 Harmonisation with Surface Traction. 
 
 80. Noting that the mechanical force on the elastically 
 electrizable body is situated where the change of permittivity 
 occurs, and is in the direction of this change, it may be inferred 
 that when the permittivity varies continuously there is a bodily 
 translational force due to the stress variation which is in the 
 direction of the most rapid change of permittivity. This is, in 
 fact, what the stress vector indicates when c varies continuously, 
 viz., the force represented by 
 
 -JEV=-VJJ, ..... (8) 
 
 where yc means the vector rate of fastest increase of c round- 
 about the point considered. Since U = J C E 2 , the second form 
 in (8) will be understood, meaning the vector- slope of U as 
 dependent upon the variation of c only. 
 
 This bodily force, and the previous surface force may be har- 
 monised by letting c vary not abruptly, but continuously from 
 the value Cj on one side to c 2 on the other side of the surface of 
 discontinuity, through a thin skin, and summing up the trans- 
 lational forces in the skin by the formula (8). Thus, if x be 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 99 
 
 measured normal to the skin, the translational force per unit 
 volume is - JE 2 (dc/dx) in the direction of the normal ; or 
 
 , dc 
 
 ~ 2-kf -j- 
 
 dx dx 
 
 where E ft and D n are the normal components of E and D, and 
 E 4 the tangential component of E. Now D n and E f are con- 
 stant ultimately, for the reason before given, so we can inte- 
 grate (9) with respect to x immediately, giving 
 
 ..r id*.- 
 
 between the limits ; or, since E n and E t are proportional to the 
 cosine and the sine of 6 respectively. 
 
 [JcE 2 cos2<9] ..... (11) 
 
 between the limits, which is the same as (6), which was to be 
 verified. 
 
 Movement of Insulators in Electric Field. Effect on the 
 Stored Energy. 
 
 81. Since a sphere of uniform permittivity placed in a uniform 
 field causes the external lines of electric force to be symmetrically 
 distorted fore and aft, it has no tendency to move, but is merely 
 strained. But if the body be not in an initially uniform field, 
 or be not spherical, complex calculations are usually needed 
 to determine the effect. If, however, it be only a small piece, 
 the tendency is for it to move in the direction in which the 
 energy, or the stress, in the field increases most rapidly, inde- 
 pendent of the direction of the electric force, when its permit- 
 tivity exceeds that of the gaseous medium. The total electric 
 energy will be diminished by permitting the motion when the 
 charges are constant ; but increased should the field be kept up 
 by constant sources of voltage. 
 
 These properties are rendered particularly evident by taking 
 the extreme cases of infinite and zero permittivity of a small 
 body placed in a widely varying field, that surrounding a charged 
 sphere, for example, the electric force varying in intensity as 
 
 H 2 
 
100 ELECTROMAGNETIC THEORY. CH. II. 
 
 the inverse square of the distance from its centre. Here the 
 non-permittive body is repelled, for the lines of force go round 
 it, and the lateral pressure comes fully into play, and is greater 
 on the side next the charged sphere. On the other hand, with 
 the infinitely permittive body, concentrating the displacement, 
 it is the tension that comes fully into play, and this being 
 greater on the side next the charged sphere, the result is an 
 attraction. The permittance of the sphere, also, is increased in 
 the latter case, and decreased in the former that is, when the 
 natural motion of the body to or from the sphere is allowed ; 
 so, since the total electric energy is JSV 2 where S is the permit- 
 tance and V the voltage, or, equivalently, JVQ, if Q is the 
 charge, we have always a diminution of energy when the 
 natural motion is allowed, whether resulting from attraction or 
 repulsion, if the charge is constant ; but an increase of energy 
 if the voltage is constant. 
 
 In intermediate cases the tension is dominant when the c of 
 the body exceeds, and the pressure is dominant when it is less 
 than that of the air, there being perfect equilibrium of a piece 
 of any shape in any field if there be equality of permittivity, 
 and therefore no disturbance of the field. Here we see the part 
 played by the lateral pressure in the case of conductors in 
 equilibrium. It has no influence on them immediately, and 
 might be thought wholly unnecessary, but it is equally 
 important with the tension in the non-conducting dielectric 
 itself. 
 
 Magnetic Stress. Force due to Abrupt or Gradual Change of 
 Inductivity. Movement of Elastically Magnetised Bodies. 
 
 82. Passing now to the corresponding magnetic side of the 
 stress question, we may observe that the analogy is an imperfect 
 one. Thus, proceeding as at the beginning of 74, to have a 
 stationary magnetic field without impressed forces, we shall 
 find that there must first be no magnetic conduction current ; 
 and next, that the gaussage in any circuit must be zero. The 
 magnetic force we then conclude to be confined entirely to the 
 magnetically non-conducting regions, and to terminate perpen- 
 dicularly upon their boundaries. Thus we come to the 
 conception of a number of detached magnetic conductors im- 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 101 
 
 mersed in a magnetically non-conductive medium, these con- 
 ductors having magnetic charges on them measured by the 
 amount of induction leaving or terminating upon them, with 
 possible associated volume magnetification in the non-conduct- 
 ing medium. 
 
 The magnetic stress will exert a normal traction 
 
 . . , . . (12) 
 per unit area on the conductors, and a force 
 
 Hcr = Hdiv.B ..... (13) 
 
 per unit volume on tridimensional magnetification, of density cr, 
 measured by the divergence of the induction. These are analo- 
 gous to the forces on surface and volume electrification. 
 
 Also, when the inductivity /* varies, we shall have a normal 
 surface traction of amount, 
 
 T 1 cos2^ 1 -T 2 cos2^, .... (14) 
 
 per unit area, analogous to (6), when p changes value abruptly 
 at the interface of two media ; and a force 
 
 -iH 2 W ...... (15) 
 
 per unit volume, when /z varies continuously. 
 
 The forces (12), (13), however, are absent, because of the ab- 
 sence of magnetic conductivity, and, in connection therewith, 
 the absence of " magnetification." But (12) may be sometimes 
 used, nevertheless, when it is the stress across any surface that 
 is in question, and we create surface magnetification by regard- 
 ing one side only. 
 
 We are, therefore, left with the forces (14), (15), depending 
 upon variation of inductivity, abrupt or gradual. These ex- 
 plain the mechanical action upon elastically magnetised 
 media, e.g., the motions of bodies to or from a magnet pole, 
 according as they are paramagnetic or diamagnetic, which, it 
 should be remembered, depends fundamentally upon the varia- 
 tion in the intensity of magnetic force near the pole ; and the 
 axial equilibrium of a paramagnetic bar, and equatorial equi- 
 librium of a diamagnetic bar. Faraday's remarkable sagacity 
 led him to the essence of the explanation of these and other 
 allied phenomena, as was later mathematically demonstrated by 
 
102 ELECTROMAGNETIC THEORY. CH. II. 
 
 Sir W. Thomson. Questions relating to diamagnetic polarity 
 are, in comparison, mere trifling. 
 
 Force on Electric Current Conductors. The Lateral Pressure 
 becomes prominent, but no Stress Discontinuity in general. 
 
 82a. But the magnetic stress has other work to do than to- 
 move elastically magnetised matter under the circumstances 
 stated. Wholly independent of magnetisation, it produces the 
 very important moving force on conductors supporting electric- 
 current, first mathematically investigated by Ampere. 
 
 The lateral pressure of the stress here comes prominently 
 into view, when we ignore the stress in the interior of the con- 
 ductors, so that the stress vector in the air at the boundary of 
 a conductor represents the moving force on it per unit area. 
 Thus, two parallel conducting wires supporting similar currents 
 attract one another, because their magnetic forces are additive 
 on the sides remote from one another, rendering the lateral 
 pressure on them greater there than on the sides in proximity. 
 But when the currents are dissimilar the magnetic force is- 
 greater on the sides in proximity, and, therefore, the lateral 
 pressure of the stress is greater there, producing repulsion. 
 
 Proceeding further, and considering the stress within ih& 
 conductor also, according to the same law, we find this pecu- 
 liarity. In the case of unmagnetisable conductors (typified 
 practically by copper), there is no superficial discontinuity in 
 the stress, and therefore no surface forcive of the kind stated. 
 This may be easily seen from 79, translating the results from 
 the electric to the magnetic stress. There is no tangential 
 discontinuity in the stress because the normal induction and 
 tangential magnetic force are continuous ; and there is no dis- 
 continuity in the normal component of the stress because (since 
 there is no difference of inductivity) the normal magnetic force 
 and the tangential induction are also continuous. 
 
 In (5), (6), (7) turn E to H, and D to B, and U to T, and 
 note that (5), as transformed, are true when the force and flux 
 are exchanged, so that the transformed expressions (6) or (7) 
 vanish. 
 
 In the case of a real conductor, therefore, with finite volume 
 density of electric current, the moving force is distributed 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 103 
 
 throughout its substance ; and the variation of the stress indi- 
 cates that the translational force per unit volume is expressed 
 
 by F = VOB, . ..... . (16) 
 
 where is the current density. This is what Maxwell termed 
 "the electromagnetic force," and it is what is so extensively 
 made use of by engineers in their dynamos, motors, and things 
 of that sort. It is probable, I think, that in grinding away at 
 the ether they also stir it about a good deal, though not fast 
 enough to produce sensible disturbances due to etherial dis- 
 placement. 
 
 Force on Intrinsically Magnetised Matter. Difficulty. Max- 
 well's Solution probably wrong. Special Estimation of 
 Energy of a Magnet and the Moving Force it leads to. 
 
 83. There is next the force on intrinsically magnetised 
 matter to be considered in connection with the magnetic stress. 
 This is, perhaps, the most difficult part of magnetic science. 
 Although, so far as the resultant effect on a magnet is con- 
 cerned, we need not trouble about its internal state, but, as 
 before, merely regard it as being pushed or pulled by the stress 
 in the surrounding air, such stress being calculable from the 
 distribution of magnetic force immediately outside it, as 
 modified by the magnet itself, yet it is impossible that the real 
 forcive can be represented merely by the surface traction P N . 
 
 Now, it is possible to find a distribution of magnetification 
 over the surface, which shall be externally equivalent to the 
 interior magnetisation, or to whatever other source of induction 
 there may be. Then we may substitute for the surface trac- 
 tion P N , another traction, namely, upon the surface magnetifi- 
 cation. 
 
 Or, we may find a distribution of fictitious electric current 
 upon the boundary of the magnet, which shall be externally 
 equivalent to the interior sources, and then represent the 
 forcive by means of fictitious electromagnetic force on this 
 current, 82a. 
 
 Or we may combine these methods in various ways. Evi- 
 dently, however, such methods are purely artificial, and that 
 to obtain the real forcive we must go inside the magnet. This 
 can only be done hypothetically, and with precarious validity. 
 
104 ELECTROMAGNETIC THEORY. CH. II. 
 
 One way of exhibiting the mechanical action on a magnet, or 
 on magnetised matter generally, is that given by Maxwell in his 
 chapter on the stresses (Vol. II.). This I believe to be quite 
 erroneous for many reasons, the principal being that it does 
 not harmonise with his scheme generally, and that it lumps to- 
 gether intrinsic and induced magnetisation, which have essen- 
 tial differences and are physically distinct. There are many 
 other ways of exhibiting the resultant force and torque as made 
 up of elementary forces, acting upon magnetisation, or on free 
 magnetism, or on the variation of magnetism estimated in 
 different ways. It is unnecessary to enter into detail regarding 
 them. Nobody would read it. It will be sufficient to point out 
 the particular way which harmonises with Maxwell's scheme in 
 general, in the form in which I display it, with a special esti- 
 mation of magnetic energy. Proportionality of force and flux 
 is assumed. The want of this proportionality is quite a separate 
 question. Given a definite relation between force and flux, the 
 accompanying change in the stress vector, in accordance with 
 the continuity of energy, can be estimated. This I have re- 
 cently shown how to do in another place, which shall not be 
 more explicitly referred to. 
 
 Intrinsic magnetisation possesses the peculiarity that it is, in 
 a manner, outside the dynamical system formulated in the 
 electromagnetic equations, inasmuch as it needs to be ex- 
 hibited in them through the medium of an impressed force, 
 although this is disguised in the ordinary mode of representa- 
 tion. Calling this intrinsic magnetic force h (any distribution), 
 as before, the induction due to it in a medium of any inductivity 
 (varying continuously or abruptly, if required) is found in the 
 same way as the displacement due to intrinsic electric force in 
 a non-conducting medium of similarly distributed permittivity ; 
 or as the conduction current due to the same in a medium of 
 similar conductivity ; or, to make a fourfold analogy, as the 
 magnetic conduction current due to h in a (fictitious) medium 
 of similar magnetic conductivity. 
 
 I may here point out that a clear recognition of the correct 
 analogies between the electric and magnetic sides of electro- 
 magnetism is essential to permanently useful work. Many 
 have been misled in this respect, especially in comparing 
 Maxwell's displacement with magnetic polarisation. The true 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 105 
 
 analogue of D is B. Investigations based upon the false 
 foundation mentioned can lead to nothing but confusion. 
 There are enough sources of error without bringing in gratuit- 
 ous ones. 
 
 Now, presuming we have induction set up by h , how is the 
 energy to be reckoned ? I reckon its amount per unit volume 
 to be JHB generally, whether outside or within the magnet ; 
 or, in the usual isotropic case, J/*.H 2 or J/ir^B 2 . This makes 
 the total work done by h to be h B per unit volume, if h is 
 suddenly established ; of which one half is wasted, and the 
 rest remains as stored magnetic energy. That is, 
 
 , .... (17) 
 
 if the 2 indicates space-summation. If h be suddenly des- 
 troyed, the energy 2T is set free and is dissipated, mainly by 
 the heat of currents induced within the magnet itself and sur- 
 rounding conductors. This is the meaning of JHB being the 
 stored energy per unit volume. But it may not be immediately 
 available. To take an extreme case, if we have a complete 
 magnetic circuit, so magnetised intrinsically that there is no 
 external field, the energy, as above reckoned, is the greatest 
 possible, since H = h . But it is now not at all available, unless 
 h be destroyed. On the other hand, Maxwell would appear to 
 have considered the energy of a magnet to be 2 J/*(H - h ) 2 , 
 which is zero in the just mentioned case. This reckoning does 
 not harmonise with the continuity of energy, although it has 
 significance, considered as energy more or less immediately 
 available without destruction of h . The connection of the two 
 reckonings is shown by 
 
 lio); . (18) j 
 or, when H and B are parallel, 
 
 2|h B = 2i J uh 2-2^(H-li )2. . . (19). 
 
 Now, according to the reckoning (17) of the stored energy, 
 and the consequent flux of energy, the stress vector derived 
 therefrom indicates that the moving force per unit volume is 
 
 F = Vj B ....... (20), 
 
 where j = curl h n . . . , , . . (21). 
 
106 ELECTROMAGNETIC THEORY. CH. II. 
 
 The interpretation is that the vector J represents the distri- 
 bution of (fictitious) electric current, which would, under the 
 same circumstances as regards inductivity, set up, or be asso- 
 ciated with, the same distribution of induction B as the intrinsic 
 magnet is. It may be remarked that the induction due to an 
 intrinsic magnetic force does not depend upon its distribution 
 primarily, but solely upon that of its curl. 
 
 Substituting this current system for the intrinsic magnetise 
 tion, equation (20) indicates that the moving force is " the 
 electromagnetic force " corresponding thereto, according to (16), 
 82. The accompanying surface distribution of current, repre- 
 senting the abrupt cessation of h , must not be forgotten; or 
 we may let it cease gradually, and have a current layer in the 
 skin of the magnet. It may be far more important than 
 the internal current, which may, indeed, be non-existent. 
 For instance, in the case of a uniformly longitudinally mag- 
 netised bar, the equivalent current forms a cylindrical sheet 
 round the magnet. In general, the surface representative of 
 
 Jo is 
 
 VNh , (22) 
 
 where N is a unit normal drawn from the boundary into the 
 magnet. It should also not be forgotten that if the inductivity 
 changes, there is also the moving force (15) or (14) to be 
 reckoned, besides that dependent upon the intrinsic magnetisa- 
 tion. 
 
 Force on Intrinsically Electrized Matter. 
 
 84. The electric analogue of intrinsic magnetisation is in- 
 trinsic electrisation, represented in a solid dielectric in which 
 "absorption " has occurred, and perhaps in pyroelectric crystals. 
 It seems very probable that there is a true electrisation, quite 
 apart from complications due to conduction, surface actions, 
 and electrolysis. When formulated in a similar manner to in- 
 trinsic magnetisation, by means of intrinsic electric force e 0> 
 producing displacement according to the permittivity of the 
 medium (which displacement, however, is now also affected by 
 the presence of conducting matter), and with a similar reckon- 
 ing of the stored energy, viz., JcE 2 per unit volume, where R 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 107 
 
 includes e , we may expect to find, and do find, a moving force 
 analogous to (20), viz., 
 
 (23) 
 
 where g = - curl e . ... . . . (24). 
 
 That is, magnetoelectric force on the fictitious magnetic cur- 
 rent g which is equivalent to the intrinsic force e . And, similarly 
 to before, we may remark that the flux due to e depends solely 
 upon its curl. 
 
 Summary of the Forces. Extension to include varying States 
 in a Moving Medium. 
 
 85. Now bring together the different moving forces we have 
 gone over. On the electric side we have 
 
 , . (25), 
 and on the magnetic side 
 
 F m = [H.r]-iH2v M + VOB + Vj B, . . (26), 
 
 where the third term on the right of (25), and the first on the 
 right of (26), in square brackets, are zero ; Ho- being force on 
 magnetification and VDK the magnetic analogue of VOB. 
 The other component forces can all separately exist, and in 
 stationary states. 
 
 Passing to unrestricted variable states, with motion of the 
 flux-supporting media also, the electric and magnetic stress 
 vectors indicate that the moving forces arising therefrom are 
 obtainable from those exhibited in (25), (26), by simply 
 changing the meaning of the electric current and the 
 magnetic current symbols in the third terms on the right. 
 Above, they stand for conduction current only, and one of 
 them is fictitious. They must be altered to G and J , the 
 " true " currents, as explained in 66. Thus 
 
 . . (27) 
 . . (28) 
 
 express the complete translational forces due to the electric and 
 magnetic stresses (31), (32), 72. 
 
108 ELECTROMAGNETIC THEORY. CH. II. 
 
 The division of the resultant translational force into a 
 number of distinct forces is sometimes useless and artificial. 
 But as many of them can be isolated, and studied separately, 
 it is not desirable to overlook the division. 
 
 The equilibrium of a dielectric medium free from electrifica- 
 tion and intrinsic forces, which obtains when the electric and 
 magnetic forces are steady, is upset when they vary. There is 
 then the electromagnetic force in virtue of the displacement 
 (electric) current and the magnetoelectric force in virtue of the 
 magnetic current ; that is, 
 
 F = VDB + VDB (29) 
 
 (30), 
 dt v* dt 
 
 where W is the flux of energy. Here we neglect possible small 
 terms depending on the motion of the medium. 
 
 That there should be, in a material dielectric, moving force 
 brought into play under the action of varying displacement and 
 induction does not present any improbability. But it is less 
 easy to grasp the idea when it is the ether itself that is the 
 dielectric concerned. Perhaps this is, for some people, because 
 of old associations the elastic solid theory of light, for in- 
 stance, wherein displacement of the ether represents the 
 disturbance. 
 
 But if we take an all round view of the electromagnetic con- 
 nections and their consequences, the idea of moving force on 
 the ether when its electromagnetic state is changing will be 
 found to be quite natural, if not imperatively necessary. We 
 do know something about how disturbances are propagated 
 through the ether, and we can, on the same principles, allow 
 for bodily motion of the ether itself. Further, reactions on the 
 ether, tending to move it, are indicated. But -here we are 
 stopped. We have no knowledge of the density of the ether, 
 nor of its mechanical properties in bulk, so, from default of real 
 data, are unable to say, except upon speculative data, what 
 motions actually result, and whether they make any sensible 
 difference in phenomena calculated on the supposition that the 
 ether is fixed. 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 
 
 Union of Electric and Magnetic to produce Electromagnetic 
 Stress. Principal Axes. 
 
 86. From the last formula we see that to have moving force 
 in a non-conductor (free from electrification, &c.) requires not 
 merely the coexistence of electric and magnetic force, but also 
 that one or other of them, or both, should be varying with the 
 time. That is, when the energy flux is steady, there is no 
 moving force, but when it varies, its vector time-variation, 
 divided by -y 2 , expresses the moving force. 
 
 The direction of W is a natural one to choose as one of the 
 axes of reference of the stress, being perpendicular to both E 
 and H, which indicate the axes of symmetry of the electric and 
 magnetic stresses. The two lateral pressures combine together 
 to produce a stress on the W plane 
 
 Pw^-W^U + T), .... (31) 
 
 where W x is a unit vector parallel to W. (Take N = W l in the 
 general formula (30) 72 for P N ). 
 
 If, further, E and II are perpendicular to one another, the 
 stresses on the planes perpendicular to them are 
 
 P E1 =E 1 (U-T) J (32) 
 
 P^H^T-U), (33) 
 
 which are also entirely normal. Here E L and H x are unit 
 vectors. Thus, W x is always a principal stress axis, while 
 E : and Hj_ become the other pair of principal axes when they 
 are perpendicular, as in various cases of electromagnetic waves. 
 Thus when a long straight wire supports a steady current, or 
 else is transmitting waves, the principal axes of the stress at a 
 point near the wire are respectively parallel to it and perpen- 
 dicular to it, radially and circularly. The first one has a 
 pressure (U + T) acting along it, the second (parallel to E) a 
 pressure (T U), and the third (parallel to H) a pressure 
 (U-T). (This legitimate use of pressure must not be con- 
 founded with the utterly vicious misuse of pressure to indi- 
 cate E.M.F. or voltage, by men who are old enough to know 
 oetter, and do.) In general, U and T are unequal. But in the 
 
110 ELECTROMAGNETIC THEORY. CH. II. 
 
 case of a solitary wave or train of waves with negligible distor- 
 tion U and T are equal, and there is but one principal stress, 
 viz., that with axis parallel to W, a pressure 2U or 2T. It is 
 this pressure (or its mean value) that is referred to as the 
 pressure exerted by solar radiation, and its space-variation con- 
 .stitutes the moving force before mentioned. This matter is 
 .still in a somewhat speculative stage. 
 
 It is natural to ask what part do the stresses play in the 
 propagation of disturbances ? The stresses and accompanying 
 strains in an elastic body are materially concerned in the trans- 
 mission of motion through them, and it might be thought that 
 it would be the same here. But it does not appear to be so 
 from the electromagnetic equations and their dynamical con- 
 . sequences that is to say, we represent the propagation of dis- 
 turbances by particular relations between the space- and the 
 time-variations of and H ; and the electromagnetic stress 
 and possible bodily motions seem to be accompaniments rather 
 than the main theme. 
 
 Dependence of the Fluxes due to an Impressed Forcive upon 
 its Curl only. General Demonstration of this Property. 
 
 87. In 83 it was remarked that the flux induction due to 
 an intrinsic magnetic forcive depends not upon itself directly, 
 but upon its curl, and in 84 a similar property was pointed 
 out connecting the flux displacement and intrinsic electric 
 force. That is to say, the fluxes depend upon the vectors J 
 and g , not upon e and h . This remarkable and, at first 
 sight, strange property, which is general, admits of being 
 demonstrated in a manner which shall make its truth evident 
 in a wider sense, and lead to a connected property of consider- 
 able importance in the theory of electromagnetic waves. 
 
 Let there be any impressed forcive e in a stationary medium. 
 Its activity is e J per unit volume, where J is the electric cur- 
 rent, and the total activity is 2e J, where the 2 indicates sum- 
 mation through all space, or at any rate so far as to include 
 every place where e exists. Its equivalent is Q , the total rate 
 of waste of energy, and the rate of increase of the total stored 
 energies, say U and T . Thus, 
 
 T (34). 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. Ill 
 
 Now, the value of the summation is zero if e has no curl, or 
 is irrotational that is, an irrotational forcive does no work 
 upon a circuital flux. This proposition may be rendered evident 
 by employing a particular method of effecting the space sum- 
 mation, viz., instead of the Cartesian method of cubic sub- 
 division of space, or any method employing co-ordinates, divide 
 space into the elementary circuital tubes belonging to the flux. 
 (Here it is J.) Fixing the attention upon any one of these 
 circuits, in which the flux is a constant quantity, we see at 
 once that the part of the summation belonging to it is the 
 product of the impressed voltage in the circuit and the flux 
 therein. But there is no impressed voltage, because e has no 
 curl ; hence the circuit contributes nothing to the summation. 
 Further, since this is true for every one of the elementary cir- 
 cuits, and inclusion of them all includes all space, we see that 
 the summation 2e J necessarily vanishes under the circum- 
 stance stated of e being irrotational. 
 
 Now return to equation (34), and suppose that the initial 
 state of things is absence of E and H everywhere, so that Q , U 
 and T are all zero. Next start any irrotational e , and see 
 what will happen. The left side of (34) being zero, the right 
 side must also be and continue zero. But Q , U and T when 
 not zero are essentially positive. Now if (U + T ) becomes 
 positive, Qo should become negative, in order to keep the right 
 number of (34) at zero. This negativity of Q being impos- 
 sible, these quantities Q , U and T must all remain zero. Con- 
 sequently E and H must remain zero. That is, an irrotational 
 forcive can produce no fluxes at all, if the flux corresponding 
 to the force is restricted to be circuital. 
 
 It should be observed that this proposition applies not merely 
 to the steady distribution appropriate to the given forcive, but 
 to all intermediate stages, involving both electric and magnetic 
 force, and flux of energy. Nothing happens, in fact, when any 
 distribution of impressed force is made to vary in time and in 
 space, provided it be restricted to be of the irrotational type, 
 so that the voltage (or the gaussage) in every circuit is nil. 
 
 Notice further the dependence of the property upon cir- 
 cuitality of the factor with which the impressed force is 
 associated (thus J with e , and similarly G with h ), and the 
 positivity of energy ; and, more strikingly, the independence 
 
112 
 
 ELECTROMAGNETIC THEORY. 
 
 CH. II. 
 
 of such details as are not concerned in equation (34), such as 
 the distribution of conductivity, permittivity, &c., or of the 
 forces being linear functions of the fluxes. 
 
 Identity of the Disturbances due to Impressed Forcives 
 having the same Curl. Example: A Single Circuital 
 Source of Disturbance. 
 
 . 88. Returning to (34) again, we see that any two impressed 
 forcives produce the same results in every particular, as regards 
 the varying states of and H gone through, if their curl is 
 the same. For the difference of the two forcives is an irrota- 
 tional forcive, and is inoperative. Here it is desirable to take 
 
 FIG. 6. 
 
 an explicit example for illustration of the meaning and effect. 
 Describe a linear circuit in space, and a surface bounded by the 
 circuit (Fig 6). Over this surface, which call S, let an impressed 
 force V act normally, V being the same all over the surface. 
 This system of force is irrotational everywhere except at the 
 bounding circuit, where there is a circuital distribution of g 
 the curl of e , of strength V. 
 
 Now, our present proposition asserts that the disturbances 
 due to V over the surface S are in every respect the same as 
 those due to V (the same normal impressed force), spread 
 over any other surface bounded by the same circuit. The 
 comprehensiveness of this property may be illustrated by sup- 
 posing that one surface is wholly in a non-conductor, whilst 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 113 
 
 the other passes through a conductor ; or that one surface is 
 wholly within a conductor, whilst the other passes out- of it 
 and through another conductor insulated from the first. Thus, 
 in the diagram, let S be the first surface (in section), and S t 
 the second, passing through a conductor represented by the 
 shaded region. The points A, B are where the common 
 boundary of the two surfaces cuts the paper, whilst the arrows 
 serve to show the direction of action of the impressed force. 
 S 2 is another surface of impressed force, also reaching into the 
 conductor. Now all these forcives (each by itself) will produce 
 the same final state of displacement in the dielectric and 
 electrification of the conductor, and will do so in the same 
 manner; that is, the electromagnetic disturbances generated 
 will be the same when expressed in terms of E and H. The 
 distribution of energy, for example, and the stresses, will be 
 the same. 
 
 But there must be some difference made by thus shifting the 
 source of energy. Obviously the nature of the flux of energy 
 is changed. This being 
 
 W = V(E-e )(H-h ), 
 
 where we deduct the intrinsic forces to obtain the forces effec- 
 tive in transferring energy, we see that every change made in 
 the distribution of the intrinsic sources affects the flux of 
 energy, in spite of the independence of E and H of their distri- 
 butions (subject to the limitation mentioned). In our example, 
 however, the only change is in the sheet of impressed force 
 itself. 
 
 Production of Steady State due to Impressed Forcive by 
 crossing of Electromagnetic Waves. Example of a Cir- 
 cular Source. Distinction between Source of Energy 
 and of Disturbance. 
 
 89. It may be readily suspected from the preceding, that, 
 as far as the production of electromagnetic disturbances goes, 
 we may ignore e and h altogether, and regard the circuital 
 vectors g and j as the real sources. This is, in fact, the case 
 when ultimately analysed. In the example just taken the cir- 
 cuit ABA is the source of the disturbances. That is, they 
 emanate from this line. If disturbances were propagated 
 
 i 
 
114 
 
 ELECTROMAGNETIC THEORY. 
 
 CII. II. 
 
 infinitely rapidly there might be some difficulty in recognising 
 the property, because the steady state appropriate to the 
 instantaneous state of the impressed force would exist 
 (if the conductor were away) ; but, as we shall see later, 
 the speed of propagation is always finite, depending upon the 
 values of c and /x- in the medium ; and conduction does not alter 
 this property, although by its attenuating and distorting effects 
 it may profoundly alter the nature of the resultant phenomena. 
 With, then, a finite speed of propagation, we have merely to 
 cause the impressed force to vary or fluctuate sufficiently 
 rapidly to obtain distinct evidence of the emanation of waves 
 from the real sources of disturbance. Thus, let the source be 
 
 FIG. 7. 
 
 circular in a plane perpendicular to the paper (Fig. 7), and A, B 
 the points where it cuts the paper. When the source is suddenly 
 started, the circle ABA is the first line of magnetic force. At 
 any time t later, such that vt is less than the distance JAB, 
 the electromagnetic disturbance will be confined wholly within 
 a ring-shaped region having the circular source for core and of 
 radius vt round A or round B. But when the distance vt in- 
 creases to JAB overlapping commences, and a little later there 
 is (as in the shaded part of the figure) a region occupied by two 
 coincident waves crossing one another. Now, the union of 
 these waves produces the steady state of displacement without 
 magnetic force tha* e, within the shaded region the magnetic 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 115 
 
 force vanishes, and the displacement is that which belongs to 
 the final steady state. As time goes on, of course the shaded 
 region enlarges itself indefinitely, although outside it is still a 
 region occupied by electromagnetic disturbances going out to 
 infinity. This supposes that there is no conductor in the 
 way. Should there be one, then, as soon as it is struck by the 
 initial electromagnetic wave, it becomes and continues to be 
 a secondary source of disturbance, and the final steady state, 
 different from the former, now arises from the superimposition 
 of the primary waves and the secondary. 
 
 It is, of course, impossible to go into detail at present, the 
 object being merely to point out the difference between sources 
 of energy and sources of disturbance. The source of energy 
 only works when there is electric current at the spot, and this 
 comes to it from the vortex lines (the circuital sources of dis- 
 turbance). Thus, in the last figure, at time vt, when the ring 
 of disturbance is of thickness 2vt, if the impressed force be in a 
 circular disc whose trace is the straight line AB, the force is 
 working in Aa and in B6, but inoperative elsewhere. Again, 
 after overlapping has begun it is still working in Aac and B6c?, 
 but is inoperative in cd, having done its work. Similarly 
 when the sheet of impressed force has any other shape. The 
 impressed force only works when it is allowed to work by the 
 electromagnetic wave reaching it. 
 
 To emphasize the matter, take another case. Let the 
 impressed force in any telegraph circuit be confined to a single 
 plane section across the conductor, so that the vortex line 
 is a line on its surface, going round it. If this be at 
 Valencia, we may shift the " seat of the E.M.F." to Newfound- 
 land, provided we preserve continuity of connexion with the 
 vortex line, as before explained; for instance, by extending 
 the sheet over the whole surface of the conductor between the 
 two places. The sheet at Newfoundland and the surface sheet 
 will together produce the same effects as the original sheet at 
 Valencia. Or, we may have the sheet entirely outside the 
 circuit, provided only that it is bounded by the original vortex 
 line, in touch with the conductor. 
 
 What the practical interpretation of this extraordinary 
 property is in connection with the "seat of E.M.F." of galvanic 
 batteries and in electrolysis generally still remains obscure. 
 
 i2 
 
116 ELECTROMAGNETIC THEORY. CH. II. 
 
 All impressed voltages and gaussages are more or less difficult 
 to understand. But there need be no doubt as to the general 
 truth of the property if the circuitality of the current can be 
 trusted. 
 
 The Eruption of "47r"s. 
 
 90. It may have been observed that the equations employed 
 by me in the preceding differ from those in use in all mathe- 
 matical treatises on the subject in a certain respect (amongst 
 others), inasmuch as the constant 47r, which is usually so 
 obtrusively prominent, has been conspicuous by its absence. 
 This constant 4?r was formerly supposed to be an essential part 
 of all electric and magnetic theories. One of the earliest 
 results to which a student of the mathematics of electricity 
 was introduced in pre-Maxwellian days was Coulomb's law of 
 the relation between the density of the electric layer on a 
 conductor and the intensity of electric force just outside it, say 
 
 (1) 
 
 and, since this was proved by mathematics, it seemed that the- 
 4?r was an essential ratio between two physical quantities, viz.^ 
 electricity and the force it exerted on other electricity. Never 
 a hint was given that this 4:r was purely conventional ; it was- 
 not, indeed, even recognised to be conventional, and is not at 
 the present day in some quarters. Then, again, at the begin- 
 ning of magnetism, was Gauss's celebrated theorem proving 
 mathematically that the total flux of magnetic force outward 
 through a closed surface equalled precisely 4?r times the total' 
 amount of magnetism enclosed within the surface ; and, for all 
 that might appear to the contrary, this remarkable result 
 flowed out necessarily from the properties of the potential 
 function and its derivatives, and of the three direction cosine* 
 of the normal to an element of the surface. It was funny 
 very funny. How ever the 4?r managed to find its way in was- 
 the puzzle in these and similar results ; for instance, in the 
 well-known 
 
 , ...... (2) 
 
 where //, is the permeability and K another physical property, 
 the susceptibility to magnetisation of a substance. The dark 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 117 
 
 mystery was carefully covered up by the mathematics. It was 
 as hard to understand as the monarch found it to explain the 
 presence of the apple in the dumpling how did the goodwife 
 manage to get it in ? Nor was the matter rendered plainer by 
 Maxwell's great treatise. Maxwell thought his theory of 
 electric displacement explained the meaning of the 47r (in the 
 corresponding electric theorem), as if it were a matter of 
 physics, instead, as is the fact, of irrationally chosen units. 
 
 As the present chapter has been mainly devoted to a general 
 outline of electromagnetic theory expressed in formulae involv- 
 ing rational units, it will be fitting, in bringing it to a conclu- 
 sion, to explain here the relation these rational units bear to 
 the ordinary irrational units. 
 
 The Origin and Spread of the Eruption. 
 
 91. The origin of the 4?r absurdity lay in the wisdom of 
 our ancestors, literally. The inverse square law being recog- 
 nised, say that two charges q l and q 2 repelled one another with 
 a force varying inversely as the square of the distance between 
 them; thus, 
 
 ...... (3) 
 
 where a is a constant ; what was more natural than to make 
 the expression of the law as simple as possible by giving the 
 constant a the value unity, if, indeed, it were thought of at all ? 
 Our ancestors could not see into the future, that is to say, 
 beyond their noses, and perceive that this system would work 
 out absurdly. They were sufficiently wise in their generation, 
 and were not to blame. 
 
 But, after learning that certain physical quantities bear to 
 one another invariable relations, we should, in forming a syste- 
 matic representation of the same, endeavour to avoid the intro- 
 duction of arbitrary and unnecessary constants. This valuable 
 principle was recognised to a small extent by our ancestors, as 
 above ; it was emphasised by Maxwell and Jenkin in their 
 little treatise on units in one of the Reports of the B. A. 
 Committee on Electrical Standards (1863, Appendix C; p. 59 
 of Spon's Reprint). Thus, referring to the magnetic law of 
 inverse squares, we have the following : 
 
118 ELECTROMAGNETIC THEORY. CII. II. 
 
 " The strength of the pole is necessarily defined as propor- 
 tional to the force it is capable of exerting on any other pole. 
 Hence the force / exerted between two poles of the strengths m 
 and m v must be proportional to the product mm r The force 
 f is also found to be inversely proportional to the square of the 
 distance, D, separating the poles, and to depend on no other 
 quantity ; hence, we have, unless an absurd and useless coefficient 
 be introduced, 
 
 f=mmJD*. ..... ..... (4) 
 
 Observe the words which I have italicised. When it is con- 
 sidered what Maxwell had then done in the way of framing a 
 broad theory of electromagnetism, it is marvellous that he 
 should have written in that way. By mere force of habit one 
 might, indeed, not consider there to be anything anomalous 
 about the 4?r in Coulomb's and Gauss's theorems. But did not 
 Maxwell's electrostatic energy K^ 2 /8?r and his magnetic energy 
 pffi/Sir per unit volume loudly proclaim that there was some- 
 thing radically wrong in the system to lead to such a mode of 
 expression, which fault should be attended to and set right ab 
 initio, especially in framing a permanent system of practical 
 units, which was what Maxwell and his colleagues were about ? 
 It would seem that the proclamation fell upon deaf ears, for 
 not only were the units irrationally constructed, but in his 
 Treatise, which followed some years later, we find the following 
 statement (p. 155, second edition). After an account of his 
 theory of electric displacement, we are told that " the theory 
 completely accounts for the theorem of Art. 77, that the total 
 induction through a closed surface is equal to the total quantity 
 of electricity within the surface multiplied by 4?r. For what 
 we have called the induction through the surface is simply the 
 electric displacement multiplied by 47r, and the total displace- 
 ment outward is necessarily equal to the total charge within 
 the surface." That is, his theory of electric displacement 
 accounts for the 47r. So it seems to do; and yet the 4?r has no 
 essential connection with his theory, or with any one else's. 
 Though by no means evident until it is pointed out, it is entirely 
 a question of the proper choice of units, and is independent of 
 all theories of electricity. It depends upon something much 
 more fundamental. 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 119 
 
 The Cure of the Disease by Proper Measure of the Strength 
 of Sources. 
 
 92. When looked into carefully, the question is simply 
 this : What is the natural measure of the strength of a source ? 
 Suppose, for instance, we have a source of heat in a medium 
 which does not absorb heat, how should we measure the 
 intensity of the source ? Plainly by the amount of heat emitted 
 per second, passing out through any surface enclosing the 
 source. If the flux of heat be isotropically regular, its density 
 will vary inversely as the square of the distance from a point 
 source, giving 
 
 ...... (5) 
 
 if C be the heat flux (per unit area) at distance r from the 
 source of strength S. If we knock out the 4?r we shall 
 obviously have an unnatural measure of the strength of the 
 source. 
 
 Similarly, if we send water along a pipe and let it flow from 
 its end, which may therefore be regarded as a source, we 
 should naturally measure its strength in a similar manner, viz., 
 by the current in the pipe, or by the total outflow. 
 
 Now in an electric field, or in a magnetic field, or in the 
 field of any vector magnitude, we have everywhere mathe- 
 matically analogous cases. We find, for instance, that electro- 
 static force is distributed like velocity in an incompressible 
 liquid, except at certain places, where it is, by analogy, 
 generated, or has its source. If, then, we observe that the 
 flux of force through a closed surface is not zero, there must 
 be sources within the region enclosed, and the natural measure 
 of the total strength of the sources is the total flux of force 
 itself. I have put this in terms of electric force rather than of 
 electric displacement, merely to exemplify that the matter has 
 no particular connexion with electric displacement. In the 
 former case it is a source of "electric force" that is considered; 
 in the latter, it would be of displacement ; and the principle 
 concerned in a rational reckoning of the strength of a source is 
 the same in either case. It is a part of the fitness of things, 
 and holds good in the abstract theory of the space-variation of 
 vector magnitudes, apart from all physical application. 
 
120 ELECTROMAGNETIC THEORY. CH. II. 
 
 Using, temporarily, the language of lines of force, or of tubes 
 of force (which, however, as here, sometimes works out rather 
 nonsensically), we may say that a unit pole sends out one line 
 of force, or one tube, when rationally estimated. 
 
 Next there is the proper measure of the strength of cir- 
 cuital fluxes to be considered; electric current, for instance. 
 The universal property here is that the circulation of the 
 magnetic force is proportional to the current through the 
 circuit ; and the natural pleasure of the strength of the current 
 is the circuitation itself, without, as usual, dividing by 4n-. 
 Now this division by 4?r arises out of the irrational reckoning 
 of the strength of point sources. We may therefore expect 
 that when the point sources are measured rationally, the 4r 
 will disappear from the reckoning of electric current, making 
 the circuitation of magnetic force the proper reckoning. This 
 is so, as may be easily seen by substituting for a linear 
 electric current an equivalent magnetic shell, and so bringing 
 in point sources distributed over its two faces. 
 
 Thus, in rational units, if we have a point source q of dis- 
 placement, and a point source m of induction, we have 
 
 D = ?/47rr 2 , B = m/47rr 2 , . . . (6) 
 
 to express the displacement and induction at distance r, when 
 the fluxes emanate isotropically ; and 
 
 . . (7) 
 
 if q/c and m/p are the measures of the sources of electric and 
 magnetic force respectively. In the magnetic case m may repre- 
 sent the strength of a pole, on the understanding that (since 
 the induction is really circuital) we ignore the flux coming to the 
 pole (as along a filamentary magnet), and consider only the 
 diverging induction. Also, 
 
 H = C/27rr ...... (8) 
 
 expresses the intensity of magnetic force at distance r from a 
 long solitary straight current of strength C. 
 
 Obnoxious Effects of the Eruption. 
 
 93. Considering merely the formulae belonging to point 
 sources with uniform divergence, we see that the effect of 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 121 
 
 changing from irrational to rational units is to introduce 4?r. 
 If this were all, we might overlook the fundamental irration- 
 ality and use irrational units for practical convenience. But, 
 as a matter of fact, it works out quite differently. For the 
 unnatural suppression of the 4:r in the formulae of central 
 force, where it has a right to be, drives it into the blood, there 
 to multiply itself, and afterwards break out all over the body 
 of electromagnetic theory. The few formulae where 4?r should 
 be are principally scholastic formulae and little used; the many 
 formulae where it is forced out are, on the contrary, useful 
 formulae of actual practice, and of the practice of theory. A 
 practical theorist would knock them out merely from the 
 trouble they give, let alone the desire to see things in their 
 right places. Furthermore, it should be remarked that the 
 irrationality of the formulae is a great impediment in the way 
 of a clear understanding of electromagnetic theory. The 
 interpretation of equation (2) above, for instance, or of the 
 similar well-known equation 
 
 $ = ) + 47r|ji, 
 
 presents some difficulty even to a student of ability, unless he 
 be given beforehand a hint or two to assist him. For if K is 
 a rational physical quantity, then u. cannot be ; or if /A is right, 
 then K must be wrong. Or ^, ^, and ^L must be incongruous. 
 The 4?r is also particularly inconvenient in descriptive matter 
 relating to tubes of force or flux, and in everything connected 
 with them. 
 
 This difficulty in the way of understanding the inner mean- 
 ing of theory is still further increased by the 4n- not entering 
 into the magnetic formulae in the same way as into the electric. 
 Thus, it is jp and ^/4w which are analogues. Again, in many 
 of the irrational formulae the irrationality appears to disappear. 
 For instance, in J^/o, in J31C/, in J($5 * n ^j an( i i n some 
 others. This is because both the factors are irrational, and the 
 two irrationalities cancel. It looks as if J) were the flux 
 belonging to ($, but it is not. In reality, we have, if x = (47r)*, 
 
 therefore, ED = 
 
 whilst c is the same in both irrational and rational units. 
 
122 ELECTROMAGNETIC THEORY. CH. II. 
 
 A Plea for the Removal of the Eruption by the Radical Cure. 
 
 94. The question now is, what is to be done? Are we 
 modern pigmies, who by looking over the shoulders of the 
 giants can see somewhat further than they did, to go on perpe- 
 trating and perpetuating their errors for ever and ever, and 
 even legalising them ? If they are to be enforced, it is to b& 
 hoped that it will not be made a penal offence not to use the 
 legal and imperial units. 
 
 The " brain-wasting perversity " of the British nation in sub- 
 mitting year after year to be ruled by such a heterogeneous 
 and incongruous collection of units as the yard, foot, inch, mile,, 
 knot, pound, ounce, pint, quart, gallon, acre, pole, horse-power, 
 etc., etc., has been repeatedly lamented by would-be reformers, 
 who would introduce the common-sense decimal system ; and 
 amongst them have been prominent electricians who hoped to- 
 insert the thin end of the wedge by means of the decimal sub- 
 division of the electrical units, and their connection with the 
 metre and gramme, and thus lead to the abolition of the present 
 British system of weights and measures with its absurd and 
 useless arbitrary connecting constants. But what a satire it 
 is upon their labours that they should have fallen into the very 
 pit they were professedly avoiding ! The perverse British 
 nation practically the British engineers have surely a right 
 to expect that the electricians will first set their own house in 
 order. 
 
 The ohm and the volt, etc., are now legalised, so that, as I 
 am informed, it is too late to alter them. This is a non seq., 
 however, for the yard and the gallon are legalised ; and if it is 
 not too late to alter them, it cannot be too late to put the new- 
 fangled ohm and its companions right. It is never too late to 
 mend. No new physical laboratory determinations will be 
 needed. The value of TT has been calculated to- hundreds of 
 places of decimals ; so that rational ohms, volts, etc., can be now 
 fixed with the same degree of accuracy as the irrational ones, 
 by any calculator. 
 
 When I first brought up this matter in The Electrician ia 
 1882-3, explaining the origin of the 4?r absurdity and its cure, 
 I did not go further than to use rational units in explaining the 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 123 
 
 theory of potentials, scalar and vector, and similar matters; then 
 returning to irrational units in order to preserve harmony with 
 the formulae in Maxwell's treatise, and I did not think a change 
 was practicable, on account of the B. A. Committee's work, and 
 the general ignorance and want of interest in the subject. But 
 much has happened since then. The spread of electrical know- 
 ledge has been immense, concurrent with the development of 
 electrical industries, to say nothing of theoretical and experi- 
 mental developments. I therefore now think the change is 
 perfectly practicable. At any rate, some one must set the 
 example, if the change is to occur. I have, therefore, in the 
 preceding, wholly avoided the irrational units ab initio ; and 
 shall continue to use rational units in the remainder of this 
 work. 
 
 So far as theoretical papers and treatises are concerned, 
 there is no difficulty. Every treatise on Electricity should be 
 done in rational formulae, their connexion with the irrational 
 (so long as they exist) being explained separately (in a chapter 
 at the end of the book, for instance), along with the method of 
 converting into volts, amperes, etc., the present practical units. 
 At present you have to first settle whether to use the electro- 
 static or the electromagnetic units, and then introduce the 
 appropriate powers of 10. If rational formula) are used, then, 
 in addition, you must first insert the constant 4?r in certain 
 places, so long as the irrational units last. 
 
 When, however, the real advantages of the rational system 
 become widely recognized and thoroughly assimilated, then 
 will come a demand for the rationalisation of the practical 
 units. Even at present the poor practician is complaining 
 that he cannot even pass from magnetomotive force to am- 
 pere-turns without a " stupid " 4?r coefficient getting in the 
 way. That the practical electrical units should be reformed 
 as a preliminary to the general reform of the British units 
 requires no argument to maintain. That this general reform is 
 coming I have not the least doubt. Even the perversity of the 
 British engineer has its limits. 
 
 Rational v. Irrational Electric Poles. 
 
 95. We may now briefly compare some of the more impor- 
 tant formulae in the two systems. Let us denote quantities in 
 
124 ELECTROMAGNETIC THEORY. CH. II. 
 
 rational units as in the preceding (except that when vector rela- 
 tions are not in question we need not employ special type), and 
 the corresponding quantities in irrational units by the same 
 symbols with the suffix f ; thus, q and &. Also denote (47r) J by x. 
 Let q i be the irrational charge which repels an equal 
 irrational charge at distance r with the same force F as a 
 rational charge q repels an equal clyirge at the same distance. 1 
 Then, 
 
 .... (9) 
 
 Therefore 
 
 2 = *% ....... (10) 
 
 As regards the ratio of the units, it is sufficient merely to 
 observe that the magnitude of the number expressing a 
 quantity varies inversely as the size of the unit. This 
 applies throughout, so that we need not bring in units at all, 
 but keep to the concretes appearing in formulae. 
 By (10) we shall have 
 
 if p is volume-density of electrification, a- surface density, 
 D displacement, C electric current density (or else the total 
 displacement and current). 
 
 Since D = cE, and D^cEJiTr, .... (12) 
 whilst D = #Di by (11), we have also 
 
 _E< A, V, e t P, x 13 x 
 
 -fi-A-V-7-F 
 
 where A is the time-integral of E, V the line-integral of E, or 
 voltage, e impressed electric force, P electric potential or poten- 
 tial difference. 
 
 The permittivity c is the same in both systems. So is the 
 ratio c/c of the permittivity to that of ether, or the specific 
 inductive capacity (electric). The rational permittance of a 
 unit cube condenser is c, and the irrational is cjkir. If S is 
 the permittance of any condenser 
 
 S-^S* ...... (14) 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 125 
 
 and its energy is 
 
 . . . (15) 
 if Q is its charge and V its voltage. We also have 
 
 JED = JE i D i = JcE2 = icE t 2/47r, . . . (16) 
 
 (17) 
 (18) 
 
 the first and second sets relating to electric energy, the third 
 to magnetic energy. 
 By Ohm's law 
 
 V = RC, and V = RC,; . . (19) 
 
 therefore by (11) and (13) 
 
 * 2 =l=-^> ..... (20) 
 where r is resistivity, so that 
 
 . . (21) 
 
 Rational v. Irrational Magnetic Poles. 
 
 96. If the repulsion between two magnetic poles m and m 
 is F at distance r, and this is also the repulsion between imu- 
 tional poles m i and m i5 we have 
 
 (22) 
 
 where /x. is the inductivity of the medium, common to the two 
 systems, as is likewise /A//A O the ratio of p. to that of the ether, 
 or the permeability. So 
 
 m = xm i ....... (23) 
 
 Here some discrepancies come in. For 
 
 B = /*H, and 3, = ^; . . (24) 
 
 , Tfl Hi Bv l2v lit /ner\ 
 
 so we have *=_ = _! = _!= - ^ . . . (25) 
 
 where ft is magnetic potential or gaussage. If I be intrinsic 
 magnetisation (intensity of), then 
 
 and l^phjtf, . . . (26) 
 
126 ELECTROMAGNETIC THEORY. CH. II. 
 
 from which, and by (25), we have 
 
 I = *Ii , - - (27) 
 
 The equation B = /*H, in the sense used by me, expands to 
 B = /x(h + F), (28) 
 
 where h is intrinsic, and F is the magnetic force of the field. 
 Also 
 
 ^ = ^ (1+K) = ^(1+4,), . . . (29) 
 
 .0 we have, by (28) and (26), 
 
 B = I + fi F + /v<F, (30) 
 
 where /* KF is the induced magnetisation. 
 
 If L is inductance, ^LC^L^ 2 , whence *=* = J^ if 
 
 L M 
 
 M is mutual inductance. 
 In the common equation, 
 
 B i = F i + 47rI i (31), 
 
 the intrinsic and induced magnetisations are lumped together 
 for one thing, and it is assumed that fi = l. The quantity K 
 is thus essentially a numeric, whilst /* is only a numeric by 
 assumption. But whilst p = i*>i, we have K = x*K i . Although 
 magnetisation, whether intrinsic or induced, are of the same 
 kind as induction, yet the reckoning is discrepant. Compare 
 (27) with (25) for B and B f . In (27) also, I may be either 
 intrinsic or induced, so far as the ratio x goes. 
 The common equation, div "D i = p { , becomes 
 
 divD=/>, ...... (32) 
 
 and the characteristic equation of Poisson becomes 
 
 V2 P= _ /) / c , (33) 
 
 the 4n- going out by the rationalisation. But P itself is given 
 
 by P = 2/>/47iTC (34) 
 
 The definition of current density, 
 
 curl H f = 471-0^, 
 becomes, curlH = C (35) 
 
OUTLINE OP ELECTROMAGNETIC CONNECTIONS. 127 
 
 Other changes readily follow. But now that I have explicitly 
 stated the relation between my rational formulae and the 
 ordinary, I leave the irrationals for good, I hope and return 
 to the rational and simplified formulae, which are so much 
 superior. 
 
 APPENDIX. 
 
 THE ROTATIONAL ETHER IN ITS APPLICATION TO 
 ELECTROMAGNETISM. 
 
 According to Maxwell's theory of electric displacement, 
 disturbances in the electric displacement and magnetic induc- 
 tion are propagated in a non-conducting dielectric after the 
 manner of motions in an incompressible solid. The subject is 
 somewhat obscured in Maxwell's treatise by his equations of 
 propagation containing A, , J, all of which are functions 
 considerably remote from the vectors which represent the state 
 of the field, viz., the electric and magnetic forces, and by some 
 dubious reasoning concerning M* and J. There is, however, no 
 doubt about the statement with which I commenced, as it 
 becomes immediately evident when we ignore the potentials 
 and use E or H instead, the electric or the magnetic force. 
 
 The analogy has been made use of in more ways than one, 
 and can be used in very many ways. The easiest of all is to 
 assume that the magnetic force is the velocity of the medium, 
 magnetic induction the momentum, and so on, as is done by 
 Prof. Lodge (Appendix to "Modern Views of Electricity"). 
 I have also used this method for private purposes, on account 
 of the facility with which electromagnetic problems may be 
 made elastic-solid problems. I have shown that when impressed 
 electric force acts it is the curl or rotation of the electric force 
 which is to be considered as the source of the resulting distur- 
 bances. Now, on the assumption that the magnetic force is 
 the velocity in the elastic solid, we find that the curl of the 
 impressed electric force is represented simply by impressed 
 mechanical force of the ordinary Newtonian type. This is 
 very convenient. 
 
128 ELECTROMAGNETIC THEORY. Cn. II. 
 
 But the difficulties in the way of a complete and satisfactory 
 representation of electromagnetic phenomena by an elastic- 
 solid ether are insuperable. Recognising this, Sir W. Thomson 
 has recently brought out a new ether ; a rotational ether. It 
 is incompressible, and has no true rigidity, but possesses a quasi' 
 rigidity arising from elastic resistance to absolute rotation. 
 
 The stress consists partly of a hydrostatic pressure (which I 
 shall ignore later), but there is no distorting stress, and its 
 place is taken by a rotating stress. It gives rise to a trans- 
 lational force and a torque. If E be the torque, the stress on 
 any plane N (unit normal) is simply VEN, the vector product 
 of the torque and the normal vector. 
 
 The force is curl E. We have therefore the equation of 
 motion 
 
 - curl E = juH, 
 
 if H is the velocity and p the density. But, alas, the torque 
 is proportional to the rotation. This gives 
 
 
 
 curl H = cE, 
 
 where c is the compliancy, the reciprocal of the quasi-rigid ity. 
 
 Now these are the equations connecting electric and magnetic 
 force in a non-conducting dielectric, when p, is the inductivity 
 and c the permittancy. We have a parallelism in detail, not 
 merely in some particulars. The kinetic energy JftH 2 repre- 
 sents the magnetic energy, and the potential energy icE 2 the 
 electric energy. The vector-flux of energy is VEH, the activity 
 of the stress. 
 
 This mode of representation differs from that of Sir W. 
 Thomson, who represents magnetic force by rotation. This 
 system makes electric energy kinetic, and magnetic energy 
 potential, which I do not find so easy to follow. 
 
 Now let us, if possible, extend our analogy to conductors. 
 Let the translational and the rotational motions be both fric- 
 tionally resisted, and let the above equations become 
 
 -curl E= 
 
 where g is the translational frictionality ; Tc will be considered 
 later. We have now the equations of electric and magnetic 
 force in a dielectric with duplex conductivity, k being the 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 129 
 
 electric and g the magnetic conductivity (by analogy with elec- 
 tric force, hut a frictionality in our present dynamical analogy). 
 
 We have, therefore, still a parallelism in every detail. We 
 have waste of energy by friction <?H 2 (translatibnal) and &E 2 
 (rotational). If g/fj> = k/c the propagation of disturbances 
 will take place precisely as in a non-conducting dielectric, 
 though with attenuation caused by the loss of energy. 
 
 To show how this analogy works out in practice, consider a 
 telegraph circuit, which is most simply taken to be three 
 co-axial tubes. Let, A, B, and C be the tubes ; A the inner- 
 most, C the outermost, B between them; all closely fitted. 
 Let their material be the rotational ether. In the first place, 
 suppose that there is perfect slip between B and its neighbours. 
 Then, when a torque is applied to the end of B (the axis of 
 torque to be that of the tubes), and circular motion thus 
 given to B, the motion is (in virtue of the perfect slip) trans- 
 mitted along B, without change of type, at constant speed, and 
 without affecting A and C. 
 
 This is the analogue of a concentric cable, if the conductors 
 A and C be perfect conductors, and the dielectric B a perfect 
 insulator. The terminal torque corresponds to the impressed 
 voltage. It should be so distributed over the end of B that the 
 applied force there is circular tangential traction, varying 
 inversely as the distance from the axis ; like the distribution 
 of magnetic force, in fact. 
 
 Now, if we introduce translational and rotational resistance 
 in B, in the above manner, still keeping the slip perfect, we 
 make the dielectric not only conducting electrically but also 
 magnetically. This will not do. Abolish the translational re- 
 sistance in B altogether, and let there be no slip at all between 
 B and A, and B and C. Let also there be rotational resistance 
 in A and C. 
 
 We have now the analogue of a real cable : two conductors 
 separated by a third. All are dielectrics, but the middle one 
 should have practically very slight conductivity, so that it is 
 pre-eminently a dielectric ; whilst the other two should have 
 very high conductivity, so that they are pre-eminently con 
 ductors. The three constants, ft, c, k, may have any value iii 
 the three tubes, but practically k should be in the middle tubs 
 a very small fraction of what it is in the others. 
 
 K 
 
130 ELECTROMAGNETIC THEORY. CH. II. 
 
 It is remarkable that the quasi-rotational resistance in A and 
 C should tend to counteract the distorting effect on waves of 
 the quasi-rotational resistance in B. But the two rotations, 
 it should be observed, are practically perpendicular, being 
 axial or longitudinal (now) in A and C, and transverse or 
 radial in B ; due to the relative smallness of Tc in the middle 
 tube. 
 
 To make this neutralising property work exactly we must 
 transfer the resistance in the tubes A and C to the tube B, at 
 the same time making it translational resistance. Also restore 
 the slip. Then we can have perfect annihilation of distortion 
 in the propagation of disturbances, viz., when k and g are so 
 proportioned as to make the two wastes of energy equal. In 
 the passage of a disturbance along B there is partial absorption, 
 but no reflection. 
 
 But as regards the meaning of the above k there is a diffi- 
 culty. In the original rotational ether the torque varies as the 
 rotation. If we superadd a real frictional resistance to rotation 
 we get an equation of the form 
 
 E being (as before) the torque, and H the velocity. But this 
 is not of the right form, which is (as above) 
 
 d 
 
 therefore some special arrangement is required (to produce the 
 dissipation of energy &E 2 ), which does not obviously present 
 itself in the mechanics of the rotational ether. 
 
 On the other hand, if we follow up the other system, in 
 which magnetic force is allied with rotation, we may put g = 0, 
 let - E be the velocity and H the torque ; /* the compliancy, 
 c the density, and k the translational frictionality.. This gives 
 
 curl H = &E + cE. 
 
 We thus represent a homogeneous conducting dielectric, with 
 a translational resistance to cause the Joulean waste of energy e 
 
OUTLINE OF ELECTROMAGNETIC CONNECTIONS. 131 
 
 But it is now seemingly impossible to properly satisfy the con- 
 ditions of continuity at the interface of different media. For 
 instance, the velocity -E should be continuous, .but we do not 
 have normal continuity of electric force at an interface. In 
 the case of the tubes we avoided this difficulty by having the 
 velocity tangential. 
 
 Either way, then, the matter is left, for the present, in an 
 imperfect state. 
 
 In the general case, the d/dt of our equations should receive 
 an extended meaning, on account of the translational motion 
 of the medium. The analogy will, therefore, work out less 
 satisfactorily. And it must be remembered that it is only an 
 analogy in virtue of similitude of relations. We cannot, for 
 instance, deduce the Maxwellian stresses and mechanical forces 
 on charged or currented bodies. The similitude does not ex- 
 tend so far. But certainly the new ether gees somewhat 
 further than anything known to me that has been yet proposed 
 in the way of a stressed solid. 
 
 [P.S. The special reckonings of torque and rotation in the 
 above are merely designed to facilitate the elastic-solid and 
 electromagnetic comparisons without unnecessary constants.] 
 
CHAPTER III. 
 
 THE ELEMENTS OF VECTORIAL ALGEBRA AND 
 ANALYSIS. 
 
 Scalars and Vectors. 
 
 97. Ordinary algebra, as is well known, treats of quantities 
 and their relations. If, however, we examine geometry, we 
 shall soon find that the fundamental entity concerned, namely 
 a straight line, when regarded as an entity, cannot be treated 
 simply as a quantity in the algebraical sense. It has, indeed, 
 size, viz., its length ; but with this is conjoined another impor- 
 tant property, its direction. Taken as a whole, it is a Vector. 
 In contrast with this, an ordinary quantity, having size only, is- 
 a Scalar. 
 
 Again, if we consider the mathematics of physical questions, 
 we find two distinct kinds of magnitudes prominently present. 
 All such magnitudes as mass, density, energy, temperature, are 
 evidently quantities in the simple algebraical sense; that is, 
 scalar magnitudes, or simply scalars. A certain (it may be 
 an unstated) unit of density, for instance, being implied, any 
 density is expressed by a simple number. (The question of the 
 "dimensions " of physical magnitudes is not in question.) All 
 magnitudes whatever which have no directional peculiarity 
 and which are each specified by a single number are scalars, 
 and subject to scalar algebra. 
 
 But such magnitudes as displacement, velocity, acceleration^ 
 force, momentum, electric current, &c., which have direction as 
 well as size, and which are fully specified by statement of the 
 size and direction, are vector magnitudes, or simply vectors. 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 133 
 
 Now, just as there is an algebra and analysis for scalars, so is 
 there a vector algebra and analysis appropriate to vectors; and it 
 is the object of the present chapter to give a brief account of 
 the latter, especially in respect to its application to electromag- 
 netism. 
 
 Characteristics of Cartesian and Vectorial Analysis. 
 
 98. Algebraical or analytical geometry in the usual Cartesian 
 form, though dealing ultimately with vectors, is not vectorial 
 algebra. It is, in fact, a reduction to scalar algebra by resolu- 
 tion of every vector into three rectangular components, which 
 are manipulated as scalars. Similarly, in the usual treatment 
 of physical vectors, there is an avoidance of the vectors them- 
 selves by their resolution into components. That this is a 
 highly artificial process is obvious, but it is often convenient. 
 More often, however, the Cartesian mathematics is ill-adapted 
 to the work it has to do, being lengthy and cumbrous, and 
 frequently calculated to conceal rather than to furnish and 
 exhibit useful results and relations in a ready manner. When 
 we work directly with vectors, we have our attention fixed 
 upon them, and on their mutual relations ; and these are 
 usually exhibited in a neat, compact, and expressive form, 
 whose inner meaning is evident at a glance to the practised 
 eye. Put the same formula, however, into the Cartesian form, 
 and what a difference ! The formula which was expressed by 
 a few letters and symbols in a single line, readable at once, 
 sometimes swells out and covers a whole page ! A very close 
 study of the complex array of symbols is then required to find out 
 what it means ; and, e v en though the notation be thoroughly 
 symmetrical, it becomes a work of time and great patience. 
 In this interpretation we shall, either consciously or uncon- 
 sciously, be endeavouring to translate the Cartesian formulae 
 into the language of vectors. 
 
 Again, in the Cartesian method, we are led away from the 
 physical relations that it is so desirable to bear in mind, to the 
 working out of mathematical exercises upon the components. 
 It becomes, or tends to become, blind mathematics. It was 
 once told as a good joke upon a mathematician that the poor 
 man went mad and mistook his symbols for realities ; as M for 
 the moon and S for the sun. There is another side to the 
 
134 ELECTROMAGNETIC THEORY. CH. IIL 
 
 story, however. If our object be ultimately physical, rathei 
 than mathematical, then the more closely we can identify the 
 symbols with their physical representatives the more usefully 
 can we work, with avoidance of useless though equally true 
 mathematical exercises. The mere sight of the arrangement 
 of symbols should call up an immediate picture of the physics 
 symbolised, so that our formulae may become alive, as it were. 
 Now this is possible, and indeed, comparatively easy, in 
 vectorial analysis ; but is very difficult in Cartesian analysis, 
 beyond a certain point, owing to the geometrically progressive 
 complexity of the expressions to be interpreted and manipu- 
 lated. Vectorial algebra is the natural language of vectors, 
 and no one who has ever learnt it (not too late in life, how- 
 ever) will ever care to go back from the vitality of vectors to 
 the bulky inanimateness of the Cartesian system. 
 
 Abstrusity of Quaternions and Comparative Simplicity 
 gained by ignoring them. 
 
 99. But supposing, as is generally supposed, vector 
 algebra is something "awfully difficult," involving meta- 
 physical considerations of an abstruse nature, only to be 
 thoroughly understood by consummately profound metaphysico- 
 mathematicians, such as Prof. Tait, for example. Well, if so, 
 there would not be the slightest chance for vector algebra and 
 analysis to ever become generally useful ; and I should not be 
 writing this, nor should I have, for several years past, persisted 
 in using vector algebra in electromagnetic theory a prophet 
 howling in the wilderness. It will readily be concluded, then, 
 that I believe that the vector analysis is going to become 
 generally used in scientific work, and that what is needed is 
 not "awfully difficult." There was a time, indeed, when I, 
 although recognising the appropriateness of vector analysis in 
 electromagnetic theory (and in mathematical physics gene- 
 rally), did think it was harder to understand and to work 
 than the Cartesian analysis. But that was before I had 
 thrown off the quaternionic old-man-of-the-sea who fastened him- 
 self on my shoulders when reading the only accessible treatise 
 on the subjects-Prof. Tait's Quaternions. But I came later 
 to see that, so far as the vector analysis I required was 
 concerned, the quaternion was not only not required, but was 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 135 
 
 a. positive evil of no inconsiderable magnitude ; and that by its 
 avoidance the establishment of vector analysis was made quite 
 simple and its working also simplified, and that it could be con- 
 veniently harmonised with ordinary Cartesian work. There is 
 not a ghost of a quaternion in any of my papers (except in 
 one, for a special purpose). The vector analysis I use may be 
 described either as a convenient and systematic abbreviation of 
 Cartesian analysis ; or else, as Quaternions without the quater- 
 nions, and with a simplified notation harmonising with Car- 
 tesians. In this form, it is not more difficult, but easier to 
 work than Cartesians. Of course you must learn how to work 
 it. Initially, unfamiliarity may make it difficult. But no 
 amount of familiarity will make Quaternions an easy subject. 
 
 Maxwell, in his great treatise on Electricity and Magnetism, 
 whilst pointing out the suitability of vectorial methods to the 
 treatment of his subject, did not go any further than to freely 
 make use of the idea of a vector, in the first place, and to 
 occasionally express his results in vectorial form. In this way 
 his readers became familiarised with the idea of a vector, and 
 also with the appearance of certain formula) when exhibited in 
 the quaternionic notation. They did not, however, derive any 
 information how to work vectors. On the whole, I am inclined 
 to think that the omission of this information has not tended 
 to impede the diffusion of a knowledge of vector analysis. 
 For, had he given an account of the theory, he would certainly 
 have followed the Hamilton-Tait system; and this would 
 probably, for reasons I shall shortly mention, have violently 
 prejudiced his readers against the whole thing. 
 
 But the diffusion of vector analysis has, undoubtedly, in 
 my opinion, been impeded by the absence of sufficiently ele- 
 mentary works on the subject, with a method of establishment 
 of principles adapted to ordinary minds, and with a con- 
 veniently workable notation. For the reader of Maxwell's 
 treatise who desired to learn to work vectors in analysis had 
 either feo go to Hamilton's ponderous volumes, or else to Prof. 
 Tait's treatise. The former are out of the question for initiatory 
 purposes. But the latter is excessively difficult, although de- 
 scribed as " an elementary treatise " not the same thing as 
 " a treatise on the elements." The difficulty arises in a great) 
 measure from the quaternionic basis. 
 
136 ELECTROMAGNETIC THEORY. Cfl. III. 
 
 Elementary Vector Analysis independent of the 
 Quaternion. 
 
 100. Suppose a sufficiently competent mathematician 
 desired to find out from the Cartesian mathematics what vector 
 algebra was like, and its laws. He could do so by careful 
 inspection and comparison of the Cartesian formula). He would 
 find certain combinations of symbols and quantities occurring 
 again and again, usually in systems of threes. He might 
 introduce tentatively an abbreviated notation for these com- 
 binations. After a little practice he would perceive the laws 
 according to which these combinations arose and how they 
 operated. Finally, he would come to a very compact system 
 in which vectors themselves and certain simple functions of 
 vectors appeared, and would be delighted to find that the rules 
 for the multiplication and general manipulation of these vectors 
 were, considering the complexity of the Cartesian mathematics 
 out of which he had discovered them, of an almost incredible 
 simplicity. But there would be no sign of a quaternion in his 
 results, for one thing ; and, for another, there would be no 
 metaphysics or abstruse reasoning required to establish the 
 rules of manipulation of his vectors. Vector analysis is, in its 
 elements, entirely independent of the exceedingly difficult 
 theory of quaternions ; that is, when the latter is treated 
 quaternionically ab initio. 
 
 " Quaternion " was, I think, defined by an American school- 
 girl to be " an ancient religious ceremony." This was, however, 
 a complete mistake. The ancients unlike Prof. Tait knew 
 not, and did not worship Quaternions. The quaternion and 
 its laws were discovered by that extraordinary genius Sir W. 
 Hamilton. A quaternion is neither a scalar, nor a vector, but 
 a sort of combination of both. It has no physical representa- 
 tives, but is a highly abstract mathematical concept." It is the 
 " operator " which turns one vector into another. It has a 
 stretching faculty first, to make the one vector become as long 
 as the other ; and a rotating faculty, to bring the one into 
 parallelism with the other. 
 
 Now in Quaternions the quaternion is the master, and lays 
 down the law to the vector and scalar. Everything revolves 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 137 
 
 round the quaternion. The laws of vector algebra themselves 
 ^re_established through quaternions, assisted by the imaginary 
 N/^l. But I am not sure that any one has ever quite under- 
 -stood this establishment. It is done in the second chapter of 
 Tait's treatise. I never understood it, but had to pass on. 
 That the establishment is not demonstrative may be the reason 
 of the important changes made therein in the third edition. 
 But it is still undemonstrative to me, though much improved. 
 Now this relates to the very elements of the subject, viz., the 
 scalar and vector products of a pair of vectors, the laws of 
 which are quite plain in the Cartesian mathematics. Clearly, 
 then, the quaternionic is an undesirable way of beginning the 
 subject, and impedes the diffusion of vectorial analysis in a way 
 which is as vexatious and brain-wasting as it is unnecessary. 
 
 Tait v. Gibbs and Gibbs v. Tait. 
 
 101. Considering the obligations I am personally under to 
 Prof. Tait (in spite of that lamentable second chapter), it does 
 seem ungrateful that I should complain. But I have at heart 
 the spread of a working knowledge of elementary vector 
 analysis quite as much as Prof. Tait has the extension of the 
 theory of quaternions. Besides, Prof. Tait has assumed a very 
 conservative attitude in relation to Hamilton's grand system. 
 For instance, to "more than one correspondent" who had 
 written for explanation of something they found obscure and 
 the same thing occurred to me described in his treatise by 
 "It is evident, . . .", he, "on full consideration," decides not 
 to modify it, but to italicise the words ! He also told them 
 that if they did not see it, in the light of certain preceding 
 parts of the treatise, then they had "begun the study of 
 Quaternions too soon" (Third edition, p. 110). This is as 
 characteristic of the sardonic philosopher as a certain heavy 
 kind of " flippancy " is of the Cockney. Again, in his Preface 
 he states one cause of the little progress made in the develop- 
 ment of Quaternions to be that workers have (especially in 
 France) been more intent on modifying the notation or the 
 mode of presentation of the elementary principles, than in 
 extending the application of the calculus. "Even Prof. 
 Willard Gibbs must be ranked as one of the retarders of 
 quaternionic progress, in virtue of his pamphlet on Vector 
 
138 ELECTROMAGNETIC THEORY. CH. III. 
 
 Analysis, a sort of hermaphrodite monster, compounded of the 
 notations of Hamilton and Grassmann." Grassmann, I may 
 observe, established, inter alia, a calculus of vectors, but not 
 of quaternions. 
 
 Prof. W. Gibbs is well able to take care of himself. I may, 
 however, remark that the modifications referred to are evidence 
 of modifications being felt to be needed ; and that Prof. Gibbs's 
 pamphlet (NOT PUBLISHED, Newhaven, 1881-4, pp. 83), is not 
 a quaternionic treatise, but an able and in some respects 
 original little treatise on vector analysis, though too condensed 
 and also too advanced for learners' use ; and that Prof. Gibbs, 
 being no doubt a little touched by Prof. Tait's condemnation, 
 has recently (in the pages of Nature) made a powerful defence of 
 his position. He has by a long way the best of the argument, 
 unless Prof. Tait's rejoinder has still to appear. Prof. Gibbs 
 clearly separates the quaternionic question from the question 
 of a suitable notation, and argues strongly against the 
 quaternionic establishment of vector analysis. I am able (and 
 am happy) to express a general concurrence of opinion with 
 him about the quaternion, and its comparative uselessness in 
 practical vector analysis. As regards his notation,' however, I 
 do not like it. Mine is Tait's, but simplified, and made to 
 harmonise with Cartesians. 
 
 Abolition of the Minus Sign of Quaternions. 
 
 102. In Quaternions, the square of a unit vector is 1. 
 This singular convention is quaternionically convenient. But in 
 the practical vector analysis of physics it is particularly incon- 
 venient, being indeed, an obtrusive stumbling-block. All 
 positive scalar products have the minus sign prefixed ; there is 
 thus a want of harmony with scalar investigations, and a 
 difficulty in readily passing from Cartesians to Vectors and 
 conversely. My notation, on the other hand, is expressly 
 arranged to facilitate this mutual conversion. 
 
 As regards the establishment of the elementary vector algebra, 
 that is quite simple (freed from the quaternion) ; it all follows 
 from the definitions of a vector and of the scalar and vector 
 products of a pair of vectors. 
 
 Now I can imagine a quaternionist (unless prejudiced) admit- 
 ting the simplicity of establishment and of operation, and the 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 139 
 
 convenience of the notation, and its sufficiency for practical re- 
 quirements up to a certain point ; and yet adding the inquiry 
 whether there is not, over and above this vector analysis, a theory 
 of Quaternions which is overlooked. To this I would reply, Cer- 
 tainly, but it is not food for the average mathematician, and 
 can, therefore, never be generally used by him, his practical 
 requirements being more suitably satisfied by the rudimentary 
 vector analysis divested of the mysterious quaternion. This 
 does not exclude the important theory of y and its applications. 
 Prof. Gibbs would, I think, go further, and maintain that the 
 anti- or ex-quaternionic vectorial analysis was far superior to 
 the quaternionic, which is^ uniquely adapted to three dimen- 
 sions, whilst the other admits of appropriate extension to more 
 generalised cases. I, however, find it sufficient to take my 
 stand upon the superior simplicity and practical utility of the 
 ex-quaternionic system.. 
 
 We may, however, if we wish to go further, after ex- 
 quaternionic establishment of vector algebra, conjoin the scalar 
 and vector, and make the quaternion, and so deduce the whole 
 body of Quaternions. But sufficient for the day is the labour 
 thereof ; and we shall* now be concerned with scalars and 
 vectors only. The reader should entirely divest his mind of 
 any idea that we are concerned with the imaginary ^^ "1 in 
 vector analysis. Also, he should remember that unfamiliarity 
 with notation and processes may give an appearance of difficulty 
 that is entirely fictitious, even to an intrinsically easy matter ; 
 so that it is necessary to thoroughly master the notation and 
 ideas involved. The best plan is to sit down and work ; all 
 that books can do is to show the way. 
 
 Type for Vectors. Greek, German, and Roman Letters un- 
 suitable. Clarendon Type suitable. Typographical 
 Backsliding in the Present Generation. 
 
 103. We should, in the first place, fix how to represent 
 vectors, although in reality this is the outcome of experience. 
 A vector may obviously be denoted by a single letter ; and, 
 having defined certain letters to stand for scalars and others 
 for vectors, it is certainly unnecessary that the vectors should 
 be distinguished from the scalars by the use of different kinds 
 of types. But practical experience shows that it is very 
 
140 ELECTROMAGNETIC THEORY. CH. III. 
 
 desirable that this should be done, in order to facilitate the 
 reading of vectorial work, by showing at a glance which letters 
 are vectors and which are scalars, and thus easing the stress 
 and strain on the memory. This is all the more important 
 because the manipulation of vectors sometimes differs from that 
 of scalars. 
 
 Now Prof. Tait usually indicates vectors by Greek letters. 
 But it is well known that a considerable familiarity with the 
 Greek letters such as is acquired by studying the literature 
 of ancient Greece is required before they can be read and 
 manipulated with facility. On the other hand, few are Greek 
 scholars, and in fact many people think it is about time the 
 dead languages were buried. Greek letters are, at any rate, 
 not very well adapted to a vector analysis which aims at 
 practicality. 
 
 Maxwell employed German or Gothic type. This was an 
 unfortunate choice, being by itself sufficient to prejudice 
 readers against vectorial analysis. Perhaps some few readers 
 who were educated at a commercial academy where the writing 
 of German letters was taught might be able to manage the 
 German vectors without much difficulty ; but to others it is a 
 work of great pains to form German letters legibly. Nor is 
 the reading of the printed letters an easy matter. Some of 
 them are so much alike that a close scrutiny with a glass is 
 needed to distinguish them, unless one is lynx-eyed. This is a 
 fatal objection. But, irrespective of this, the flourishing 
 ornamental character of the letters is against legibility. In 
 fact, the German type is so thoroughly unpractical that the 
 Germans themselves are giving it up in favour of the plain 
 Roman characters, which he who runs may read. It is a relic 
 of mediaeval monkery, and is quite unsuited to the present 
 day. Besides, there can be little doubt that the prevalent 
 shortsightedness of the German nation has (in a great measure) 
 arisen from the character of the printed and written letters 
 employed for so many generations, by inheritance and accu- 
 mulation. It became racial ; cultivated in youth, it was inten- 
 sified in the adult, and again transmitted to posterity. German 
 letters must go. 
 
 Rejecting Germans and Greeks, I formerly used ordinary 
 Roman letters to mean the same as Maxwell's corresponding 
 
HSITY 1 
 
 ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 141 
 
 Germans. They are plain enough, of course ; but, as before 
 mentioned, are open to objection. Finally, I found salvation in 
 Clarendons, and introduced the use of the kind of type so- 
 called, I believe, for vectors (Phil. Mag., August, 1886), and 
 have found it thoroughly suitable. It is always in stock ; it is 
 very neat; it is perfectly legible (sometimes alarmingly so), and 
 is suitable for use in formulae along with other types, Roman or 
 italic, as the case may be, contrasting and also harmonising 
 well with them. 
 
 Sometimes block letters have been used ; but it is sufficient 
 merely to look at a mixed formula containing them to see that 
 they are not quite suitable. I should mention here, however, 
 that it is not the mere use of special types that converts scalar 
 to vector algebra. For instance, engineers have often to deal 
 with vector magnitudes in their calculations, but not (save 
 exceptionally) in their vectorial signification. That is, it is 
 merely their size that is in question, and when this is the case 
 there is no particular reason why a special kind of type should 
 be used, whether blocks or another sort. 
 
 In connection with Clarendon type, a remark may be made 
 on a subject which is important to the community in general. 
 I refer to the retrograde movement in typography which has 
 been going on for the last 20 years or so. Many people, who 
 possess fairly good and normal eyesight, find a difficulty in 
 reading printed books without straining the eyes, and do not 
 know the reason. They may think the print is too small, or 
 that the light is not good, or that their eyes are not right. But 
 the size of type per se has little to do with its legibility ; a far 
 more important factor is the style of type, especially as regards 
 the fineness of the marks printed. The " old style," revived a 
 generation since, and now largely in use, differs from the more 
 legible " modern style " (of this page, for instance) in two re- 
 spects. It has certain eccentricities of shape, which make it 
 somewhat less easy to read ; but, more importantly, the letters 
 on the types are cut a good deal finer, which results in a pale 
 impression, as if the ink were watery. This is the main cause 
 of the strain upon the eyes, and the good light wanted. Even 
 very small print is easy to read if it be bold and black, not 
 thin and pale. It would be a public benefit if the retrograde 
 step were reversed, and the revived old style, which threatens 
 
142 ELECTKOMAGNETIC THEOEY. CH. III. 
 
 to drive out the modern style, discarded. Not that the latter 
 is perfect. A still better style would be arrived at by thicken- 
 ing the fine lines in it, producing something intermediate 
 between it and the Clarendon style (which last, of course, would 
 be too much of a good thing). As everyone who has had to 
 read MSS. knows, the most legible handwritings irrespective 
 of the proper formation of the letters are those in which the 
 writing-master's fine upstroke is discarded. Now it is the same 
 in print. Thicken the fine lines, and the effect is magical. 
 
 Notation. Tensor and Components of a Vector. Unit 
 Vectors of Reference. 
 
 104. The tensor of a vector is its size, or magnitude apart 
 from direction. Other important connected quantities are its 
 three rectangular scalar components the Cartesian compo- 
 nents which are the tensors of the three rectangular vector 
 components. It is usual, in Cartesian work, to use three sepa- 
 rate letters for these components, as F, G, H ; or u, v, w ; or 
 ^> fj'&c. One objection to this practice is that when there 
 is a large number of vectors the memory- is strongly taxed to 
 remember their proper constituents. Another is the prodigal 
 waste of useful letters. One alphabet is soon exhausted, and 
 others have to be drawn upon. 
 
 In my notation the same letter serves for the vector itself, 
 and for its tensor and components. Thus, E denoting any 
 vector, its tensor is E, and its components are Ej, E 2 , E 3 . Simi- 
 larly, the tensor of a is a, and the components are a lt a^ a y A 
 large stock of letters is thus set free for other use. 
 
 But a remark must be made concerning MS. work, as dis- 
 tinguished from printed work. In MS. work it is inconvenient 
 to be at the trouble of writing two kinds of letters ; ordinary 
 letters will suffice for both scalars and vectors. Or the ordi- 
 nary letters representing vectors may receive some conventional 
 mark to vectorise them. But, presuming ordinary letters are 
 written, something is required to distinguish between the vector 
 and its tensor. This may be satisfied by calling E the tensor 
 of E. Only when an investigation is to be written out for the 
 printers is it necessary to bring in special letters, and this is 
 most simply done by a conventional mark affixed to every 
 e) vector. Compositors are very intelligent, read mathe- 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 143 
 
 matics like winking, and carry out all instructions made by the 
 author. 
 
 If a vector a be multiplied by a scalar x, the result, written 
 afa or az, is a vector x times as big as a, and having the same 
 direction. Thus, if a 3 be a unit vector parallel to a (or, more 
 strictly, parallel to and concurrent with a), of unit length, 
 we have a = aa r It is sometimes useful to separately represent 
 the direction and length of a vector, and the above is a con- 
 venient way of doing it without introducing new letters. This 
 applies to any vector. But it need not be an absolute rule. 
 For instance, the Cartesian co-ordinates x t y, z of a point may be 
 retained. Thus, let r be the vector from the origin to any point, 
 and let i, j, k be^ unit vectors from the origin along the three 
 rectangular axes. We shall then have 
 
 x = ^i, y = yj, z = zk, . . (1) 
 
 where x is the vector projection of r on the i axis, as we know 
 by the elementary geometry of a rectangular parallelepiped or 
 brick. Also, the direction-cosines of r are x/r, yjr, z/r, and 
 
 by Euclid I., 47. 
 
 The Addition of Vectors. Circuital Property. 
 
 105. This brings us to vector addition. The vector x 
 signifies translation through the distance x in the direction i. 
 If, now, after performing this operation, we carry out the 
 operation indicated by y, viz., translation through the distance 
 y in the direction j ; and, lastly, carry out the operation z, or 
 translation through the distance z in the k direction, we shall 
 arrive at the end of the vector r. That is, starting from one 
 corner of a brick, we may reach the opposite corner by three 
 mutually perpendicular journeys along three edges of the brick. 
 The final result is the same as if we went straight across from 
 corner to corner, that' is, by carrying out the operation indicated 
 by the vector r. This equivalence is expressed by 
 
 r = x + y + z, . . , . , . (3) 
 Or, by (1), r = XL + yj + zk. 3 , , . (4) 
 
144 
 
 ELECTROMAGNETIC THEORY. 
 
 CH. III. 
 
 The meaning of addition of vectors in this example is simply 
 the carrying out of the operations implied by the individual 
 vectors added, the geometrical vector meaning a displacement 
 in space, or translation from one point to another. The order of 
 addition is indifferent, since there are six ways of going from 
 one corner to the opposite one of a brick along its edges. 
 
 We see that any vector may be expressed as the sum of 
 three mutually perpendicular vectors, viz., its vector projections 
 on the axes. Furthermore, by the use of a skew parallelepiped 
 Instead of a brick we see that any vector may be expressed as 
 the sum of three other vectors having any directions we please, 
 provided they are independent, or not all in the same plane. 
 For in the latter case the parallelepiped degenerates to a plane 
 figure. 
 
 FIG. 2. 
 
 But it is perhaps best to explain vector addition in general 
 without any reference to axes. Thus let a and b be vectors to 
 be added, a meaning translation from P to Q in the first figure, 
 or through an equal distance along any parallel line, as from 
 S to R in the second figure, whilst b means translation from Q 
 to R in the first figure, or through an equal distance along any 
 parallel line, as from P to S in the second figure. In the first 
 case, performing the operation a first, and then b, we go from 
 P to R irid Q ; in the second case, with b first and then a, we go 
 from P to R vid S. The final result in either case is equivalent to 
 direct translation from P to R, symbolised by the vector c. 
 
 Thus, 
 
 (5) 
 
 The above may be extended to any number of vectors. Or we 
 may reason thus : Let A be any given vector, translating, say, 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 145 
 
 from P to Q. We need not go from P to Q direct, but may 
 follow any one of an infinite number of paths, as for example, 
 
 I 
 
 f 
 
 P A 
 
 FIG. 3. 
 
 Q 
 
 FIG. 4. 
 
 a + b + c-fdin the figure. The final result of the successive 
 translations is always the same, viz., the direct translation A. 
 
 Or A=a+b+c+d (6) 
 
 Thus any vector A may be split up into the sum of any 
 number n of vectors, of which n 1 are perfectly arbitrary, for 
 instance ,a, b, c in the figure, The remaining one d is, of 
 course, not arbitrary. It is the vector required to complete 
 the circuit of vectors. 
 
 The vectors need not be in one plane. Nor need the path 
 followed consist of finite straight portions. It may be wholly 
 or partly curved. The curved portions are then made up of 
 infinitesimal vectors. Each curved portion may be replaced 
 by the vector joining its terminals. 
 
 Since the original vector and the substituted vectors form a 
 circuit, if the positive direction in the circuit be the same for 
 all the vectors, we may express vector addition thus: The 
 sum of any number of vectors which make a circuit is zero. 
 That is, 2a = 0, if a is the type of the vectors summed. For 
 a curved circuit we shall have 
 
 (7) 
 
 where da is the vector element of the circuit. Here s itself 
 may be taken to be the vector from any fixed point P to a 
 point Q on the circuit, Fig. 4. Then ds is the infinitesimal 
 
146 ELECTROMAGNETIC THEORY. CH. III. 
 
 change in s made in an infinitesimal step along the circuit, 
 that is, it is the vector element of the circuit itself. 
 
 In a vector equation every term is a vector, of course, how- 
 ever the individual terms may be made up, and every vector 
 equation expresses the fact symbolised by (7), or by 2a = 
 when the vectors are finite. In the latter case, a may be also 
 regarded as As, the finite change in the vector s from any fixed 
 origin produced by passing from beginning to end of the 
 vector a. 
 
 The - sign prefixed to a vector is the same as multiplication 
 by - 1, and its effect is simply to reverse the direction of trans- 
 lation. Or it may be regarded as reversing the tensor, without 
 altering the direction: thus 
 
 - a = - aa x = a x ( - a x ) = ( - a) x aj . . (8). 
 Thus, in any vector equation, for example 
 
 a + b + c + d + e + f=0, . . . . (9) 
 
 we may transfer any terms to the other side by prefixing the 
 - sign. Thus, 
 
 a + b + c= -d-e-f- -(d + e + f). . . (10) 
 
 In the form (9) we express the fact that translation in a circuit 
 is equivalent to no translation. In (10), however, we express 
 the equivalence of two translations by different paths from one 
 point to another. 
 
 If any trouble be experienced in seeing the necessary truth of 
 (9) for a circuit in whatever order the addition be made, the 
 matter may be clinched by means of i, j, k, the unit rectangular 
 vectors ; as we saw before, the resolution of vectors into 
 rectangular component vectors depends only upon the properties 
 of the right angled triangle. Thus split up the vectors a, b, c 
 into components, we have 
 
 k6 3 , ..>.';. (11) 
 c = icj + jc 2 + kc 3 , ) . 
 
 Now add up. All vectors parallel to i add in scalar fashion, 
 for there is no change of direction. Similarly for j and k ; so 
 we have 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 147 
 
 Now the scalar additions may be done in any order we 
 please. It follows that the same is true in vector addition. 
 The addition and subtraction of vectors and transfer from one 
 side of an equation to the other are thus done identically as in 
 the algebra of scalars. Addition has, it is true, not the eame 
 meaning, but the vector meaning is not inconsistent with the 
 scalar meaning, and, in fact, includes the latter as a particular 
 case. There is never any conflict between + put between 
 vectors and + put between scalars. 
 
 Application to Physical Vectors. Futility of Popular Demon- 
 strations. Barbarity of Euclid. 
 
 106. In the above we have referred entirely to the geo- 
 metrical vector. But a very important step further can be 
 made referring to physical vectors, a step which immediately 
 does away with piles of ingeniously constructed and brain- 
 wasting " demonstrations," especially compiled for the tortur- 
 ing of students and discouragement of learning. If a quantity 
 be recognised to be a directed magnitude, it is, in its mathe- 
 matical aspect, a vector, and is therefore subject to the same 
 laws as the vector, or geometrical vector. So, just as we have 
 the triangle, or parallelogram, or polygon of geometrical vectors, 
 we must have the same property exemplified in the addition of 
 all vectors, as velocity, acceleration, force, &c. This identity of 
 treatment applies not only to the addition property, but to the 
 multiplication and other properties to be later considered. 
 We may symbolise a physical vector by a straight line of given 
 length and direction. 
 
 It used to be thought necessary to give demonstrations of 
 the parallelogram of forces, perhaps even before the student 
 knew what force meant. I have some dim recollection of days 
 spent in trying to make out Duchayla's proof, which was 
 certainly elaborate and painstaking, though benumbing. Max- 
 well, in his treatise, elaborated a demonstration that electric 
 currents compounded according to the vector law. But surely 
 there is something of the vicious circle in such demonstrations. 
 Is it not sufficient to recognise that a quantity is a vector, to 
 know that it follows the laws of the geometrical vector, the 
 addition property of which does not want demonstrating, but 
 only needs pointing out, as in the polygon of vectors ? 
 
 L2 
 
148 ELECTROMAGNETIC THEORY. CH. IIL 
 
 Although this is not the place for exercises and examples, yet 
 it is worth while to point out that by means of the addition 
 property of vectors a good deal of geometry can be simply 
 done better than by Euclid, a considerable part of whose 12 
 books consists of examples of how not to do it (especially 
 Book V.). There is a Society for the Improvement of Geometrical 
 Teaching. I have no knowledge of its work ; but as to the 
 need of improvement there can be no question whilst the reign 
 of Euclid continues. My own idea of a useful course is to begin 
 with arithmetic, and then, not Euclid, but algebra. Next, not 
 Euclid, but practical geometry, solid as well as plane ; not 
 demonstrations, but to make acquaintance. Then, not Euclid, 
 but elementary vectors, conjoined with algebra, and applied to 
 geometry. Addition first; then the scalar product. This 
 covers a large ground. When more advanced, bring in the 
 vector product. Elementary calculus should go on simultane- 
 ously, and come into the vector algebraic geometry after a bit. 
 Euclid might be an extra course for learned men, like Homer. 
 But Euclid for children is barbarous.* 
 
 The Scalar Product of Two Vectors. Notation and 
 
 Illustrations. 
 
 107. Coming next to the products of vectors, it is to be 
 noted at the beginning that the ordinary idea of a product in 
 
 * From The Electrician, December 4, 1891, p. 106, I learn that the 
 correct title of the society above alluded to is the Association for the 
 Improvement of Geometrical Teaching. " It was founded, we believe, 
 about ten years ago by a few teachers, who realised that Euclid for 
 children is, as Mr. Heaviside puts it, simply barbarous. Three pamphlets 
 have been published by Messrs. Macmillan and Co. a syllabus of plane 
 geometry, corresponding to Euclid, Books I. to VI. ; another of modern 
 plane geometry ; and another of linear dynamics. Messrs. Swan, Sonnen- 
 schein and Co. have published an elementary geometrical conies, and there 
 the labours of the Association appear to have stopped. The Association 
 has had to struggle against the stubborn conventionalism of the modern 
 schoolmaster, who pleads that he cannot make any change, because of the 
 Universities. After a considerable fight, Mr. Hamblin Smith's common- 
 sense proofs of Euclid's problems were accepted by Cambridge examiners. 
 Small as the visible results of the Association have been, there is a distinct 
 change of feeling taking place with regard to geometry, both as an educa- 
 tional subject and as an implement of scientific work. At present geometry 
 is taught as badly as Greek, even in the best public schools ; and the 
 educational value of Greek is in many respects higher than that of Euclid." 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 149 
 
 arithmetic and in scalar algebra does not apply to vectors, 
 because they are not scalars. We cannot, therefore, say before- 
 hand what the product of two vectors ought to be, or deter- 
 mine this question by any prior reasoning of a legitimate nature. 
 We must examine in what way vectors enter into combination 
 as products, and then introduce definitions and conventions of 
 notation to give expression to the facts in the simplest and 
 most convenient way. After this, the work is deductive. 
 
 By this examination we are led to recognise two distinct 
 kinds of products of a pair of vectors, the scalar product and 
 the vector product. It is with the former of these that we are 
 now immediately concerned. 
 
 We define the scalar product of a pair of vectors A and B 
 whose tensors are A and B, and whose included angle is 6, to 
 be the scalar AB cos 0, and we denote it by AB. Thus 
 
 AB = ABcos0 (12) 
 
 defines the scalar product and its notation. 
 
 To see its full significance and how it works out, let us first 
 apply (12) to the unit reference vectors, i, j, k, which are oeper- 
 pendicular, by taking A and B to be one or other of them in 
 turn. First multiplying each of i, j, k by itself, we get 
 
 12 = 1, J2 = l, k2=l, . . . . (13) 
 
 with the convention borrowed from scalar algebra that ii is 
 equivalently denoted by i 2 . This may be called the square 
 of i. In each of the three cases (13) the vectors multiplied 
 together are of unit length, and are parallel and concurrent, so 
 that the scalar product, according to (12), is unity. 
 
 Similarly, the square of any unit vector is unity. And since 
 A = AA 15 the square of any vector is the same as the square of 
 its tensor ; or 
 
 A 2 = A 2 (14) 
 
 Also, the scalar product of any two parallel vectors is the 
 product of their tensors, for AB = AB A 1 B 1 , and A 1 B 1 is now 
 unity. 
 
 Next make scalar products of i, j, k in pairs. We get 
 
 ij = 0, jk = 0, ti = (15) 
 
150 ELECTROMAGNETIC THEORY. CH. III. 
 
 That is, the scalar product of two perpendicular unit vectors is 
 zero. The same is, of course, true of any two perpendicular 
 vectors. Thus the equation AB = means that A is perpen- 
 dicular to B ; unless, indeed, one or other of them is zero. 
 We have also ^ 
 
 AA-cosfl, (16) 
 
 by dividing (12) by AB ; or, the scalar product of any two 
 vectors is the cosine of the included angle. This is reckom 
 positively from concurrent coincidence, so that as ge2s from 
 to 2 TT, AjBi goes from 1 through to 1, and then through 
 again to + 1. 
 
 There is, strictly, no occasion to introduce trigonometry. Or 
 we might make the trigonometry be a simultaneously developed 
 subject. It is, in fact, a branch of vectorial algebra, being 
 scalar developments of parts thereof. We may employ the idea 
 of perpendicular projection simply. rrThus, we may say that the 
 scalar product of a pair of unit vectors is the length of the 
 projection of either upon the other ; and that the scalar 
 product of any vector A and a unit vector i is the projection of 
 A upon the axis of i ; and, comprehensively, that the scalar 
 product AB of any two vectors A and B is the product of the 
 tensor of either into the projection (perpendicularly) upon it of 
 the other. It is the effective product, so to speak H In physical 
 mathematics scalar products frequently have reference to 
 energy, or activity, or connected quantities. Thus, if F be a 
 force and v the velocity of its point of application, their scalar 
 product Fv is the activity, of the force ; it is the product of the 
 speed and the effective force. When the force and the velocity 
 are perpendicular, the activity is nil, although the velocity may 
 be changing a fact which familiarity does not render less 
 striking. When F and v are parallel (whether concurrent or 
 not), Fv becomes the same as Fi;, in the common meaning of a 
 product. A notation^that harmonises in this way is obviously 
 a convenient one. 
 
 Notice, also, that AB = BA, as in scalar algebra. 
 
 A frequently occurring operation is the surface integral of 
 the normal component of a vector ; for example, to express the 
 amount of induction through a surface. Here the idea of a 
 unit normal vector N is useful. The normal component of B 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 151 
 
 is then NB, and: the integral is 2 NB, the summation extending 
 over the surface. 
 
 Similarly, to express the line-integral of the effective 
 component of a vector along a line, we may let T be the unit 
 element of curve ; that is, the unit tangent ; then TE is the 
 tangential component of the vector E, and 2TE is the integral ; 
 for example, the voltage between two points along the path to 
 which the summation refers, if E be the electric force. 
 
 Since ab is a scalar, it behaves as a scalar, when considered 
 as a whole. Thus, when multiplied by a scalar, x, the result, 
 #ab or aba:, is scalar, being simply x times ab. When multi- 
 plied by a vector the result is a vector ; thus, c.ab or ab.c 
 means ab times the vector c . The dot here acts rather as a 
 separator than as a sign of multiplication. Thus, to illustrate, 
 ca.b means ca times the vector b ; and, similarly, a.bc is be 
 times the vector a. But, instead of the dot, we may use 
 brackets to indicate the same thing, thus c.ab may be written 
 c(ab). This is, perhaps, preferable in initiatory work, but I 
 think the dot plan is more generally useful. 
 
 We have an example of this combination of three vectors in 
 the stress formulae. Thus, the electric stress vector on the 
 plane whose unit normal is N, is expressed by 
 
 E.DN - N.JED, 
 
 (equation (31), 72) ; that is, the sum of two vectors, parallel 
 to E the electric force and to N respectively, whose tensors are 
 EDN and -JED respectively, where D is the displacement. 
 The interpretation as a tension along E combined with an equal 
 lateral pressure, obtained by taking N parallel to and then per- 
 pendicular to E or D, has been already discussed. 
 
 Fundamental Property of Scalar Products, and Examples. 
 
 108. As all the preceding is involved in the definition of a 
 scalar product, and is obvious enough, it may be regarded 
 merely as illustrative. The reader might, in fact, say that ha 
 knew it all before in one form or another, trigonometrical or 
 geometrical, and that he did not see any particular advantage 
 in the way of stating it in the notation of scalar products. 
 But we now come to a very striking and remarkable property 
 
152 ELECTROMAGNETIC THEORY. CH. 111. 
 
 of scalar products, which will go far to justify them as working 
 utilities. 
 
 We know that in scalar algebra, when we have a product 
 xy t we may express x by the sum of any number of other 
 quantities, and similarly as regards y, and then obtain the 
 complete product xy by adding together all the component 
 products obtained by multiplying every element of x into every 
 element of y ; and that this process may be carried out in any 
 order. 
 
 Now, in vector algebra, we know already that there is a 
 partial similarity, viz., that we can decompose a vector A into 
 the sum of any number of others, and similarly as regards B. 
 The question now is whether the scalar product AB is the sum 
 of all the scalar products made up out of the components of A 
 paired with the components of B, taken in any order. For 
 example, if 
 
 and B = c + d, 
 
 is AB = a + bc + d ) 
 
 J * 
 
 The answer is Yes, and the process of manipulation is the 
 same as in scalar algebra, that is, as if all the vectors were 
 scalars. Moreover, this property admits of demonstration in 
 a sufficiently simple manner as to enable one to see its truth. 
 
 Start with the vector A, and first split it up into 
 
 A = a + b + c, ..... (18) 
 
 two or three vector components being sufficient for illustration. 
 Now project the vector A perpendicularly upon any axis, say 
 that of i. According to our definition of a scalar product, the 
 projection is Ai. ' Now it requires no formal demonstration, 
 but becomes evident as soon as the meaning of the proposition 
 is correctly conceived, that the projection of A upon any axis 
 is the same as the sum of the projections of its component 
 vectors, a, b, c, on that axis. That is, the latter projections 
 are either all positive or all negative, and fit together to make 
 up the projection of A ; or else some may be negative and 
 others positive, when there is overlapping and cancelling, but 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 153 
 
 till with the same result algebraically. This is expressed by 
 Ai = (a + b + c)i = ai + bi + ci, . . . (19) 
 
 ; got by multiplying (18) by i. 
 
 If we now multiply (19) by any scalar B, so that Bi=B, 
 which is any vector, since i may have any direction, we obtain 
 
 AB = (a + b + c)B = aB + bB + cB, . . (20) 
 
 the same as if we multiply (18) by B direct. 
 
 Similarly, we may split up B into the sum of any number of 
 other vectors, say, 
 
 (21) 
 
 If we substitute this in (20) we have 
 
 Now make use of the same reasoning which established (19) 
 and (20), applied to the bracketed vectors, and we establish 
 the property fully, with the result 
 
 = ad + ae + af + bd + be + bf + cd + ce + cf ; 
 
 the expansion being done formally as in scalar algebra in every 
 respect ; since the various terms may be written in any order, 
 and each may be reversed, thus, ad = da. 
 
 I have already remarked that a good deal of geometry may 
 be done by the addition property alone. The range of applica- 
 tion is greatly extended by the use of the scalar product and the 
 fundamental property (17). 
 
 To obtain the Cartesian form of AB, put the vectors in terms 
 of i, j, k; thus 
 
 AB = (A 1 i + A 2 j+A 3 k) (B^ + BJ + Bak). . (22) 
 
 Now effect the multiplications, remembering (13) and (15). 
 The result is 
 
 AB = A 1 B 1 + A 2 B 2 + A 3 B 3 . . . . (23) 
 
 For example, the activity of a force is the sum of the 
 activities of its component forces in any three coperpendicular 
 directions. Or, if A and B are unit vectors, we express the 
 cosine of the angle between them in terms of the products of 
 
154 ELECTROMAGNETIC THEORY. CH. 111- 
 
 the direction cosines of the vectors, each product referring to 
 one axis. In (23) we may, if we please, vectorise the six scalars 
 on the right side. 
 
 Since the square of a vector is the square of its tensor, we 
 may express the tensor at once in terms of the tensors and the 
 cosines of the angles between a series of vectors into which -the 
 original vector is resolved. Thus, for two, 
 
 (24) 
 
 (A-B) 2 = A 2 -2AB + B 2 ; . . . (25) 
 and similarly for three, 
 
 (26) 
 
 Here (24) and (25) apply to a parallelogram, and (26) to a 
 parallelepiped. In (24), (25), if A and B be the vector sides of 
 a parallelogram, then (A + B) and (A-B) are the vector 
 diagonals ; so (24) gives the length of one diagonal and (25} 
 that of the other. 
 
 Adding (24) and (25) we obtain 
 
 -B) 2 = 2(A 2 + B 2 ), . . (27) 
 
 expressing that the sum of the squares of the diagonals equals 
 the sum of the squares of the four sides. Similarly, by sub- 
 traction 
 
 (A + B) 2 -(A-B) 2 = 4AB, . . . (28) 
 
 expressing the difference of the squares of the diagonals as four 
 times the scalar product of two vector sides. 
 
 Equation (27) also shows that the sum of the squares of the 
 distances of the ends of any diameter of a sphere from a fixed 
 point is constant. For if A is the vector from the fixed point 
 to the sphere's centre, and B the vector from the centre to one 
 end of the diameter, then A + B and A-B are the vectors- 
 from the fixed point to the ends of the diameter. Whence, by 
 (27), the proposition. 
 
 There is an application of this in the kinetic theory of gases* 
 For when two elastic spheres collide they keep the sum of their 
 kinetic energies constant. If, then, their velocities before 
 collision be A + B and A-B, their velocities after collision are 
 indicated by vectors from the origin (in the velocity diagram) 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 155 
 
 to the ends of some other diameter of the sphere described 
 upon the line joining their original positions (in the velocity 
 diagram) as diameter ; which is actually the new diameter 
 depending upon the circumstances of impact. 
 
 Reciprocal of a Vector. 
 
 109. It is occasionally useful to employ the reciprocal of a 
 vector in elementary vector algebra. We define the reciprocal 
 of a vector a to be a vector having the same direction as a, 
 and whose tensor is the reciprocal of that of a. We may denote 
 the reciprocal of a by a -1 or I/a. Thus as a = aa p we have 
 
 (29) 
 
 Any unit vector is, therefore, its own reciprocal. 
 
 The reciprocal of a vector, being a vector, makes scalar pro- 
 ducts with other vectors. Thus ab- 1 or a/b means the scalar 
 product of a and b- 1 , and we therefore have 
 
 i-^-S-H-S- ' (30) 
 
 The tensor of a/b is (a/b) cos 6, where 6 is the angle between 
 a and b, or between their reciprocals, 'or between either and the 
 reciprocal of the other. 
 
 So a/a or aa' 1 or a' 1 a equals unity. 
 
 In using reciprocals the defined meaning should be at- 
 tended to, especially when put in the fractional form. Thus we 
 easily see that aV 1 = ab/6 2 , because the tensor of b is b 2 times 
 the tensor of b' 1 . But we cannot equivalently write a 2 /ab, 
 because this is (a/6)/cos 0, which is quite a different thing. 
 
 Notice that a" 1 b- 1 . is not the same as (ab)- 1 . The first is 
 a- 1 6' 1 cos 6, whilst the latter is or 1 ft^/cos 0. 
 
 Expression of any Vector as the Sum of Three Independent 
 
 Vectors. 
 
 110. We know that in the equation 
 
 i yd -i "^"j 
 
 where r is the vector distance of a point from the origin, the 
 scalars are the lengths of the projections of r upon the axes. 
 
156 ELECTROMAGNETIC THEORY. CH. III. 
 
 How should we, however, find them in terms of r algebraically ? 
 To find x we must operate on the equation in such a manner as 
 to cause the j and k terms to disappear. Now this we can do 
 by multiplying by i. For i is perpendicular to j and k, so that 
 multiplication by i gives 
 
 Similarly, rj = y and rk = z. 
 
 From this obvious case we can conclude what to do when the 
 reference vectors are not perpendicular ; for instance, in 
 
 ...... (31) 
 
 where a, b, c are any independent vectors. To find / we must 
 multiply by a vector perpendicular to b and c, say 1, so that 
 Ib = 0, and Ic = 0. Then 
 
 rl=/al, therefore /=rl/al 
 Similarly to isolate g and A, so that we get 
 
 '-i'OO ..... (32> 
 
 where 1, m, n are vectors normal to the three planes of b,c, c,a 
 and a,b. This exhibits explicitly the expansion of any vector 
 in terms of any three independent vectors a, b, c, as the three 
 edges of a skew parallelepiped. This ease, of course, reduces 
 to the preceding Cartesian case by taking a, b, c to be i, j, k, 
 when 1, m, n will also be i, j, k, or any scalar multiples of the 
 same. 
 
 Observe the peculiarity that auxiliary vectors are used, each 
 of which is perpendicular to two others, that is, to the plane 
 containing them. These auxiliary vectors bring us to the 
 study of the vector product. 
 
 The Vector Product of Two Vectors. Illustrations. 
 
 111. The auxiliary vectors just employed in the expansion of 
 a vector into the sum of three vectors having any independent 
 directions, are examples of vector products. Two vectors being 
 given, their vector product is perpendicular to both of them. 
 Of course, disregarding magnitude, there is but one such 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 
 
 157 
 
 vector, viz., the normal to the plane containing the given 
 vectors. Nor is the tensor of any consequence in the example 
 in question, for it will be observed that one of the vectors, 
 1, m, n, appears both in the numerator and in the denomi- 
 nator of one of the three fractions in equation (32), so that 
 the values of the fractions are independent of the tensors of 
 the auxiliary vectors. 
 
 But the vector product of two vectors has a strictly-fixed 
 tensor, depending upon those of the component vectors and 
 their inclination. There is particular advantage in taking the 
 tensor of the vector product of a and b to be ab sin 0. Thus 
 we define the vector product of two vectors a and b whose 
 tensors are a and &, and whose included angle is 0, to be a 
 
 b 
 
 FIG. 5. 
 
 
 FIG. 6. 
 
 third vector c whose tensor c equals ab sin 0, and whose direc- 
 tion is perpendicular to the plane of a and b ; the positive 
 direction of c being such that positive or right-handed rotation 
 about c carries the vector a to b. This vector product is 
 denoted by 
 
 c = Vab, ...... (33) 
 
 and its tensor may be denoted by V ab, so that we have 
 
 (34) 
 
 Similarly, V x ab may be used to denote the unit vector 
 parallel to Vab. 
 
 The only troublesome part is to correctly fix which way along 
 the perpendicular to the plane of a and b is to be considered 
 positive. Examples will serve to clinch the matter. Thus, let 
 
158 ELECTROMAGNETIC THEORY. CH. III. 
 
 a be towards the north, and b towards the east on the earth's 
 surface ; then c is straight downwards. Again, in Fig. 5, the 
 direction of c is downwards through the paper, and its tensor 
 is the area of the parallelogram upon a and b. 
 
 Let b be fixed, whilst a moves round so as to vary the angle 
 Q. Starting from coincidence, with 6 = 0, the tensor of c is zero. 
 It reaches a maximum (downwards) when is a right angle, 
 and falls to zero again when reaches two quadrants, and a is 
 in the same line with b, though discurrent. After this, in the 
 next two quadrants, the same numerical changes are gone 
 through ; but now, the sine being negative, c must be upwards 
 from the paper. 
 
 The unit reference vectors i, j, k are so arranged that when, 
 in (33), a is i and b is j, then c is k ; noting in Fig. 6, that k 
 is supposed to go downwards, or away from the reader. Thus, 
 in accordance with our definition we have 
 
 Vij = k, Vjk = i, Vki=j, . . (35) 
 
 because the mutual angles are quadrants and the tensors unity. 
 Observe the preservation of the cyclical order i, j, k in (35). 
 Also, we have 
 
 Vii = 0, Vjj = 0, Vkk = 0, . . (36) 
 
 because the vectors paired are coincident. 
 
 By the definition, the reversal of the order of the letters in 
 a vector product negatives it. Thus . 
 
 Vab= -Vba. ..... (37) 
 
 Since the tensor of Vab is ab sin and the scalar product 
 .ab is ab cos 0, we have 
 
 (38) 
 
 Combinations of Three Vectors. The Parallelepipedal 
 Property. 
 
 112. A vector product, being a vector, of course combines with 
 other vectors to make scalar and vector products again. Thus 
 cVab, where c is any new vector, means the scalar product of c 
 and Vab; and VcVab means the vector product of c and Vab. 
 These are both important combinations, which occur frequently, 
 and their interpretations and expansions will be given presently. 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 159 
 
 As illustrative of notation, it may be mentioned that cV ab, 
 where, as before explained, V ab is the tensor, is obviously a 
 vector parallel to c, but V ab times as long. On the other 
 hand, cVjab, where V x ab is the unit vector, is the same as 
 cVab/V ab; since by dividing by its tensor we unitise a 
 vector. Similarly as regards VcVjab. We may also have 
 V^Vab and V^V^b and V cVab, and various other modifica- 
 tions, whose meanings follow from the definition of a vector 
 product and its notation. 
 
 We do not often go further in practical vector algebra than 
 combinations in threes ; for instance, on to dVaVbc, the scalar 
 product of d and the previously explained VaVbc. 
 
 The scalar product cVab has an important geometrical illus- 
 tration. Its value is given by 
 
 cVab = V abxccos<, .... (39) 
 
 where < is the angle between c and Vab. This is by the defi- 
 nition of the scalar product, and of the tensor of a vector. 
 
 Now refer to Fig. 5 again. We know that V ab is the area 
 of the parallelogram. We also know that the volume of a 
 parallelepiped is the product of its base and altitude. Construct, 
 then, a parallelepiped whose three edges meeting at a corner 
 are a, b, and c. The area of one of its bases is V ab, and the 
 corresponding altitude is c cos <. Therefore, by (39), cVab is 
 the volume of the parallelepiped. 
 
 But there are two other bases and two other altitudes to 
 correspond, so there are two other ways of expressing the 
 'volume, giving the equalities 
 
 .... (40) 
 
 in which observe the preservation of cyclical order, done to 
 keep the sign the same throughout, as will be verified a little 
 later. 
 
 Semi-Cartesian Expansion of a Vector Product, and Proof 
 of the Fundamental Distributive Principle. 
 
 113. The semi-Cartesian expansion of c = Vab is 
 
 C = i(a 2 6 3 -a 3 6 2 )+j(a 3 6 1 -a 1 6 3 )+k(a 1 6 2 -a 2 6 l ), . (41) 
 in terms of i, j, k and the scalar components of a and b. 
 
160 ELECTROMAGNETIC THEORY. OH. Ill* 
 
 To prove this, multiply (41) by a and b in turns to form the 
 scalar products ac and be. Do this by the rule embodied in 
 equation (23). We get 
 
 ac = aj (a 2 6 3 
 be = 6j (a 2 6 ;i 
 
 But, by cancelling, all the terms on the right disappear, 
 That is, 
 
 ac = 0, be = 0. 
 
 From these we know that c is perpendicular to a and to b. 
 It is, therefore, Vab itself, or a multiple of the same. To find 
 its tensor, square (41). We get 
 
 c 2 = (aj), - 0362)8 + (a j h _ ai j 3 )2 + (ai&2 _ a2 j i)a . (42) 
 But this may, by common algebra, be transformed to 
 
 c 2 = fa* + a* + a 3 2 ) (V + Z> 2 2 + & 3 2 ) - (a^ + a 2 & 2 + 3 5 3 ) 2 . (43) 
 That is, c 2 = a 2 b 2 - (ab) 2 , ..... (44) 
 
 or, c 2 = a 2 6 2 - a 2 6 2 cos 2 6 = (ab sin 0) 2 . 
 
 The tensor c is, therefore, that of Vab itself, or else its 
 negative. Equation (44) is the same as (38), in a slightly 
 different form. Equation (41) is proved, except that it remains 
 to be seen whether the right member represents Vab or its 
 negative. 
 
 To do this, we may remark that the angle between two 
 planes is the same as the angle between the normals to the 
 planes, if the normals coincide when the planes do, so that the 
 three quantities in ( )'s in (41), which are the scalar pro- 
 jections of c on the axes, are also the areas of the projections 
 of the area V ab on the planes perpendicular to them. We 
 have therefore only to verify that the projection of any plane 
 on the plane normal to one of the axes is given correctly in 
 sign by (41). Take a rectangle, parallel to the j, k plane, 
 sides parallel to j and k, that is, 2 j and 6 3 k ; so that 
 
 Here 2 6 3 is the projection of the rectangle on a parallel plane, 
 and is correctly positive. 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 161 
 
 Now go back to (41). We have 
 
 Vab = i(a 2 6 3 -a 3 6 2 )-i- (45) 
 
 Similarly Vac = i (a 2 c 3 - a 3 c 2 ) + (46) 
 
 Add these together. We get 
 
 Vab + Vac = i[a 2 (6 3 + c 3 )-a 3 (6 2 + c 2 )] + (47) 
 
 But the right member, by definition of a vector product, or by 
 (41 ), means Va (b + c). That is, 
 
 Va(b + c) = Vab + Vac. .... (48) 
 
 Similarly Vd (b + c) = Vdb + Vdc (49) 
 
 So, by adding these again, 
 
 V (a + d) (b + c) = Vab + Vac + Vdb + Vdc, . (50) 
 
 a very remarkable and important formula. It shows that the 
 vector product of two vectors, A and B, equals the sum of all 
 the vector products which can be made up out of the com- 
 ponent vectors [(a + d) and (b + c) in (50)] into which we may 
 divide A and B, provided we keep the components of A always 
 before those of B. The necessity of this proviso of course 
 follows from the reversal of Vab with the order of a and b. 
 Although (50) only shows this for two components to each 
 primary vector, yet the process by which it was obtained evi- 
 dently applies to any number of components. 
 
 Subject to the limitation named, the formula (50) is pre- 
 cisely similar to (17) 108, relating to the scalar products. 
 Now the truth of (17) could be seen without much trouble. 
 A similar proof of (50), on the other hand, would not be at all 
 easy to follow, owing to the many changes of direction in- 
 volved in the vector products. This is why I have done it 
 through i, j, k, which are auxiliaries of the greatest value. 
 When in doubt and difficulty, fly to i, j, k. 
 
 On the other hand, although the establishment of (50) by 
 geometry without algebra is difficult, and there is preliminary 
 trouble in fixing the direction of a vector product, yet we see 
 from (50) that vector products are nearly as easily to be 
 manipulated algebraically as scalar products. 
 
162 ELECTROMAGNETIC THEORY. CH. III. 
 
 Examples relating to Vector Products. 
 
 114. By means of the formula (41) we can at once obtain 
 the cartesian expansion of the parallelepipedal cVab. Use 
 equation (23), applied to (41), remembering that c is now any 
 vector, and we get 
 
 cVab = ^(0363 - a 3 6 2 ) + c^a^ - a^) + c^a^ - aj)^ . (51) 
 
 By rearrangement of terms on the right side of this, putting 
 the a's outside the brackets, we obtain aVbc, and putting the 
 b s outside, we obtain bVca, and thus verify (40) again. 
 
 The vector VcVab is, perhaps, most simply expanded through 
 i, j, k. First write d for Vab, then by (41) we have 
 
 Vcd = icd 
 
 Next put for d v d 2 , d 3 their values in terms of a's and b's, 
 given in (41), and we have 
 
 Next, add and subtract a^c^ and we get 
 
 VcVab = ifc^bc - ^ac] + . . . . 
 Lastly, reconvert fully to vectors, and we have 
 
 VcVab = a.bc-b.ca ..... (52) 
 
 2 *^ "i. ^ % I 2* 
 
 This important formula should be remembered. That VcVab 
 is in the plane of a and b is evident beforehand, because Vab 
 is perpendicular to this plane, and multiplying by Vc sends it 
 back into the plane. It is, therefore, expressible in terms of a 
 and b, as in (52), which shows the magnitude of the com- 
 ponents. 
 
 In some more complex formulae it is sufficient to remember 
 the principle upon which they are founded, as by its aid they 
 can be recovered at any time. Thus in the case of 
 
 r=/a+#b + Ac, ..... (53) 
 
 already treated, equation (31), if we use now Vbc for 1, which 
 was, before, any multiple of it, and similarly for m and n, we 
 
 have r 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 163 
 
 There is no occasion to put a formula like this in the 
 memory, because it is so simply obtained at any time from 
 (53) by multiplying by the auxiliary vectors so as to isolate /, 
 <7, h in turn and give their values. 
 
 Notice that the three denominators in (54) are equal. Also 
 that by exchanging a and Vbc, b and Vca, c and Vab, we 
 obtain 
 
 r- " Vbc+ * Vca+--Vab, . (55) 
 aVbc bVca cVab 
 
 and this is also true, as we may at once prove by multiplying 
 it in turns by a, b, and c, each of which operations nullifies two 
 terms on the right. (55) is the expression of r in terms of 
 three vectors, which are normal to the three planes of any 
 three independent vectors, a, b, c, taken in pairs. Here, again, 
 such a formula can be immediately recovered if wanted by 
 attending to the principle concerned. 
 
 To obtain the cartesian expansion of any formula containing 
 vectors is usually a quite mechanical operation. The cartesian, 
 or semi-cartesian, representatives of a few fundamental functions 
 being remembered, their substitution in the vector formula is all 
 that is required. If the formula be scalar (although involving 
 as many vectors as we please) the result is a single scalar for- 
 mula in cartesians. But if it be a vector formula, it reduces 
 to a semi-cartesian vector formula involving i, j, k, giving 
 three scalar equations, one for each component. 
 
 The converse process, to put a scalar cartesian investigation 
 into vectorial form, is less easy, though dependent upon the 
 same principles. Here the three component equations have to 
 be reduced to one vector equation. It is very good practice to 
 take a symmetrically written-out cartesian investigation and 
 go through it, boiling it down to a vector investigation, using 
 the unit reference vectors i, j, k, whenever found to be con- 
 venient. 
 
 The Differentiation of Scalars and Vectors. 
 
 115. In the preceding account of vector addition, and of the 
 
 scalar product and the vector product, the reader has nearly all 
 
 that is needed for general purposes in geometry and in the usual 
 
 physical mathematics involving vectors, so far as the algebra 
 
 M 2 
 
164 ELECTROMAGNETIC THEORY. CH. III. 
 
 itself is concerned. For the addition of differentiations does 
 not usually introduce anything new into the algebra. Thus, in 
 the analysis of varying vectors, we have the same vector 
 algebra with new vectors introduced, these being derived from 
 others by the process of differentiation. 
 
 The ideas concerned in the differentiation of a vector with 
 respect to a scalar are essentially the same as in the differentia- 
 tion of a scalar. Thus, u being a scalar function of x, i.e., a 
 quantity whose value depends on that of x, we know that if 
 AM is the increment in u corresponding to the increment A# 
 in x, then the ratio Aw/A# usually tends to a definite limiting 
 value when A# is infinitely reduced, which limit, denoted by 
 du/dxy is called the differential coefficient of u with respect to x. 
 This I should be strongly tempted to call the " different iant " 
 of u to x, were I not informed on the highest authority that 
 the expression " differentiant " for "differential coefficient" is 
 objectionable. But the differentiating operator djdx which 
 acts on the operand u may be termed the " differentiator," as 
 has perhaps been done by Sylvester and others. This way of 
 regarding a differential coefficient, splitting du/dx into (d/dx) 
 and u, sometimes leads to great saving of labour, though we are 
 not concerned with it immediately. 
 
 The differential coefficient is thus strictly the rate of increase 
 of the function with the variable, or the increase of the func- 
 tion per unit increase of the variable, .on the tacit assumption 
 that the rate of increase of the function keeps the same value 
 throughout the whole unit increment in the variable as it has 
 for the particular value of the variable concerned. This plan, 
 referring to unit increment, is often very useful. The reservation 
 involved, though it must be understood, need not be mentioned. 
 
 There is also the method of differentials, which is of some 
 importance in vector analysis, as it can be employed when 
 differential coefficients do not exist. Thus, u being a function 
 of x, whose differential is dx, the corresponding differential of 
 
 du=f(x + dx)-f(x); . .'.. . (56) 
 which, by expanding/ (x + dx), reduces to 
 
 dudx, (57) 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 165 
 
 provided we neglect the squares and higher powers of dx, that 
 is, regard dx as infinitesimal, and therefore du also. And if 
 there be two variables, as in u=f(x, y), then we have 
 
 f(x,y), . . (58) 
 
 leading to du = dx + dy, ..... (59) 
 
 dx dy 
 
 expressing the differential of u in terms of those of x and y. 
 
 Now let the operand be a vector function of x t say, E. For 
 instance, x may mean distance measured along a straight line 
 or axis in an electrostatic field, where E, the electric force, will 
 usually change as we pass along the line. It may change con- 
 tinuously in direction as well as in magnitude ; in any case, the 
 change itself in E between two points on the line is a vector, 
 being the vector AE which must be added to the E at the first 
 point to produce that at the second point. Dividing by Ax, the 
 increment in x, we get the vector AE/A# ; and this, when the 
 increments are taken smaller and smaller, approximates towards 
 a definite limiting vector denoted by dE/dx, which is, by the 
 manner of its construction, the rate of increase of E with x, or 
 the increase in E per unit increase in x. 
 
 We have also, in differentials, 
 
 -f(x), .... (60) 
 
 leading to dE=^dx, ...... (61) 
 
 dx 
 
 understanding that the differentials dx and dH are infinitesimal. 
 Similarly, the rate of increase of the vector d'E/dx with x is 
 the second differential coefficient d 2 E/dx 2 ; and so on. 
 
 Semi-Cartesian Differentiation. Examples of Differentiating 
 Functions of Vectors. 
 
 116. In semi-Cartesian form we have 
 
 E = iE 1 +jE 2 + kE 3 ..... (62) 
 
 Here the scalars E I} E 2 , E 3 are functions of x, whilst the 
 reference vectors i, j, k are constants ; so we have 
 
 ^? = i^l+j^2 + k ^3 j . (63) 
 
 dx dx dx dx 
 
 showing the components of the vector dEjdx. 
 
166 ELECTROMAGNETIC THEORY. CH. III. 
 
 Snarly, i +J . . . (64) 
 
 dx* ax 2 ax 2 ax* 
 
 shows the components of the second differential coefficient, 
 obtained by differentiating (63). 
 
 Now, this semi-Cartesian process is quite general. Any vector 
 function may be expressed in the form (62), and when the right 
 member is differentiated, the differentiations are performed 
 upon scalar functions. The same remark applies to (63) and 
 (64), &c. ; so we see that the rules for differentiating vectors, 
 and functions of vectors, with respect to scalar variables, 
 are the same as those for differentiating similar functions of 
 scalars, so far as the independent action of a differentiator on 
 the separate members of a product goes. 
 
 Thus, in differentiating a scalar product, AB, with respect to 
 a variable scalar, t, we have, by (23), 
 
 = (A! B! + A 2 B 2 + A 3 B 3 ) + (A l B x + A 2 B 2 + A 3 B 3 ) ; 
 or, re-transforming to vectors, 
 
 (65) 
 just as if A and B were scalars. Similarly, we have 
 
 .... (66) 
 
 as we may see immediately by differentiating the semi-Cartesian 
 expansion of VAB, equation (41). We have also 
 
 ^AVBC = AVBC + AVBC + AVBC, . . . (67) 
 
 and A VAVBC = VAVBC + VAVBC + VAVBC. . . (68) 
 
 But, although we proceed formally, as in the ordinary 
 differentiation of scalars, as regards the properties peculiar to 
 differentiation, yet we must always remember those which are 
 peculiar to the vector algebra. For example, we must not 
 only on the left sides, but also on the right sides of (67) and 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 167 
 
 (68), remember the reversal of the sign of a vector product with 
 the order of the letters. 
 
 Independently of i, j, k, we may proceed thus, using differen- 
 tials : 
 
 d (AB) = (A + cf A) (B + dB) - AB = AdB + Bo?A, . (69) 
 
 omitting the infinitesimal dAdE of the second order. Dividing 
 by dt, and proceeding to the limit, we obtain (65). 
 Similarly we have 
 
 d (VAB) = V(A + dA.) (B + dB) - VAB = V(</A)B + VAdB, . (70) 
 
 omitting the second differential WAc?B. So, dividing by dt, 
 we obtain (66). 
 
 In getting (69) we have, of course, used the distributive 
 law of scalar products, equation (17) ; and, in getting (70), the 
 similar law of vector products, equation (50). 
 
 Motion along a Curve in Space. Tangency and Curvature ; 
 Velocity and Acceleration. 
 
 117. Some spacial and motional examples maybe here use- 
 fully inserted to illustrate the differentiation of vectors. Let 
 
 FIG. 7. 
 
 there be any curve in space, and let s be length measured along 
 it from some point on the curve. Also let r be the vector from 
 any fixed origin to a point P on the curve. We have then 
 r =/(s). With the form of the function we have no concern. 
 
 Now, consider the meaning of the first differential coefficient 
 dijds. First, if the increment As is finite, say the arc PQ, the 
 
168 ELECTROMAGNETIC THEORY. CH. III. 
 
 corresponding increment in Ar is the vector chord PQ, the new 
 vector of the curve being OQ, that is r + Ar, instead of OP orr. 
 We see that Ar/As is a vector parallel to the chord, whose tensor 
 is the ratio of the chord to the arc. Now reduce the incre- 
 ments to nothing. In the limit the direction of dr/ds is that 
 of the tangent to the curve at P, and its tensor is unity, since 
 the chord and the arc tend to equality. That is, 
 
 J-I-irff + j^ + k*, . . . .(71) 
 as as as as 
 
 where T is the unit vector tangent, and the semi-Cartesian 
 expression (got by expanding r) is also given. In the figure the 
 size of T is quite arbitrary, since the unit of length is arbitrary. 
 
 Next, differentiate again with respect to s to obtain the 
 second differential coefficient d 2 T/ds 2 or dT/ds. It is the change 
 in the unit tangent per unit step along the curve. Now, as the 
 unit tangent cannot change its length, it can only change its 
 direction ; and, moreover, this change is a vector perpendicular 
 to the tangent. It and the two tangents, initial and final, are 
 in some plane, usually termed the osculating plane. It is the 
 plane of the curve for the time being, unless, it be a plane 
 curve, when it is the plane of the whole curve. Thus dT/ds, 
 being at right angles to T, points from the curve towards the 
 centre of curvature for the time being. Moreover, its tensor 
 measures the curvature. For the usual measure is d6/ds t 
 where dO is the angle between the tangents at the ends of ds, 
 and it may be readily seen that this is the tensor of dT/ds. 
 
 Now the reciprocal of the curvature is the radius of curva- 
 ture. If then R be the vector from the curve to the centre of 
 curvature C (Fig. 7), we have 
 
 & -5 3 
 
 and the vector from the origin to the centre of curvature is 
 r + R. 
 
 Now, referring to the same figure, let a point move along the 
 curved path. Consider its velocity and acceleration, taking t 
 the time for independent variable. We have 
 
 * = **-t,T = v,. (73) 
 
 dt ds dt l ; 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 169 
 
 by using (71), and denoting the velocity by v. Its tensor is 
 v or ds/dt. The interpretation of (73) is sufficiently obvious, 
 since it says that the motion is (momentarily) along the 
 tangent, at speed v. 
 
 Differentiate again to t. We get 
 
 . (74) 
 
 dt 2 dt dt 
 
 That is, the vector rate of acceleration is exhibited as the sum 
 of two vectors, the first being the tangential component, whose 
 tensor equals the rate of acceleration of speed, whilst the second 
 we may expand thus, by introducing the intermediate vari- 
 able s : 
 
 &-,*_* ..... (75) 
 dt ds dt R 
 
 where the third form is got by using (72). This shows the 
 rate of acceleration perpendicular to the motion. It is towards 
 the centre of curvature, of amount -y 2 /R, the well known 
 result. 
 
 Tortuosity of a Curve, and Various Forms of Expansion. 
 
 118. Again referring to Fig. 7, if the curve be not a plane 
 curve, or be a tortuous curve, the osculating plane undergoes 
 change as we pass along the curve. It turns round the tan- 
 gent, and the measure of the tortuosity is the amount of turn- 
 ing per unit step along the curve. It is d<f>/ds, if d<J> is the 
 angle between the two osculating planes at the extremities of 
 ds. It is equivalently denoted by the tensor of dN/ds, if N be 
 the unit normal to the osculating plane. This is similar to 
 the equivalence of dd/ds and the tensor of dT/ds in measuring 
 the curvature before noticed, the equivalent plane in that case 
 being the plane normal to T. 
 
 The three vectors, T, N, and R lf form a unit rectangular 
 system, for we have 
 
 N = VTR 1 , ...... (76) 
 
 since T and R x are perpendicular unit vectors. As we move 
 along the curve this system of axes moves as a rigid body, since 
 there is no relative change. They may, in fact, be imagined 
 to be three perpendicular axes fixed in a rigid body moving 
 
170 ELECTROMAGNETIC THEORY. CH. III. 
 
 along the curve in such a way that these axes keep always 
 coincident with the T, R x , and N of the curve. As it moves, 
 the body rotates. It rotates about the N axis only when the 
 curve is plane, but about the T axis as well when tortuous. 
 The vector change (infinitesimal) in T is perpendicular to it, in 
 the osculating plane that is, the plane whose normal is N, 
 and is directed towards the centre of curvature. And the 
 vector change (infinitesimal) in N is perpendicular to it, also in 
 the osculating plane, and directed towards the centre of curva- 
 ture, the vector R from the curve to the centre of curvature 
 being on the intersection of the two planes mentioned. 
 
 Various expressions for the tortuosity may be obtained. Let 
 it be Y the tensor of Y, given by 
 
 ' -< 77) 
 
 Then, by (76) and the context, (77) gives one expression. A 
 second is got by performing the differentiation, giving 
 
 i . . . (78) 
 ds ds 
 
 But the first term on the right is zero, by (72), the two vectors 
 after V being parallel. So 
 
 Y = VT^i ..... (79) 
 gives another expression. Or 
 
 Y = V T^, ..... (80) 
 
 considering the tensor only. But, since Y is parallel to R u we 
 have YRj = Y ; therefore, by (79), 
 
 - 1 . . (81) 
 
 as 
 
 is a third and entirely different expression. That Y is parallel 
 to R! we may prove by (77) and (79). Multiply (77) by N. 
 
 Then YN = N^ = j4N 2 =0, . (82) 
 
 ds ds 
 
 since N 2 = l ....... (83) 
 
 This shows Y is perpendicular to N. It is also, by (79), per- 
 pendicular to T. So its direction is that of R r 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 171 
 
 By the parallelepipedal property (40) we may also write (81) 
 thus, 
 
 (84) 
 
 as 
 Here again, we may write 
 
 R, dE, I dR, , ^ d 1 /Q ~ X 
 
 L = -- = -- + K - .... (oo) 
 
 ds ds-R R ds dsR 
 
 Substituting this compound vector in (84), we see that the 
 first of the resulting vector products vanishes because RJ and 
 R" 1 are parallel. There is then left 
 
 -^-L ..... (8G) 
 ds R 
 
 f, . . . (37) 
 s as* 
 
 by (72), or, in terms of r and derivatives, 
 
 which is a known symmetrical form, but whose meaning is less 
 easy to understand than several preceding expressions. 
 
 Since, by the definition of a vector product we have V^b 
 = 6 when B,^ and b are perpendicular, a x being a unit and b any 
 vector, we may, from the first equation (77) conclude that 
 
 , ..... (89) 
 as 
 
 which is equivalent to a form given by Thomson and Tait. 
 
 Similarly, Y = V T^, ..... (90) 
 
 as 
 
 Y being perpendicular to T as well as to N. 
 
 The reader may, perhaps, find the above relating to tortuosity 
 hard to follow, though the previous matter relating to tangency, 
 curvature, velocity, and acceleration may be sufficiently plain. 
 The hardness lies in the intrinsic nature of tortuosity, as to 
 which see works on analytical geometry. But the reader who 
 wishes to get a sound working knowledge of vectors should go 
 through the ordinary Cartesian investigations, and turn into 
 
172 ELECTROMAGNETIC THEORY. CH. III. 
 
 vectors. Also, convert the above to Cartesian form through 
 i, j, k, and verify all the results, or correct them, as the case 
 may be. Space is too short here for much detail. 
 
 Hamilton's Finite Differentials Inconvenient and Unnecessary. 
 
 119. It is now desirable to say a few words regarding the 
 method of treating vector differentiation employed by Hamilton, 
 and followed by Tait. The latter speaks in his treatise (chapter 
 L, 33, third edition) of the novel difficulties that arise in 
 quaternion differentiation ; and remarks that it is a striking 
 circumstance, when we consider the way in which Newton's 
 original methods in the differential calculus have been decried, 
 to find that Hamilton was obliged to employ them, and not the 
 more modern forms, in order to overcome the characteristic 
 difficulties of quaternion differentiation. (For the word 
 "quaternion," we may read vector or vectorial here, because a 
 vector is considered by Hamilton and Tait to be a quaternion, 
 or is often counted as one. This practice is sometimes confusing. 
 Thus the important physical operator y is called a quaternion 
 operator. It is really a vector. It is as unfair to call a vector 
 a quaternion as to call a man a quadruped ; although, four 
 including two, the quadruped might be held (in the matter of 
 legs) to include the biped, or, indeed, the triped, which 
 would be more analogous to a vector. It is also often 
 inconvenient that the name of the science, viz., Quater- 
 nions, should be a mere repetition of the name of the 
 operator. There is some gain in clearness by preserving the 
 name " quaternion " for the real quaternion the quadruped, 
 that is to say.) The matter is illustrated by the motion of a 
 point along a curve. But it does not appear, from this 
 illustration, where the obligation to depart froni common usage 
 comes in. 
 
 The subject is, however, returned to more generally in 
 Chapter IV., on Differentiation, where we are informed that 
 we " require to employ a definition of a differential somewhat 
 different from the ordinary one, but coinciding with it when 
 applied to functions of mere scalar variables." Here again, 
 however, a most searching examination fails to show me the 
 necessity of the requirement ; or, at least, that the necessity is 
 demonstrated. 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 173 
 
 To examine this matter, consider for simplicity a function 
 u of a single scalar variable a and a single vector variable r, 
 say 
 
 =/(r,). 
 
 Following the common method of infinitesimal differentials, we 
 have 
 
 du=f(r + dr,a + da)-f(r,a). ... (91) 
 
 That is, on giving the infinitesimal increments dx and da to r 
 and a we produce the infinitesimal increment du in the function. 
 Here du, di and da are the differentials, and it is important to 
 remark, in connection with the following, that they are infini- 
 tesimal, and that du is really the increment corresponding to 
 dr and da. 
 
 Now, Hamiltonian differentials have a different meaning. 
 Though they would be denoted by du, di and da, yet, being 
 different, we shall here dignify them with brackets, and denote 
 them by (du), (di), and (da), to distinguish them from the 
 differentials in (91). The Hamiltonian differential (du) is then 
 defined by 
 
 (du) - [/(r + (>, + to) -/(r, )], . (92) 
 
 on the understanding that n is infinity, and that (on which 
 stress is laid) the differentials (di) and (da) are finite and 
 perfectly arbitrary. 
 
 That this process is circuitous is obvious, but is it necessary ? 
 Let us examine into its meaning, and compare (92) with (91). 
 First, divide (92) by n. We then have 
 
 + (>-/(,, a). . . (93) 
 
 Now, we see at a glance, that (93) and (91) become iden- 
 tical if 
 
 *>-&. *-(*) *,-<*> (94) 
 
 n n n 
 
 The division of the finite differentials (da) and (di) by n pro- 
 duces infinitesimal results, and the finite differential (du) is 
 similarly reduced to an infinitesimal. Why then not employ 
 the infinitesimal differentials at once, and avoid the circuitous 
 ness? 
 
174 ELECTROMAGNETIC THEORY. OH. HI. 
 
 When we proceed to form a differential coefficient, as du/dt, 
 if r and a are functions of t, or du/da, if r is a function of a, 
 we see that their values are equivalently expressed by (du)/(dt) 
 and (du)l(da), using the finite differentials; because this merely 
 says that the value of the fraction du/dt is unaffected when 
 the numerator and denominator are each multiplied by the same 
 quantity. 
 
 In the above, the reasoning is the same, whether u is a scalar 
 or a vector function of r and a. 
 
 But whilst the differential coefficient comes out the same, yet 
 the differential (du) is not the increment in u belonging to the 
 finite increments (di) and (da\ nor is it anything like it, excep- 
 tions excepted. 
 
 The above is as fair and as clear a statement as I can make 
 of the difference between common and finite differentials as 
 applied to vectors ; and we see that the Hamiltonian plan is 
 only a roundabout and rather confusing way of expressing what 
 is done instantly with infinitesimal differentials. But then 
 the Hamiltonian plan is said to be obligatory. There's the 
 rub ! I can only give my own personal experience. After 
 muddling my way somehow through the lamentable quater- 
 nionic Chapter II., the third chapter was tolerably easy ; but 
 at the fourth I stuck again, on this very matter of the obliga- 
 tory nature of the finite differentials. I am no wiser now about 
 it than I was then. Perhaps there is some very profound and 
 occult mystery involved which cannot be revealed to the vulgar 
 in a treatise ; Hamilton is a name to conjure by. Or perhaps 
 I " began the study of quaternions too soon !" Or perhaps it 
 is only a fad, after all. Anyway, I am willing to learn, and if 
 I live long enough, may reach a suitable age. At present let 
 us return to common infinitesimal differentials. 
 
 Determination of Possibility of Existence of Differential 
 Coefficients. 
 
 120. Although we may employ the method of differentials 
 in any equation involving vectors, using the symbol d, and 
 although we may turn it into an equation of differential coeffi- 
 cients by changing d to the scalar differentiator d/dt (for 
 example), as in 116, yet we cannot usually turn differentials 
 into differential coefficients when the variable is a vector. 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 175 
 
 This is not because we cannot form the expressions du\dt or 
 dH/di, but because they are not differential coefficients when 
 the denominators are vector differentials. This will be easily 
 seen from typical examples. (The failure has nothing to do 
 with the previous question of finiteness or otherwise of 
 differentials.) Let 
 
 w = ab ........ (95) 
 
 and let the vector b vary/ Then, 
 
 ...... (96) 
 
 and, therefore, = (dT>)- l (B,db) = (db) l [a,(db) 1 l . . (97) 
 db 
 
 where (db) l means the unit db. Here the imitation differential 
 coefficient du/db is a fraud, because its expression on the right 
 side of (97) is not clear of the differential db, whose direction 
 it involves. Nor is equation (97) of any particular use, com- 
 pared with (96). 
 
 Again, in the vector equation 
 
 R = Vab ...... (98) 
 
 let b be variable. Here we have 
 
 ..... (99) 
 
 and from this equation of differentials we may form dH/db and 
 VdR/db ; but neither of them, nor any combination of them, is 
 a differential coefficient clear of differentials. Thus, 
 
 . . (100) 
 contains the differential on the right side. And 
 
 (101) 
 
 also contains the differential. As a special result the value of 
 (100) is zero (a parallelepiped in a plane) : and by equation 
 (52) we may develope (101) to 
 
 (rfb^a^b^-a ..... (102) 
 
 But it is (99) that is the really useful fundamental equation, 
 the developments (100) to (102) being useless for ditferentiating 
 purposes, although true and interpretable geometrically, 
 
176 ELECTROMAGNETIC THEORY. CH. III. 
 
 Generally, if 
 
 .... (103) 
 is a function of 
 
 (104) 
 
 then Rj, R 2 , R 3 are functions, usually different, of x, y and 2, 
 and the ratio of dR to di (even in the most general quater- 
 nionic sense), cannot possibly be freed of the differentials dx, 
 dy, dz. 
 
 Even when we reduce the generality of the problem by 
 making R and r coplanar, say in the plane of i, j, with z = 
 and R 3 = 0, dR/dr is not usually a definite differential co- 
 efficient. It only becomes one by special relations between 
 the differential coefficients of Rj and R 2 with respect to x and y, 
 viz: 
 
 =__^52. . . (105) 
 
 ^ __ 
 
 dx dy ' dy dx 
 
 This is the vectorial basis of the large and important branch 
 of modern mathematics called the Theory of Functions, which 
 appears in electric and magnetic bidimensional problems. 
 
 Variation of the Size and Ort of a Vector. 
 
 121. Sometimes it is convenient to have the variation of a 
 vector exhibited in terms of the variations of its tensor and 
 unit vector, or its size and ort. Thus, from 
 
 we produce, by differentiating, 
 
 dR = EdE l +H l dU ..... (106) 
 If there be no variation of direction, then 
 
 ^R = R^R; ..... (107) 
 
 and if the tensor does not vary, whilst the direction does, then 
 dH = ~RdR l ..... . (108) 
 
 Dividing (107) by the scalar c?R, we get a proper differential 
 coefficient 
 
 = E ...... < 109 > 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 177 
 
 provided dR 1 = 0. From (108), 
 
 provided dE = 0, which makes c?R perpendicular to E. But it 
 is as well to avoid forms of this kind (with a vector variation in 
 the denominator), even when by some limitation, as above, we 
 can get rid of the differential. Of course, in (106), without 
 any limitation, we can change d to d/dt (or other scalar diffe- 
 rentiator) ; thus, 
 
 R = RR 1 + R 1 R ...... (Ill) 
 
 It is this (scalar) kind of differentiation that nearly always 
 occurs in physical applications, and there is no abstrusity about 
 it. See 116. Nor is there about differentials, provided we 
 do not attempt to make differential coefficients with respect to 
 vectors, or make the differentials themselves finite. 
 
 We have also R 2 = R 2 , 
 
 which, on differentiation, gives 
 
 RrfR = RdR; ..... (112) 
 or, dividing by R, 
 
 ..... (113) 
 
 which says that the increment in the tensor is the component 
 along the vector of its vector change dR, whose full expression is 
 given by (106). Using (106) in (113) we get 
 
 = ....... (114) 
 
 Using (113) in (106) we get 
 
 JR = RdK 1 + R 1 (R 1 ^),. . . . (115) 
 or, dividing by R, 
 
 ^ = ( /R 1 + ^5, ..... (116) 
 
 it xx. 
 
 all of which results may be geometrically interpreted. 
 
 In differentiating the reciprocal of a vector, proceed thus : 
 
 ,< m) 
 
 where, for d may be written djdt, <kc. 
 
178 ELECTROMAGNETIC THEORY. OH. III. 
 
 Another way is 
 
 It may easily be shown, by (106), that these results are 
 equivalent. 
 
 Enough has now been said about the meaning and effect of 
 differentiating vectors. The important vector differentiator V 
 deserves and demands a separate treatment. 
 
 Preliminary on V- Axial Differentiation. Differentiation 
 referred to Moving Matter. 
 
 122. The Hamiltonian vector V occurs in all physical 
 mathematics involving three dimensions, when treated vec- 
 torially. It is denned by 
 
 V = i-^+j/+k^, .... (119) 
 dx dy dz 
 
 the second form being merely an equivalent one, often more 
 convenient, and easier to set up. We see that the Hamiltonian 
 is a fictitious vector, inasmuch as the tensors of its components 
 are not magnitudes, but are differentiators. As, however, 
 these differentiators are scalar not scalar magnitudes, but 
 scalar operators, having nothing vectorial about them the 
 Hamiltonian, in virtue of i, j, k, behaves just like any other vec- 
 tor, provided its differentiating functions are simultaneously 
 attended to. Of course, an operand is always implied, which 
 may be either scalar or vector. 
 
 The Hamiltonian has been called Nabla, from its alleged 
 resemblance to an Assyrian harp presumably, only the frame 
 thereof is meant, without any means of evoking melody. On 
 the other hand, that better known musical instrument, the 
 Triangle, perhaps equally well resembles V> and- it does not 
 want strings to play upon. 
 
 First notice that 
 
 iV = Vi, JV=-V 2 , kV = V 8 , - . (120) 
 and that these are all special cases of the scalar product 
 
 3 , . (121) 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 179 
 
 where N is any unit vector. Here NV or its equivalent 
 Cartesian expansion, means differentiation with respect to 
 length, say s, measured along the axis of N, usually denoted 
 by 
 
 ..... 
 
 ds dx dy dz 
 
 or by similar forms. This operation was termed by Maxwell 
 differentiation with respect to an axis, which we may more 
 conveniently call axial differentiation. We see, therefore, that 
 N Vj or the N component (scalar) of V> is an axial differentiator, 
 the axis being that of N. Thus, 
 
 = , . . (123) 
 ds ds 
 
 P being scalar and A vector. Also 
 
 NV.AB=A.NV.B+B.NV.A, . . (124) 
 NV.VAB=V(NV.A)B+VA.NV.B, . (125) 
 
 and so on, simply because N V is a differentiator. Expanded in 
 Cartesians, (124) contains the eighteen differential coefficients 
 of the components of A and B with respect to x, y^ and z. The 
 last two equations will also serve to illustrate notation. The 
 dots are merely put in to act as separators, and keep the proper 
 symbols connected. Mere blank spacing would do, but would 
 be troublesome to work. We may, however, use brackets 
 equivalently. 
 
 Thus, A.NV.B may be written A(NV)B, and this form may 
 be used with advantage if found easier to read. Of course, N V 
 being scalar, on its insertion between the two members of AB, 
 the result remains a scalar product. Similarly, VA.NV.B, 
 arising from VAB, is still a vector product. 
 
 A somewhat more general operator is vVj where v is any 
 vector. It is plainly v times the corresponding axial differen- 
 tiator, or / y.v 1 V- Thus, expanded, 
 
 vV-D = KVi + v a V 2 + %V 3 ) (iDi + JD 2 + kD 8 ), . (126) 
 
 which we may separate into i, j, k terms if we like. 
 
 When v is the velocity of moving matter, vV comes frequently 
 into use. Let, for example, w denote the measure of some 
 
 N2 
 
180 ELECTROMAGNETIC THEORY. CH. III. 
 
 property of the moving matter (here a scalar, but it may equally 
 well be a vector), and, therefore, a function of position and of 
 time. Its rate of change with the time at a fixed point 
 in space is dw/dt, and the matter to which this refers is changing. 
 But if we wish to know the rate of time-change of w for the 
 same portion of matter, we must go thus. Let Swfit denote 
 the result, then 
 
 Sw __ dw dw dx dw dy dw dz 
 ~St~di ~dxdi ~dydi ~ 
 
 by elementary calculus. Or 
 
 where v is the velocity of the matter. 
 
 This is the equation extensively employed in hydrokinetics. 
 In elastic solid theory the term vV- w is generally omitted, 
 being often small compared with the first term on the right. 
 But, in special applications, it may happen that any one of the 
 three terms in (127) vanishes. Thus the term on the left side 
 obviously vanishes when w keeps the same in the same matter. 
 If, however, for w we substitute a vector function, say A, even 
 though it may not suffer change at first sight in the same 
 matter, yet the rotational part of the motion will alter it by 
 turning it round, so that SA/St will not vanish. 
 
 The operator 8/8t is distributive, like dfdt and vV> a s 
 exemplified in (124), (125). 
 
 Motion of a Rigid Body. Resolution of a Spin into other 
 
 Spins. 
 
 123. If the moving matter is so connected that it moves 
 as a rigid body, we have a further development. If we set a 
 rigid body spinning about a fixed axis whose direction is defined 
 by the unit vector a 1} with angular speed a, the speed of any 
 point P in the body equals the product of its perpendicular 
 distance from the axis into the angular speed, and the direction 
 of P's motion is perpendicular to the plane through the axis 
 containing P. That is, a given particle describes a circle round 
 the axis. This is obviously necessitated by the rigid con- 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 181 
 
 nection of the parts of the body (or detached bodies). The 
 velocity of P is therefore expressed by 
 
 v = Var, ..... (128) 
 
 if a = aa 1 , and r is the vector to P from any point on the 
 axis. This is by the definition of a vector product. 
 
 Here we may observe a striking advantage possessed by the 
 vectorial method. For (128), obtained by elementary consider- 
 ations, proves without any more ado that angular velocities 
 about different axes compound like displacements, translational 
 velocities, or forces, or, iu short, like vectors. Thus, in (128) 
 we see that the velocity of P is the vector product of a and r. 
 The latter fixes the position of P. The former depends on the 
 spin, its tensor being the angular speed, and its direction that 
 of the axis of spin. Now every vector, as before seen, may be 
 expressed as the sum of others, according to the rule of vector 
 addition. But every vector has its species. Thus, r being a 
 vector distance, its components are vector distances. Similarly, 
 a being a. vector axis of spin with tensor equal to the angular 
 speed, the components of a, for instance, the three terms on 
 the right of 
 
 are vectors of the same nature. Therefore any spin may be 
 replaced by other spins about other axes, according to the 
 vector law. Of course the body cannot spin about more than 
 one axis at once, but its motion is the same as if it could and 
 did. This property is notoriously difficult to understand by 
 Cartesian mathematics. 
 
 The most general kind of motion the rigid body can possess 
 is obtained by imposing upon the rotational any translational 
 velocity common to all parts. Let q be this translational 
 velocity, then, instead of (128), 
 
 v = q + Var ...... (129) 
 
 Here observe that q is the velocity of the point 0, or of any 
 other point on the axis of rotation passing through the point 0. 
 We may infer from this that if we shift the origin to Q, a 
 point on a parallel axis, equation (129) will still be true, pro- 
 vided r means the vector from the new origin Q to P, and q 
 means the translational velocity of Q itself. Of course, the 
 
182 ELECTROMAGNETIC THEORY. CH. III. 
 
 axis of rotation is now through Q. We may formally prove it 
 thus. Take 
 
 r = h + R, (130) 
 
 where h is the vector from to Q and R the vector from Q 
 to P. Put (130) in (129), then 
 
 v = q + Vah + VaR (131) 
 
 But here a + Vah is, by (129), the velocity of Q, say u, 
 so we have 
 
 v = u + VaR, (132) 
 
 showing the velocity of P to be the sum of the translational 
 velocity u at any point Q, plus the velocity at P due to rota- 
 tion about the axis through Q. 
 
 All motion is relative. But observe the absolute character 
 of the spin a. It is absolute just because it involves and 
 depends entirely upon relative motions. 
 
 The translational velocity u of the point Q consists of a 
 motion along the axis of rotation, combined with a transverse 
 motion. The latter may be got rid of by shifting the axis. 
 For if w is the velocity of Q transverse to the axis, if we go to . 
 the distance w/a from the axis through Q, keeping in the plane 
 through the axis perpendicular to w, we shall reach a place 
 where the velocity due to the rotation about the Q axis is 
 either + w or w, according to which side of the Q axis we 
 go. Choosing the latter, where the transverse motion is can- 
 celled, and transferring the axis to it. we see that the general 
 motion of a rigid body consists of a spin about a certain 
 axis (which is termed the central axis), combined with a 
 translation along this axis. That is, it is a screw motion. It 
 may, however, be more convenient not to employ the screw 
 motion with a shifting axis, but to use (132), and let u and a 
 be the translational and rotational velocities of a point Q 
 fixed in the body. 
 
 We may also have simultaneous spins about axes which do 
 not meet. Thus, let R be the vector from any origin to a 
 point Q about which there is a spin a, and r the vector from 
 to any other point P. Then the velocity of P due to the 
 spin is, by the preceding, 
 
 v = Va(r-R) (133) 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 183 
 
 Next let there be any number of points of the type Q, each 
 with its spin vector a. The velocity at P is then the sum of 
 terms of the type (133), in which r is the same for all. The 
 result is therefore 
 
 v = VAr-2VaR, .... (134) 
 
 where A is the sum of the a's, or the spin of the body, whilst 
 the minus summation expresses the velocity at 0. For example, 
 two equal spins, a, in opposite senses about parallel axes at dis- 
 tance h combine to make a mere translational velocity perpen- 
 dicular to the plane of the axes; speed ah. I give these 
 examples, in passing, to illustrate the working of vectors. 
 These transformations are effected with a facility and a sim- 
 plicity of ideas which put to shame the Cartesian processes. It 
 may, indeed, be regarded as indicative of a mental deficiency 
 to be unable to readily work the Cartesian processes which is 
 in fact my own case. But that does not alter the fact that if 
 a man is not skilful in Cartesians, he may get along very well 
 in vectors, and that the skilful mathematician who can play 
 with Cartesians of great complexity, could as easily do far 
 more difficult work in vectors, if he would only get over the 
 elements, and accustom himself to the vectors as he had to do 
 to the other method. 
 
 Motion of Systems of Displacement, &c. 
 
 124. Going back to (127), we should, when the property 
 whose time-variation is followed belongs to matter moving 
 as a rigid body, employ in it the special reckoning of v of 
 equation (132), giving the velocity of any point P in terms of 
 u and a at an invariable point Q in the body. Thus, we have 
 
 S = i!r (n+VaIl)v - w - (135) 
 
 But equation (127) applies not merely to the case of matter 
 moving through space, carrying some property with it, but also 
 when the matter itself is fixed, whilst some measurable pro- 
 perty or quality moves through the matter, or is transferred 
 from one part of the matter to another. And here we may 
 leave out the consideration of matter altogether, and think 
 only of some stationary medium which can support and through 
 which the phenomenon can be transferred. 
 
184 ELECTROMAGNETIC THEORY. OH. III. 
 
 Thus, we may have a system of electric displacement D in a 
 dielectric, which may be moving through it independently of 
 any motion it may possess, and apply (127) to calculate D. It 
 is, however, only in relatively simple cases that this can be 
 followed up. For it is implied that we know the motion of the 
 displacement and how it changes in itself, and these may be 
 things to be found out. 
 
 But should the displacement system move as a rigid body, 
 the matter is greatly simplified. This will occur when the 
 sources of the displacement (electrification, for instance) move 
 as a rigid body, and the motion is of a steady type, so long 
 continued that the displacement itself (in the moving system) 
 is steady. Writing D for w in (135), we shall have 
 
 ~=VaD-(u + VaR)V.D .... (136) 
 at 
 
 Here VaD is the rate of time-variation of the displacement in 
 a moving element of the displacement system, caused by the 
 rotation. It is zero when the displacement is parallel to the 
 axis of spin, and a maximum when it points straight away 
 from the axis, like the spoke of a wheel. 
 
 The displacement system is, however, not necessarily or 
 usually the same when in steady motion as when at rest. 
 In the latter case only can it be regarded as known initially. 
 Set it steadily moving and it will (by reason of the self- 
 iuduction) be changed to another displacement system. Never- 
 theless (136), along with the electromagnetic equations, enable 
 us to find the new displacement system. 
 
 Should, however, the velocities of the connected sources be 
 only a very small fraction of the speed of propagation of dis- 
 turbances through the medium, the re-adjustment of the dis- 
 placement as the sources move takes place practically 
 instantaneously (as if the speed of propagation were infinite), 
 so that the displacement system remains unchanged ; preserving 
 its stationary type, whilst it moves through the medium as 
 a rigid body, in rigid connection with the sources. (This is 
 obviously quite incorrect at a great distance from the sources, 
 but there the effects themselves are insensible.) In this case, 
 too, it is clearly unnecessary that the motion of the sources 
 should be steady. Thus, in (136), D becomes the known 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 185 
 
 displacement system (of equilibrium), moving as defined by u 
 and a, which may vary anyhow. 
 
 Similar considerations apply to systems of induction, moving 
 with their sources. A general caution, however, is necessary 
 when there are conductors or other bodies in the field, not 
 containing sources. To keep them from having currents 
 induced in them, or in other wys upsetting the regularity of 
 the moving system, they should also partake in the motion, by 
 rigid connection. 
 
 Motion of a Strain-Figure. 
 
 125. Equation (136) also applies to the motion through a 
 medium of a " strain-figure," treated of by Dr. C. V. Burton 
 in the current number of the Phil. Mag. (February, 1892). It 
 is similar to that of slow motion of an electrical displacement 
 system. Imagine a stationary infinite elastic medium with 
 inertia, somehow set into a strained state, and let the speed of 
 propagation of disturbances of strain be practically infinite. 
 If then the strain-figure can move about through the medium, 
 it will do so as a rigid body, provided the sources of the 
 strain do so. In the above, D may signify the displacement 
 (ordinary), whose variation in space constitutes the strain. 
 Then D is the velocity of displacement, and is expressed in 
 terms of the velocity of the strain-figure. From the expres- 
 sion for the kinetic energy of the complete strain-figure, 
 the mechanical forces concerned can be deduced. As this 
 is to be done on dynamical principles involving Newton's 
 laws, we may expect beforehand that a strain-figure sym- 
 metrical with respect to a centre will behave as a New- 
 tonian point-mass ; as does a similar electrical displacement 
 figure. But, in general, without symmetry, the calcu- 
 lation of the forces concerned would be very troublesome 
 indeed. 
 
 But the real difficulty appears to me to be rather of a 
 physical than a mathematical nature. We have first to get 
 some idea of how the strain-figure is kept up. Let it be 
 stationary first. Then the strain-figure is referred to a 
 forced or unnatural state of the medium in certain places. 
 At any rate, we require intrinsic "sources" somewhere, and 
 perhaps it might for this reason be convenient to consider 
 
186 ELECTROMAGNETIC THEORY. CH. Ill 
 
 these portions of the medium (with the sources) to constitute 
 the atom or molecule, rather than the whole strain-figure. 
 
 Now, having got a stationary strain-figure, how is it to be 
 set moving? Three ways suggest themselves. First, a bodily 
 motion of the medium carrying the strain-figure with it. This 
 is plainly inadmissible. Next a motion of the atomic portions 
 only of the medium (with the sources) through the rest of the 
 medium, either disturbing it to some extent near by, or not 
 disturbing it at all slipping through, so to speak. This 
 would carry on the strain-figure. But it is inadmissible, if it 
 be our object not to move the medium at all in any part. 
 Thirdly, we may keep the whole medium at rest, and cause 
 the sources themselves to move through it, so that the atomic 
 portions of the medium change. But there is no means for 
 doing this presented to our consideration. It lies beyond 
 the dynamical question of the forces on an atom brought 
 into play if the strain-figure can move in the manner sup- 
 posed. The second course above, on the other hand, is more 
 intelligible, although it implies something akin to liquidity. 
 
 These objections are, however, only made suggestively. 
 The matter is sufficiently important to be deserving of a 
 thorough threshing out.* 
 
 Space- Variation or Slope VP of a Scalar Function. 
 
 126. The simplest and most easily understood effect of V 
 upon a function is when it acts upon a scalar, say P. This 
 implies that P is a scalar function of position, or of x, y> z. In 
 the most important applications P is single-valued, as when it 
 represents density, or pressure, or temperature, or electric 
 potential, or the corresponding magnetic potential of magnets. 
 But P may also be multiplex, as when it is the magnetic 
 potential outside a linear electric current. At present let P be 
 simplex. 
 
 By (119) we have 
 
 VP = LV 1 P+j.V 2 P + k.V 3 P. . . . (137) 
 
 Of course, since v is vector and P scalar, the result is a vector. 
 Its meaning is easily found. From the fact indicated in (137) 
 that the rectangular scalar components of V P are the rates of 
 increase of P along the axes of i, j, k, we may conclude that 
 
 [* In case of eolotropy, add to the right member of (26) the term Sa, 
 where S is .the torque and a the spin.] 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 187 
 
 the component of vP in any direction is the rate of increase of 
 P in that direction. Thus, N being any unit vector, we have, 
 by (137), 
 
 3 P. . (138) 
 
 Comparing with (121), we see that 
 
 . . . (139) 
 
 by (123), s being length measured along N. 
 
 The identity of NvP and Nv-P is tolerably obvious algebrai- 
 cally. It is not, however, true when for P is substituted a 
 vector, as we shall see later. 
 
 If we take P = constant, we obtain the equation to a surface. 
 If, then, T is any tangent to the surface at a chosen point, 
 that is, any line in the tangent plane perpendicular to the 
 normal, we shall have TvP = 0. The direction of vP itself 
 is therefore that of the normal to the surface; or the lines of 
 vP cut the equipotential surfaces perpendicularly, and pass 
 from one to the next (infinitely close) by the shortest paths. 
 The distance between any two consecutive equipotential sur- 
 faces of a series having a common difference of potential varies 
 inversely as the magnitude of vP, and vP itself is the vector 
 showing at once the direction and the rate of the fastest 
 increase of P. No perfectly satisfactory name has been found 
 for this slope, or space- variation of a scalar function. Com- 
 paring P with height above the level on a hillside, vP shows 
 the greatest slope upwards. But the illustration is inadequate, 
 since on the hillside we are confined to a surface. 
 
 The tensor of vP is given by 
 
 (VP)2 = (V 1 P) 2 + (V 2 P) 2 + (V 3 P) 2 , . . (140) 
 
 as with any other vector. 
 
 We may also here notice the vector product VNv in its 
 effect on a scalar. We have, by the semi-Cartesian formula 
 for a vector product, 
 
 VNV = i(N 2 V 3 - N 3 V,) + j(N 8 V 1 - NiVg) + k(N x V 2 - N^). (141) 
 Thus, when the operand is scalar, as P, we shall have 
 
 (VNV)P = VNVP ...... (142) 
 
188 ELECTROMAGNETIC THEORY. CH. III. 
 
 The result is a vector perpendicular to the plane of N and vP, 
 *s in any other vector product, vanishing when they are 
 parallel, and a maximum when they are perpendicular. 
 
 Scalar Product VD. The Theorem of Divergence. 
 
 127. When the operand of v is a vector, say D, we have 
 both the scalar product and the vector product to consider. 
 Taking the former alone first, we have 
 
 divD = VD = V 1 D 1 + V 2 D 2 + V 3 D 3 . . . (143) 
 
 This function of D is called its divergence, and is a very impor- 
 tant function in physical mathematics. Its general signification 
 will be best appreciated by a consideration of the Theorem of 
 Divergence. 
 
 Let liquid be in motion. The continuity of existence of 
 the matter imposes certain restrictions on the motion. The 
 current is mq per unit area, whore m is the density and q the 
 velocity. But, to further simplify, let m = 1, making the liquid 
 incompressible. Then the current is simply q, which measures 
 the amount of liquid crossing unit. area of any surface per- 
 pendicular to q, per second. But if the surface be not perpen- 
 dicular to q, the effective flux is only Nq, the normal component 
 of q, if N denotes the unit normal to the surface. Therefore, 
 2 Nq, or the summation of Nq over any surface, expresses the 
 total flux of liquid through the surface. 
 
 Now suppose the surface (fixed in space) is closed. Then the 
 summation represents the amount of liquid leaving the enclosed 
 region per second through its boundary, if N be the outward 
 normal. This amount is evidently zero, because of the assumed 
 incompressibility. If then we observe, or state, that 2 Nq is 
 not zero, either the fluid is compressible, or else there must be 
 sources of liquid within the region. Adopting the latter idea, 
 because it simplifies the reasoning, we see that the summation 
 2 Nq is the appropriate measure of the total strength of the 
 sources in the region, since it is the rate at which liquid is 
 being generated therein, or the rate of supply. The position of 
 the internal sources is quite immaterial, and so is. the shape of 
 the region, and its size, or the manner in which a source sends 
 out its liquid (i.e., equably or not in all directions). So we have 
 
'ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 189 
 
 if p represents the strength of a source, and the 2 includes 
 them all. 
 
 When the sources are distributed continuously, so that p is 
 a continuous function of position, its appropriate reckoning is 
 per unit volume. That this is the divergence of q is clear 
 enough, according to the explanations relating to divergence 
 already given, ( 51). That it is the same as vq, as in (143), we 
 may prove at once by the Cartesian form there exhibited. 
 
 We have to reckon the flux outward through the sides of a 
 unit cube. Take its edges parallel to i, j, k respectively. Then 
 there are two sides whose outward normals are - i and i, and 
 their distance apart is unity. The outward fluxes due to them 
 are therefore - iq, or - q v and q l + v^, whose sum is v^. 
 Similarly the two sides whose normals are j contribute 
 V 2 g 2 , and the remaining sides contribute V 3 ^ 3 . Comparing 
 with (143), we verify the Cartesian form of the divergence of 
 a vector. 
 
 But we may have any number of other special forms, accord- 
 ing to the co-ordinates we may choose to employ for calculating 
 purposes, such as spherical, columnar, &c., and the most ready 
 way to find the corresponding form is by the immediate appli- 
 cation of the idea of divergence to the volume-element concerned. 
 For purposes of reasoning, however, it is best to entirely 
 eliminate the idea of co-ordinates. Divergence is independent 
 of co-ordinates. 
 
 The sources need not be so distributed as to give rise to a finite 
 volume-density. We may have, within the region concerned, 
 surface-, line-, or point-sources, and the principle concerned 
 is the same throughout. Thus the density of a surface-source 
 is measured by the sum of the normal fluxes on its two sides 
 per unit area, that is, by 2 Nq applied to the two sides of the 
 unit area, if the flux wholly proceeds outwards. This is, 
 however, not fully general, as there may be a flux in the 
 surface itself, so that the full measure of the surface-density 
 has to include the divergence of the surface-flux, to be found by 
 calculating the flux leaving the unit area across its bounding 
 line. Similar considerations apply to linear sources. In the 
 case of a point-source the measure of the strength is the flux 
 outward through a closed surface enclosing the point infinitely 
 near it the surface of the point, so to speak. Any other- 
 
190 ELECTROMAGNETIC THEORY. CH. III. 
 
 surface enclosing the point will do, provided there are no other 
 sources brought in by the change. There may also be multiplex 
 sources. Thus, a pair of equal point-sources, one a source, the 
 other a sink, would be equivalent to no source at all if brought 
 infinitely near one another ; but if the reduction in distance be 
 accompanied by a corresponding increase in strength of the 
 sources, the final result is not zero. This is the case of a 
 magnetised particle on the theory of magnetic matter ; but it 
 is not necessary or desirable to complicate matters by entering 
 upon special peculiarities of discontinuity in considering the 
 Divergence Theorem. Its general form is 
 
 2ND = 2divD, (145) 
 
 when the vector D, to which it is applied, admits of finite 
 differentiation ; and the special meanings to be attached to the 
 divergence of D, in order to satisfy the principle concerned, 
 may be understood. 
 
 Although a material analogy, as above, is very useful, it is 
 not necessary. Any distributed vector magnitude will have 
 the same peculiarities of divergence as the flux of a liquid. If 
 it be the motion of a real expansible fluid that is in question, 
 then the divergence of its velocity represents the rate of ex- 
 pansion. It would, however, be very inconvenient to have to 
 carry out this analogy in electric or magnetic applications ; 
 an incompressible liquid, with sources and sinks to take the 
 place of expansions and contractions, is far more manageable. 
 
 Extension of the Theorem of Divergence. 
 
 128. The following way of viewing the Divergence Theorem, 
 apart from material analogies, is important. Consider the 
 summation 2 ND of the normal component of a vector D over 
 any closed surface. Divide the region enclosed into two 
 regions, Their bounding surfaces have a portion in common. 
 If, then, we sum up the quantity ND for both regions (over 
 their boundaries, of course), the result will be the original 
 2 ND for the complete region. The normal is always to be 
 reckoned positive outwards from a region, so that on the 
 surface common to the two smaller regions N is + for one 
 and - for the other region, and 2 ND for one is the negative of 
 that for the other, so far as the common surface goes. 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 191 
 
 Since this process of division may be carried on indefinitely, 
 we see that the summation 2ND for the boundary of any region 
 equals the sum of the similar summations applied to the surfaces 
 of all the elementary regions into which we may divide the 
 original. That is, 
 
 (146) 
 
 where, on the right side, we have a volume-summation whose 
 elementary part <(D) is the same quantity 2ND as before, 
 belonging now, however, to the elementary volume in question. 
 We have already identified <(D) with the divergence of D. 
 
 But if this demonstration be examined, it will be seen that 
 the validity of the process whereby we pass from a surface- to a 
 volume-summation, depends solely upon the quantity summed 
 up, viz., ND, changing its sign with N. We may therefore at 
 once give to the Divergence Theorem a wide extension, making 
 it, instead of (146), take this form : 
 
 2F(N) = 2/(N) ..... (147) 
 
 Here, on the left side, we have a surface-, and on the right side 
 a volume- summation. The function F(N), where N is the out 
 ward normal, is any function which changes sign with N. The 
 other function /(N), the element of the volume-summation, is 
 the value of 2F(N) for the surface of the element of volume. 
 Thus, by considering a cubical element, 
 
 /(N)=V 1 F(i) + V 2 F(j) + V 3 F(k) . . (148) 
 
 is the Cartesian form, should F(N) be a scalar function, or the 
 semi-Cartesian form should it be a vector function. A few 
 examples of the general theorem (147) will be given later. In 
 the meantime the other effect of v should be considered. 
 
 Vector Product WE, or the Curl of a Vector. The Theorem 
 of Version, and its Extension. 
 
 129. The vector product of v and a real vector, say E, is 
 given in semi-Cartesian form by 
 
 WE = i(V 2 E 3 - V 3 E 2 ) + j^ - 
 
 = curlE. 
 
192 ELECTROMAGNETIC THEORY. CH. III. 
 
 As before, with respect to the divergence of a vector, we can 
 best appreciate the significance of this formula by the general 
 property involved, expressed by the Theorem of Version. 
 
 On any surface draw a closed curve or circuit. Let, for 
 distinctness, E be electric force. Calculate the voltage in the 
 circuit due to E. The effective force per unit length of circuit 
 is the tangential component of E, or TE, if T is the unit 
 tangent. The voltage in the circuit is, therefore, 2 TE, the 
 summation being circuital, or a line-integration extended once 
 round (along) the closed curve. 
 
 Now draw on the surface a line joining any two points of 
 the circuit. Two circuits are thus made, having a portion in 
 common. Reckon up the voltage in each of the smaller circuits 
 and add them together. The result is the voltage in the 
 first circuit, if we rotate the same way in both the smaller 
 circuits as in the original, because the common portion 
 contributes voltage equally and oppositely to the two smaller 
 circuits. 
 
 This process may be carried on to any extent by drawing 
 fresh lines on the surface. We therefore have the result that 
 the voltage in the circuit bounding a surface equals the sum of 
 the voltages in the elementary circuits bounding the elements 
 of surface. Or 
 
 2TE = 20(E), (150) 
 
 where on the left side we have a circuital summation, and on 
 the right side an equivalent surface-summation, in which 0(E), 
 the quantity summed, is the value of 2 TE, that is, the voltage, 
 in the circuit bounding the particular element of surface con- 
 cerned. 
 
 To find the form of in terms of i, j, k, &c., we need only 
 calculate the voltages in unit square elements of surface taken 
 successively with edges parallel to j, k, to k, i, and to i, j. In 
 the first case, when the normal to the square circuit, or the axis 
 of the circuit, is i, the voltage is 
 
 V 2 E 3 -V 3 E 2 , 
 
 that is, by (149), the i component of VvE, and therefore the 
 normal component, since here N = i. Similarly when N=j, and 
 the axis of the circuit is j, the voltage in it is expressed bv the 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 193 
 
 coefficient ofj in (149). And when N = k, the voltage is the 
 coefficient of k in (149). Thus, in any case, the voltage in an 
 elementary circuit of unit area is the normal component of 
 VvE, that is, NVvE. So (150) becomes 
 
 , . . (151) 
 
 expressing the Theorem of Version, sometimes termed Stoke's 
 Theorem. 
 
 When E is not electric force, 2 TE is not circuital voltage, 
 but circuital something else ; but this does not affect the 
 general application of the theorem. 
 
 This theorem is particularly important in electromagnetism 
 because it is involved in the two fundamental laws thereof, 
 what were termed the First and Second Circuital Laws, or 
 Laws of Circuitation, connecting together the electric and 
 magnetic forces and their time-variations. (See 33 to 36, 
 Chap. II., and later.) 
 
 If the vector whose curl is taken be velocity q in a moving 
 fluid, then curl q represents twice the spin or vector angular 
 velocity of the fluid immediately surrounding the point in 
 question ; its direction being that of the axis of rotation, and 
 magnitude twice the rate of rotation. But I have not made 
 use of the fluid analogy in describing and proving the Version 
 Theorem, because it is not of material assistance. 
 
 Since the validity of the process whereby we pass from a 
 circuital summation of the tangential component of a vector to 
 an equivalent surface-summation depends upon the fact that 
 TE changes sign with T, we may generalise the theorem thus : 
 
 2F(T) = 2/(T), ..... (152) 
 
 where on the left we have a circuital and on the right a surface- 
 summation, and F(T) is such a function of T as to change 
 sign with T ; whilst /(T), the element of the surface-summa- 
 tion, is the value of the former 2 F(T) for the particular element 
 of surface in question. Of this general theorem a few examples 
 will be given later. 
 
 A few words regarding v, div and curl, terminologically con- 
 sidered, may be useful. Since divergence and curl are expres- 
 sible in terms of vex (a provisional name for v, which has been 
 suggested to me), why not use the vex operator only, like 
 
 o 
 
194 ELECTROMAGNETIC THEORY. CH. HI. 
 
 the quaternionists ? The reasons, which are weighty, should be 
 obvious. 
 
 In the first place, we require a convenient language for de- 
 scribing or referring to processes and results, expressing approxi- 
 mately their essential meaning without being too mathematical. 
 Now V alone is not convenient for this purpose. The scalar 
 product of v and D conveys no such distinct idea as does 
 divergence ; nor does the vector product of v and E speak so 
 plainly as the curl or rotation of E. 
 
 Besides, the three results of V, exemplified in vP, and vD, 
 and VvE, are so remarkably different in their algebraical 
 development and in their meaning, that it is desirable, even in 
 the algebra, to very distinctly separate them in representation. 
 Therefore, in the preceding part of the present work (as in all 
 former papers), the symbol V is only prefixed to a scalar, as in 
 V P, the space variation of P, whilst for the scalar and vector 
 product are employed div and curl, in the formulae as well as 
 in descriptive matter. 
 
 There are, however, cases when it may be desirable to use v 
 and Vv applied to vectors in formulae, namely, when the 
 combinations of symbols are not so simple that their meaning 
 and effect can be readily seen, and when it is required to 
 perform transformations also not readily recognisable. The 
 utility of V in its vectorial significance then becomes apparent, 
 for one may use it alone, temporarily if desired, and work it as 
 a vector, remembering, however, its other functions. Of this, 
 too, some examples should be given. 
 
 Five Examples of the Operation of V in Transforming from 
 Surface to Volume Summations. 
 
 130. Returning to the theorem (147), or 
 
 2/, ...".. (153) 
 
 let us take a few of the simplest cases that present themselves. 
 Given any odd function F of N, the normal outwards from a 
 closed surface over which the summation on the left side ex- 
 tends, we convert it to an equivalent summation throughout 
 the enclosed region by making/, the quantity summed, be the 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 195 
 
 value of 2 F(N) for the surface of the element of volume. This 
 last is most conveniently a unit cube, so that 
 
 /=V 1 F(i) + V 2 F(j) + V 3 F(k),. . . (154) 
 as in (148). 
 
 (a). The simplest case of all is F(N) = N itself. Then, by 
 (154), or by considering that the vector normals to the six faces 
 of a cube balance one another in pairs, we have 
 
 2N = 0, ...... (155) 
 
 expressing that a closed surface has no resultant orientation ; 
 or, that a normal pull applied to every part of a closed surface, 
 of uniform amount per unit area, has no resultant. 
 
 (b). If we multiply by p, any scalar function of position, we 
 have the same case again if p be constant. When negative, it 
 makes a well-known elementary hydrostatic result. But when 
 p is not constant, then, by (154), 
 
 Vjp, . . . (156) 
 so that we have 
 
 2Np = 2V# ...... (157) 
 
 These are, of course, vector summations. The sum of the 
 surface tractions equals the sum of the bodily forces vp arising 
 from the space- variation of the internal tension. Take p nega- 
 tive to indicate pressure. 
 
 (c). Take F(N) = ND, where D is a vector function of position. 
 This gives the most valuable theorem of divergence, 
 
 = 2divD, . . . (158) 
 already discussed. 
 
 (d). Take F(N) = NDP, where P is a scalar. Here, by (154), 
 or by (158), we shall find 
 
 2NDP = 2V(DP) ...... (159) 
 
 Here V has to differentiate both D and P, thus, 
 
 , . . . . (160) 
 02 
 
196 ELECTROMAGNETIC THEORY. CH. III. 
 
 so that the previous equation may be written 
 
 . . . (161) 
 
 which is a form of Green's Theorem relating to electrostatic 
 energy. D may be the displacement in one system of electrifi- 
 cation and P the potential in another. The quantity - 2DvP is 
 their mutual energy ; and this is, by (161), equivalently ex- 
 pressed by the sum of products of every charge in one system 
 into the potential due to the other. 
 
 (e). Take F(N) = VNH. Then, by (154), 
 
 /= ViViH + V 2 VjH + V 3 VkH. 
 
 But here we may shift the v's to the other side of the V's a 
 because they are scalars ; this produces 
 
 /=VVH, 
 so that we have 
 
 2VNH = 2VVH = 2curlH. . . . (162) 
 
 The interpretation may be more readily perceived by reversing 
 the direction of the normal. Take N = - n, so that n is the 
 normal drawn inward from the boundary. Then 
 
 2curlH + 2VnH = ..... (163) 
 
 If H be magnetic force, curl H is the .electric current-density in 
 the region. Now VnH is the surface equivalent of the bodily 
 curlH. Ignore altogether the magnetic force outside the 
 region, if there be any. Then the circuitation of H gives the 
 current through a circuit. Applied to elementary circuits 
 wholly within the region, the result is curl H. But at the 
 boundary, where H suddenly ceases, there is a surface-current 
 as well. To find its expression, apply the process of circuitation 
 to a circuit consisting of two parallel lines of unit length, 
 infinitely close together, but on opposite sides of the boundary, 
 joined by infinitely short cross-pieces. Only the unit line 
 inside the region contributes anything to the circuitation ; and 
 by taking it to coincide with H, so as to make the circuitation 
 a maximum, we find that VnH represents the surface-density 
 of current. So, if J be current, we have, by (163), 2 J = for 
 any region, by itself. The surface-distribution and the volume- 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 197 
 
 distribution of current are complementary ; that is, they are 
 properly joined together to make up a circuital distribution. 
 Thus, 
 
 nVVH=-VVnH, .... (164) 
 
 where on the left side we have the divergence (at the surface) 
 of the internal current, and on the right the equal convergence 
 of the surface-current. 
 
 Five Examples of the Operation of V in Transforming from 
 Circuital to Surface Summations. 
 
 131. Next, take a few examples of the extended Theorem 
 of Version (152), viz. : 
 
 2F(T) = E/, .' . . . . (165) 
 
 where now, on the left side, we have the circuital summation of 
 an odd function of T, and on the right an equivalent surface- 
 summation, whose elementary part / is the value of F(T) for 
 the circuit bounding the element of surface. 
 
 Taking three elements of surface to be unit squares, whose 
 normals are i, j, k, we readily see that the corresponding /'s are 
 
 v 2 F(k) - v 3 F(j), v.FW-v^k), vjfl-vf.. (166) 
 
 By means of these we can see the special form assumed by / in 
 any case. 
 
 (a). Thus, take F(T) = T itself, the unit tangent. Then we 
 have 
 
 2T = 0, ... . . . (167) 
 
 merely expressing the fundamental property of adding vectors, 
 that the sum of any vectors forming, when put end to end, a 
 circuit, is zero. 
 
 (6). Take F(T) = TP, where P is a scalar function of position. 
 Then, with normal i, we have, by the first of (166), 
 
 ; . . (168) 
 so, writing N for i, we obtain 
 
 - . . (169) 
 
198 ELECTROMAGNETIC THEORY. OH. Til. 
 
 The quantity summed over the surface is, therefore, the 
 surface representative of the curl of vP. This has no volume 
 representative, its value being then zero. 
 
 (c). Take F(T) = TH. Here, with normal i to the element 
 of surface, we have 
 
 /= V 2 kH - VgjH = V 2 H 3 - V 3 H 2 = iVVH, 
 by (149). Therefore, putting N for i, 
 
 2TH = 2NVVH = 2NcurlH . . (170) 
 
 the Version Theorem again. But observe that, by the trans- 
 formation (164), we may also write it 
 
 2TH = 2VNV.H, . . . . (171) 
 
 similarly to (169), in which the operand is a scalar. This is 
 mnemonically useful, but (170) is more practically useful. 
 
 (d). Take F(T) = VTH. Here, with normal i, the first of 
 (166) gives 
 
 /= V 2 VkH - V 3 VjH = V (kV 2 - JV 3 )H. 
 
 But here we have kV 2 - j V 3 = ViV, 
 so that /=V(ViV;H; 
 
 and therefore, generally, putting N for i, 
 
 2VTH = 2V(VNV)H. . . . (172) 
 
 (e). Let the quantity in the circuital summation be a vector 
 of length P (a scalar function of position) drawn perpendi- 
 cularly to the plane of T and N. That is, 
 
 F(T) = (VTN)P. 
 We then find, taking N = i, and using the first of (166), 
 
 /= V 2 Vki . P - V 3 Vji . P = (j V 2 + kV 3 )P = VP - i (iV) P. 
 In general, therefore, 
 2 VTN . P = 2 (VP - N . NVP) = 2 V a P = 2 V (VNV)N.P . (173) 
 
 The element of the surface-summation is V 8 P, meaning the 
 slope of P on the surface itself, disregarding any variation it 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 199 
 
 may have out of the surface. The last form of (173) involves 
 
 the transformation formula (52). 
 
 * Observe that in all the above examples, 
 
 2F(N) = 2F(V), (174) 
 
 when we pass from a closed surface to the enclosed region; 
 and that 
 
 2F(T) = 2F(VNV), .... (175) 
 
 when we pass from a circuit to the surface it bounds. Thus, 
 N becomes V, and T becomes VNv. But I cannot recommend 
 anyone to be satisfied with such condensed symbolism alone. 
 It is much more instructive to go more into detail, as in the 
 above examples, and see how the transformations occur, bearing 
 in mind the elementary reasoning upon which the passage from 
 one kind of summation to another is based ( 128, 129). 
 
 Nine Examples of the Differentiating Effects of V 
 
 132. The following examples relate principally to the modi- 
 fications introduced by the differentiating functions of V. 
 
 (a). We have, by the parallelepipedal property, 
 
 NVVE = VVEN = EVNV, . . . (176) 
 
 when V is a common vector. The equalities remain true when 
 V is vex, provided we consistently employ the differentiating 
 power in the three forms. Thus, the first form, expressing the 
 N component of curl E, is not open to misconception. But in 
 the second form, expressing the divergence of VEN, since N 
 follows V, we must understand that N is supposed to remain 
 constant. In the third form, again, the operand E precedes the 
 differentiator. We must either, then, assume that y acts back- 
 wards, or else, which is preferable, change the third form to 
 VNV.E, the scalar product of VNv and E; or (VNv)E, if that 
 be plainer. 
 
 (b). Suppose, however, that both vectors in the vector pro- 
 duct are variable. Thus, required the divergence of VEH, 
 expanded vectorially. We have 
 
 . . . (177} 
 
200 ELECTROMAGNETIC THEORY. CH. IIL 
 
 where the first form alone is entirely unambiguous. But we 
 may use either of the others, provided the differentiating power 
 of y is made to act on both E and H. But if we keep to the 
 plainer and more usual convention that the operand is to 
 follow the operator, then the third form, in which E alone is 
 differentiated, gives one part of the result, whilst the second 
 form, or rather, its equivalent -EVvH, wherein H alone is dif- 
 ferentiated, gives the rest. So we have, complete, and with- 
 out ambiguity, 
 
 div VEH = H curl E-E curl H, . . . (178) 
 
 a very important transformation. It is concerned in the de- 
 duction of the equation of activity from the two circuital laws 
 of electromagnetism. 
 
 (c). In these circuital laws we have also to consider the curl 
 of a vector product, viz., the curl of the motional electric force 
 in one law, and the curl of the motional magnetic force in the 
 other. Taking the former, we have 
 
 curlVqB = VVVqB, . ... (179) 
 
 where B is the induction and q the velocity of the medium 
 supporting it. Apply the elementary transformation (52) to 
 (179). It gives 
 
 VVVqB = q.VB-B.Vq, ... (180) 
 
 when y is a mere vector. But on the left side both q and B 
 have to be differentiated ; therefore the same is true in both 
 terms on the right side. This gives 
 
 VVVqB = qdivB + BV.q 
 
 -Bdivq-qV.B, . . . (181) 
 
 without ambiguity or need of reservation. That is to say, as 
 in the q. vB of (180) both q and B have to be differentiated, we 
 get qdivB when B alone, and Bv.q when q alone is differ- 
 entiated. Similarly for the other term in (180). 
 
 Or we might write V q when q alone, and V B when B alone 
 suffers differentiation. Then, fully, 
 
 VVVqB = VV q VqB + VV B VqB, . . (182) 
 VV q VqB = BV.q-Bdivq, . . (183) 
 
 W B VqB = qdivB-qV.B. . . (184) 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 201 
 
 Here the sum of (183) and (184) gives (181). The inean- 
 iftg of By and qy has been already explained. 
 
 (d). Equation (181) may be applied to the circuital laws. 
 Take the second, for example, in the form (4), 66, 
 
 - curl (E - e ) = K + B + wo- - curlVqB, . (185) 
 
 and suppose that w = q, or that sources move with the medium. 
 Then, by (181), we cancel the convective term wo-. Further, 
 
 we have B + qy.B = SB/fo, b 7 ( 127 )> 122 so that ( 185 ) 
 becomes 
 
 - curl (E -e ) = K + SB/8* + Bdivq-BV.q, . (186) 
 
 and the corresponding form of the first law (equation (3), 
 66), is 
 
 curl(E-li ) = C + oD/o-* + Ddivq-Dy.q. . (187) 
 
 The time-variations refer to the same (moving) portion of the 
 medium now. But if we wish to indicate the movement of 
 electrification, &c., through the medium, that is, have relative 
 motion u - q of p (and w - q of <r) with respect to the ether 
 then to the right side of (187) add the term (u-q)/>, and to 
 the right ride of (186) add (w - q)<r. 
 
 It is desirable to preserve the velocities u and w, or else the 
 relative velocities, as well as the velocity q of the medium, in 
 order to facilitate the construction and comprehension of 
 problems relating to electromagnetic waves, which, although 
 abstract and far removed from practice, are of a sufficiently 
 simple nature to enable one to follow the course of events. 
 
 (e). We have already had the divergence of the product of a 
 scalar and vector under consideration. Now examine its curl. 
 Thus, 
 
 curlDP = VV(DP) 
 
 = VVD.P-VDV.P 
 
 = PcurlD-VDVP. . . . (188) 
 
 Here V can only make a vector product with D, because P is 
 scalar. On the other hand, both P and D suffer differentiation. 
 So in the second line we have V both before and after D. 
 
 (/). The divergence of the curl of any vector is zero. That 
 is, divcurlH = 0, or VWH = 0. . - (189) 
 
202 ELECTROMAGNETIC THEORY. CH. III. 
 
 If y here were a real vector (189) would mean that the 
 volume of a parallelepiped vanishes when two edges coincide. 
 
 (g). A somewhat similar case is presented by the vanishing 
 of the curl of a polar force. Thus, 
 
 curlVP = 0, or VV-VP = 0. . . . (190) 
 
 Of course Vw is zero. But the scalar product VV, or v 2 , is 
 the Laplacean operator, 
 
 V2 = V 1 2 + V 2 2 + V 3 2 , .... (191) 
 which occurs frequently. 
 
 (h). Let the operation curl be done twice on a vector. Thus, 
 
 (curl) 2 A = VVVVA 
 
 = V-VA-V 2 A, . . . (192) 
 by the transforming formula (52). Or 
 
 V 2 A = VdivA-curl 2 A. . . . (193) 
 
 Thus there are two principal forms. If the vector A has no 
 curl, then V 2 A is the slope of its divergence. If, on the other 
 hand, it has no divergence, then - V 2 has the same effect exactly 
 as taking the curl twice. 
 
 (i). In the case of the operand being a scalar, then we have 
 
 V 2 P = divVP, (194) 
 
 the divergence of the slope of the scalar. 
 
 The Potential of a Scalar or Vector. The Characteristic 
 Equation of a Potential, and its Solution. 
 
 133. The last equation brings us to the theory of potentials. 
 There are several senses in which the word potential has been 
 employed, to enumerate which would be valueless here. For 
 our present purpose we may conveniently fix its meaning by 
 denning the potential at A of a quantity p at B to be the quantity 
 />/47ir, where r is the distance from B to A. This is the rational 
 potential, of course. 
 
 When p is distributed throughout space, whether at points, 
 or over surfaces, or throughout volumes, the potential at any 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 203 
 
 pofnt- is the sum of the potentials of all the elements of p. 
 That is, 
 
 , .... (195) 
 
 if P is the potential of p. 
 
 We may use the same definition when it is a vector that has 
 to be potted, or potentialised. Thus, if A is the potential of 
 0, then 
 
 ..... (196) 
 
 The summation is now a vector summation. Also, pot means 
 " potential," or " the potential of," and has no more to do with 
 kettle than the trigonometrical sin has to do with the un- 
 mentionable one. It seems unnecessary to say so, but one 
 cannot be too particular. 
 
 We may connect these potentials with v as follows : Given 
 that 
 
 divF = />, ..... (197) 
 
 that is to say, that the divergence of a vector F is p. The 
 meaning of divergence has been explained more than once; 
 both its intrinsic and its vectorial meaning. Now, if the vector 
 F be explicitly given, it is clear that p is known definitely, 
 since it is derived from F by differentiation, which should, 
 perhaps, be regarded as a direct process, rather than inverse. 
 But if it be p that is given, F is not immediately deter minable, 
 unless we subject F to limitations. For we may construct any 
 number of different F's to satisfy (197). Let every elementary 
 source p of F send out the quantity p of F, according to my 
 rational theory of sources already explained ; that is, the 
 Unitarian system of one "line of force" to the unit "pole." 
 Then, by the manner of construction, the resultant F will satisfy 
 (197), and it will do so independently of the way we choose 
 to let a source send out the flux it generates, whether equably 
 or not. 
 
 But if it be done equally in all directions, so that p/4nr 2 is 
 the intensity of the " force " at distance r from the point-source 
 p, and r 1 /o/47rr 2 the vector force to correspond, where r x is a 
 unit vector drawn from p towards the point under considera- 
 tion, making the resultant F be 
 
 1 , ..... (198) 
 
204 ELECTROMAGNETIC THEORY. CH. IIL 
 
 we obtain a special solution of (197) which has a remarkable 
 property, viz., 
 
 curl F = 0, ..... (199; 
 
 so that if F be electric force, the voltage in any circuit is zero. 
 The meaning of curl, I may observe, has been explained more 
 than once ; both its intrinsic and its vectorial meaning. Those 
 who seek can find, If they will not take the trouble to seek 
 or to remember it is of no consequence to them. There are 
 plenty of other things they may concern themselves about ; 
 perhaps more profitably. 
 
 The property (199) is visibly true in the case of a single 
 source. It is therefore separately true for the fields of all the 
 sources, and therefore, by summation, is true for the com- 
 plete F. 
 
 But (198) does not give the only vector which has no curl 
 and a given divergence. For a constant vector (that is, constant 
 throughout all space), has no curl and no divergence, unless we 
 go to the very end of space to find the sources. Of course this 
 constant solution has no relation to the sources p, and may be 
 wholly ignored. If allowed, F would not vanish at an infinite 
 distance from the sources. Remembering this, and excluding 
 the constant solution, we may say that (198) is the solution of 
 (197) and (199). 
 
 That there is no other solution may be proved analytically 
 by Green's Theorem. But we do not really need any 
 appeal to analysis of that kind, if the intrinsic meanings of 
 divergence and curl are understood. For the admission that 
 there could be a second solution, say, F + f, where F is the solu- 
 tion (198;, would, by (197) and (199), imply that the vector 
 f had no divergence, and also no curl anywhere. But the first 
 of these conditions means that f is entirely circuital, if existent 
 at all. The second denies that it is circuital. So f is non- 
 existent. 
 
 Now observe that 
 
 or the slope of the scalar l/47rr is the vector with tensor l/4?rr 2 
 and direction T r It follows from this that 
 
 -Vpotp = F ...... (200) 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 205 
 
 when a single point-source is in question. Therefore, by sum- 
 mation, the same is true for any distribution of sources, or 
 
 F= -VP= -Vpotp, .... (201) 
 
 where P is the potential of /o, as denned by (195), and F is as 
 in (198). The slope of P, if by this we understand slope down- 
 wards, or vector rate of fastest decrease, is therefore the same 
 vector as was constructed to solve (197) subject to (199). 
 Taking the divergence of (201), we have, by (197) and (194), 
 
 /o = divF= -V 2 P= -V 2 pot/x . . . (202) 
 A solution of the characteristic equation of P, or 
 
 V 2 P=-/o, (203) 
 
 is therefore (195), and it is the solution vanishing at an infinite 
 distance from the sources. 
 
 If we start from (203), we should first use (194), and make it 
 
 div(-VP) = /> (204) 
 
 Then, by (190), we see that - VP has no curl, so that we have 
 again the two equations (197), (199) to consider, as above. 
 
 The consideration of F rather than of P has many advan- 
 tages for purposes of reasoning, as distinguished from calcula- 
 tion. This is true even in statical problems; for instance, 
 when F is electrostatic force, and P the corresponding potential. 
 When we proceed further, to kinetic problems, when F can no 
 longer be wholly expressed as the slope of a potential, the 
 utility of considering P at all, even for calculating purposes, 
 becomes sometimes very questionable, and the consideration is 
 sometimes certainly useless and misleading. 
 
 From (202) we see that - v 2 and pot are reciprocal. In 
 another form, -V~ 2 and pot are equivalent; or (pot)" 1 and 
 - v 2 are equivalent. The property has only been proved for a 
 scalar function, having a scalar potential. But since any 
 vector may be written iC 1 +jC 2 + kC 3 , and the property is 
 true for the three scalars C lf &c., it is also true for the vector 
 0. Thus, explicitly, 
 
 - V 2 pot = - V 2 pot (iCi + jC 2 + kC 3 ) 
 = i ( - V 2 pot CJ + j ( - V 2 pot CJ + k ( - V 2 pot C 3 ), 
 
206 ELECTROMAGNETIC THEORY. OH. III. 
 
 because the reference vectors i, <fec., are constant vectors. So, 
 if Aj is the potential of C 15 A 2 of C 2 , and A 3 of C 3 , which makes 
 A be the potential of C, according to (196), we shall have 
 
 A = potC, (205) 
 
 -V 2 A = C, (206) 
 
 -V 2 potC = C (207) 
 
 In short, the characteristic equation of the vector A merely 
 unites, from the above point of view, the characteristics of the 
 components, so that pot and - V 2 are reciprocal when the 
 operand is a vector, as well as when it is a scalar. 
 
 Connections of Potential, Curl, Divergence, and Slope. Separa- 
 tion of a Vector into Circuital and Divergent Parts. A 
 Series of Circuital Vectors. 
 
 134. But the above gives a very partial and imperfect view 
 of the general theory of potentials. There are numerous other 
 relations between a vector and its associated functions. For 
 instance, if in (206), A be circuital, then the Laplacean y 2 
 may, by (193), be replaced by -(curl) 2 . That is, if A x be cir- 
 cuital, and be the potential of C 15 then, 
 
 curl 2 A! = 0!, . . . . (208) 
 or curl 2 pot C^Cj (209) 
 
 Here, then, we have replaced the scalar operation - V 2 by the 
 vector operation curl done twice. Of course, C x is also cir- 
 cuital, as is proved by (189). 
 
 Again, let the A of (206) be polar, or wholly divergent, and 
 be now called A^ the potential of C 2 ; then, by (193), we shall 
 have 
 
 -VdivA 2 = C 2 , . . . . (210) 
 
 or -VdivpotC 2 = C 2 (211) 
 
 Here again we have replaced V 2 by a double operation, first 
 div and then V. This is similar to the passage from (203) to 
 (204), only done in the reverse manner. In (210), by (190), 
 C 2 is polar, because A 2 is. 
 
 Conversely, we see that the potential of a circuital vector is also 
 circuital, and that the potential of a polar vector is also polar. 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 207 
 
 Now Aj has no divergence, so it may be added on to the 
 divergent A 2 in (210) without affecting its truth. Thus, if 
 A = Aj + A 2 , we have 
 
 -VdivA = C 2 ..... (212) 
 
 Similarly, A 2 has no curl, so may be added to the Aj in (208), 
 giving 
 
 cur! 2 A = C 1 ...... (213) 
 
 Here remember that A is the potential of or Oj + 2 . 
 
 These equations supply one way of effecting the division of a 
 vector A of general type (having both curl and divergence) into 
 two vectors, one of which, A I} is circuital, whilst the other A 2 
 is polar. For A 2 is the potential of 0. 2 , so, by (212), 
 
 -potVdivA = A 2 . . . . (214) 
 
 separates A, from A. Similarly, A l is the potential of C 1? so 
 by (213), 
 
 pot curl 2 A = Aj .... (215) 
 
 separates A x from A, and therefore A 2 from A by a different 
 method. There are many other ways of splitting A into cir- 
 cuital and divergent parts. The one most easily understood, 
 apart from the mathematics, is the following. Go over the 
 whole field of A and measure its divergence. If we find that 
 there is no divergence, then we do not need to go further, for 
 we know that A is circuital already; that is, A = A 1} and 
 A 2 = 0. But should there be divergence, say B 2 , so that 
 
 divA = divA 2 = B 2 , .... (216) 
 
 then construct the flux A 2 corresponding to the divergence B 2 
 according to the method already explained with respect to 
 (197); thus, 
 
 = -VpotB 2 , ..... (217) 
 
 by (198) and (200). Knowing A 2 , we know A 1? or A - A 2 . 
 
 Or we might vary the process thus. First measure the curl 
 of A. This is the same as the curl of A l because the curl of 
 A 2 is zero. Let, then, 
 
 B 1 ..... (218) 
 
208 ELECTROMAGNETIC THEORY. CH. III. 
 
 and construct the circuital solution of this equation ; that is to 
 say, regarding B x as given, find A r It is given by 
 
 A! = curl pot B! (219) 
 
 For Aj as thus defined is evidently circuital, in the first place ; 
 and next, by taking the curl, we produce 
 
 curl AJ = curl 2 pot B^Bj, . , . (220) 
 
 which is the given datum. Here we use curl 2 pot = l, because 
 the operand is circuital, as in (209). 
 But instead of (219) we may write 
 
 A 1 =potcurlB 1 , . . . . (221) 
 
 showing an entirely different way of going from B x to A r For 
 A! as thus constructed is circuital ; and, since curl Bj = G v 
 (221) is the same as 
 
 A 1= pot Op 
 
 which was our definition of A l in terms of C r 
 
 Thus pot curl and curl pot are equivalent when the operand 
 is circuital, as above. They are, however, also equivalent when 
 the operand is general, or both circuital and divergent, because 
 if any divergent vector be added to the B x in the right members 
 of (219) or (221), the operation of curl to which it is subjected 
 renders its introduction inoperative. We therefore have 
 
 pot curl C = curl pot 0, .... (222) 
 
 where is any vector. We have also the similar exchange- 
 ability, 
 
 pot curl 2 C = curl 2 pot 0, .... (223) 
 
 where C is any vector. For, either way, the result is the 
 circuital part of C, or C r 
 
 These results, though puzzling at first from their variety, 
 are yet capable of being brought under rapid- mental control 
 by bringing them together in a compact form. Thus, start 
 with any circuital vector A r Let B x = curl Ap O x = curl Bp 
 Dj =cur!0 1 , &c. We have a series of vectors 
 
 Aj, B I} Op Dp Ep . . . 
 
 which are all circuital, and any one of which is the eurl of the 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 209 
 
 preceding. We thus pass down the series one step at a time 
 by means of the operation of curling ; for example, 
 
 D^curlOi (224) 
 
 If, however, we wish to go down two steps, we do not need 
 to go first one step, as above, and then another, also as above ; 
 but can make a double step in one operation by means of the 
 Laplacean v 2 . Thus, 
 
 -VSB^Dj (225) 
 
 Now go the other way. If we wish to rise up two steps, 
 we can do it in one operation by potting ; thus, 
 
 B^potDi (226) 
 
 If we wish to go up only one step, we may do it by (224), 
 (225), (226) combined ; that is, either go down one step first, 
 and then up two, as in 
 
 B 1 = pot Dj = pot curl G l - ) ... (227) 
 or else, first go up two steps and down one j thus, 
 
 B x = curl A! = curl pot O r . . . (228) 
 
 There are other less important combinations. But if we 
 wish to make one step up directly, without making use of the 
 double step, we must do it by the Amperean formula, already 
 used, whereby we pass direct from electric current to its 
 magnetic force, which, in rational units, is (when applied to 
 any pair of neighbours G l and D x in the above series), 
 
 C 1 = 2(VD 1 r 1 )/47rr2, .... (229) 
 
 where i^ is a unit vector from the element D 1 to the point at 
 distance r therefrom, where G 1 is reckoned. We have now a 
 complete scheme, so far as the circuital vectors are concerned. 
 
 A Series of Divergent Vectors. 
 
 135. Deferring temporarily a vectorial proof of the last 
 formula (229), which is the only unproved formula in the con- 
 nections of the series of circuital vectors, it will now be 
 convenient to bring together the connections of the divergent 
 vectors and associated quantities. We saw the advantage of 
 the systematic arrangement of the connected circuital vectors 
 
210 ELECTROMAGNETIC THEORY. CH. III. 
 
 to be like producing a harmonious chord out of apparently 
 disconnected tones. The advantage is much greater in the 
 divergent series, on account of the less uniform relations in- 
 volved and the greater need of a system to bring them under 
 rapid mental control. In the circuital series, four kinds of 
 operation were involved ; but in the divergent series there are 
 six. The chord will be found to be perfect, though of greater 
 complexity. 
 Thus, let 
 
 A 2 , B 2 , C 2 , D 2 , E,, . . . . 
 
 be a series of vectors and scalars connected as follows : Start 
 with A 2 , which is to be any divergent vector ; that is, havinr 
 no curl. Let B 2 be its divergence ; 2 the slope of B 2 ; D 2 
 the divergence of 2 ; E 2 the slope of D 2 , &c. Then A 2 , C 2 , 
 E 2 , . . . are all divergent vectors. But they are separated 
 from one another by two steps instead of one, as was the case 
 in the circuital series last treated. The intermediate quantities 
 are scalars. Instead, also, of the single operation of curl 
 which suffices, in the circuital series, to carry us from any 
 vector to the following one, we now have two distinct opera- 
 tions ; viz., that of slope, when we pass from a scalar to the 
 next vector, as in 
 
 C 2 =-VB 2 ; (230) 
 
 and that of divergence, when we pass from a vector to the next 
 scalar, as in 
 
 D 2 = div0 2 (231) 
 
 But if we wish to go down two steps at once, we can do so 
 by means of the Laplacean operator, whether the operand be a 
 scalar, as in 
 
 -V 2 B 2 = D 2 , (232) 
 
 or else a vector, as in 
 
 -V 2 C 2 = E 2 . ...... (233) 
 
 In this respect, then, we have the same property as in the 
 circuital series. 
 
 We have also identity of operation in going up two steps at 
 once, whether from a scalar to the next higher scalar, as in 
 
 B 2 = potD 2 , (234) 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 211 
 
 or from a vector to the next higher vector, as in 
 
 C 2 = potE 2 ; (235) 
 
 any member of the series being the potential of the second 
 after, as in the circuital series. 
 
 Next, to go up one step only, we may utilise the preceding 
 in two ways. First go up two steps and then down one, as in 
 
 B 2 = divA 2 = divpotC2, . . . (236) 
 
 where we pass from the vector C 2 to the scalar B 2 by pot 
 first (up two steps), and then by div (down one step) ; and 
 also as in 
 
 2 = -VB 2 = -VpotD 2 , . . . (237) 
 
 where we pass from the scalar D 2 to the vector 2 by pot first 
 (up two steps), and then by - y (down one). 
 
 Or, secondly, we may first go down one step and then up 
 two, as in 
 
 B 2 = potD 2 = potdiv0 2 , . . . (238) 
 
 when rising from the vector C 2 to the scalar B 2 ; or as in 
 
 2 = potE 2 = -potVD 2 , . . . (239) 
 
 when rising from the scalar D 2 to the vector C 2 . 
 
 Finally, if we wish to rise up one step at once, without using 
 the double step either up or down, we can do it by means of 
 
 C 2 = (div)- 1 D 2 = 2r 1 D 2 /47rr 2 , . . (240) 
 
 when we rise from a scalar D 2 to a vector 2 , which is, in fact, 
 the fundamental formula of the inverse- square law upon which 
 our potential investigations are based. But in rising from a 
 vector C 2 to the scalar B 2 just above it in the series, we require 
 to use a different process, namely, 
 
 B 2 = (-V)- 1 C 2 =-2r 1 2 /47rr2. . . (241) 
 
 In these formulas (240), (241), r x is a unit vector from the 
 element in the summation towards the place of the resultant ; 
 that is, from D 2 to C 2 in (240), and from C 2 to B 2 in (241). 
 
 We now have a complete scheme for the divergent vectors 
 as we had before for the circuital series. On comparing them 
 we see that they are alike in the double steps, either up or 
 down, but differ in the single steps. There is but one kind of 
 
 p2 
 
212 ELECTROMAGNETIC THEORY. CH. III. 
 
 step up and but one kind down in the circuital series, whereas 
 there are two kinds up and two kinds down in the divergent 
 series. The down step in the circuital series is always done by 
 curl; the up step, shown in (229), may be denoted by (curl)" 1 . 
 In the divergent series the down steps are done by - v and by 
 div; their inverses may be denoted by (-v)- 1 and (div)- 1 . 
 It is now the nature of these inverse operations (229), (240), and 
 (241) that remains to be elucidated vectorially. The first is 
 the Amperean formula rationalized, whereby we rise from 
 electric current to its magnetic force ; by the second we rise 
 from (for instance) electrostatic force to the electrostatic 
 potential ; or, with a slight change (of sign), from intensity of 
 magnetisation to magnetic potential ; in the third we rise 
 from (for instance), electrification to electrostatic displacement. 
 
 The Operation inverse to Divergence. 
 
 136. Let p and q be scalar functions, and consider the 
 space-variation of their product. We have 
 
 .... (242) 
 
 a formula not previously used, but which is seen to be true 
 by observing that it is true for each of the three components 
 of V. Now integrate through any region. We know that 
 
 2Np2 = SV(p2), ..... (243) 
 
 if N is the unit normal outwards from the boundary of the 
 region, so that the left member is a surface-summation, whilst 
 the right member is a volume-summation throughout the 
 region bounded by the surface. Equation (243) is, in fact, a 
 case of (157), with pq substituted for p. If, then, the surface- 
 summation vanishes, we shall have a simultaneous evanescence 
 of the right member of (243), and therefore, by (242), 
 
 2pVq=-2qVp. ...'.. (244) 
 
 All the work done by a vector-analyst is exhibited in (244) 
 itself, viz., the transfer of the symbol v from one operand 
 to the other with change of sign, converts the integral of 
 pvq into that of -gvp. The previous remarks contain the 
 justification of the process, 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 213 
 
 Now - vq is a polar or divergent vector, so may be any one 
 of our divergent series, say C 2 , when q itself becomes B 2 . Then 
 
 2^C 2 = 2B 2 V^. ..... (245) 
 
 Lastly, let p have the special value l/4;rr ; then (245) is the 
 same as 
 
 potC 2 = A 2 = 2B 2 V^, .... (246) 
 
 which exhibits the divergent vector A 2 in terms of its diver- 
 gence B 2 . It is the same as (240), since v# ij/lnr 2 , if r x is 
 the nnit vector from B 2 to A 2 . 
 
 The Operation inverse to Slope. 
 
 137. Next, substitute for q in pq a vector, say g. The 
 new quantity pg has, being a vector, both curl and divergence^ 
 in general. Considering the latter first, we have 
 
 div^?g =p div g + gVp, . . . (247) 
 
 which is an example of (160). Integrating throughout any 
 volume, we have 
 
 SNpg = 2divjpg, .... (248) 
 
 as in (159), where N is as before. So, if the surface-integral 
 vanishes, we obtain, by (247), 
 
 2^divg= -SgV^, .... (249) 
 
 and, in this transformation, all the vector-analyst has to do is 
 to shift the operator v from one operand to the other, and 
 change the sign. 
 
 The vector g here has no restriction imposed upon it. It 
 may therefore be of the general type = 0! + C 2 , giving 
 
 2^D 2 = - 2 GVp = - 2 G 2 Vp . . . (250) 
 
 Here the portion 2 G l Vp vanishes because Cj is circuital and yp 
 is polar, which is one of the important theorems in analysis 
 that become visibly true by following the tubes of the circuital 
 flux in performing the summation, when the summation is seen 
 to vanish separately for every tube. (See 87.) 
 
 If in (250) we give^? the special value l/4?rr, viz., the poten- 
 tial due to a unit source at distance r, we obtain 
 
 pot D 2 = B 2 = - 2 CW = - 2 G 2 Vp, . . (251) 
 
214: ELECTROMAGNETIC THEORY. CH. III. 
 
 showing how to pass up one step in the divergent series from 
 a vector to the preceding scalar. It is the same as (241), 
 remembering the value of Vp. 
 
 The Operation inverse to Curl. 
 
 138. Thirdly, we have the curl of pg to consider. Here, 
 by (188), 
 
 curing =#> curl g- VgVp, . . . (252) 
 
 Integrating throughout any region, we obtain 
 
 2VN^g = 2curl^g, .... (253) 
 
 which is a case of (162), with pgput for H. So, if the surface- 
 summation vanishes, we obtain, by (252), 
 
 2^curlg = 2VgVp, .... (254) 
 
 where the symbol VV is moved from g to p, with a change of 
 sign, as before. In this, take > = l/47rr; then, since there is 
 no restriction upon g, we get, taking g = C, 
 
 . . (255) 
 
 which is the companion to (251), showing how to pass up one 
 step in the circuital series, from O x to B r This is equivalent 
 to (229). The divergent part of contributes nothing. That 
 is to say, for example, the magnetic force due to a completely 
 divergent distribution of electric current, according to Ampere's 
 formula for the magnetic force of a current element, is zero. 
 We might, indeed, argue from this, that there could not be 
 such a kind of electric current ; that is to say, that the current 
 must be circuital, since the mathematical machinery itself* 
 constructed on old ideas, automatically rejects the want of 
 circuitality, and refuses to admit the purely divergent part 
 as contributory to magnetic force. This is a perfectly valid 
 argument, provided the test of the existence of electric current 
 be the existence of magnetic force, which is tantamount to 
 what Maxwell insisted upon, in another form. 
 
 For instance, if we calculate by (255) the magnetic force 
 due to a supposititious current element at a point, simply by re- 
 moving the sign of summation, we obtain the magnetic force of 
 a rational current element, a system of circuital current resem- 
 bling the induction due to a magnetised particle. (See 62.) 
 
ELEMENTS OP VECTOIUAL ALGEBRA AND ANALYSIS. 215 
 
 Remarks on the inverse Operations. 
 
 139. Returning to the three single up-step formulae, it will 
 be observed that in every case we base the proof upon the 
 vanishing of a surface-integral, viz., 2 Npq, or 2 Npg, or 2 VN>g. 
 Now it is obvious that these are true if the surface be taken 
 wholly outside the region occupied by the quantity q or g. The 
 value pq or pg is made zero all over the surface of summation. 
 This is what occurs in the applications made. In (246) the 
 space-summation has to include all the B 2 , which repre- 
 sents q; it is sufficient, therefore, for the surface to com- 
 pletely enclose all B 2 . Again, in (255), the enclosure by the 
 surface of all 0, which represents g, will ensure the vanish- 
 ing of the surface-integral; and in (251) the same. Now, 
 it is usual to imagine the surface to be at infinity, so that 
 the space- summations extend over all space. This is also most 
 convenient, in general. But caution is sometimes necessary 
 when the quantity to be summed extends to infinity. The 
 surface-summation then may, or may not, vanish, according to 
 circumstances. Even if the quantity summed becomes in- 
 finitely small at infinity, the summation may still not vanish. 
 To illustrate this, it is sufficient to mention the case of a single 
 point-source. The surface-integral of the flux it produces is 
 finite always, being the measure of the strength of the source. 
 At infinity the flux may be infinitely attenuated, but the sur- 
 face is simultaneously infinitely magnified. But in the appli- 
 cation of the above processes to practical cases in electro- 
 magnetism it is usually quite easy to see that the integral over 
 the surface at infinity vanishes, owing either to the actual 
 absence of anything to be summed, or else, when there is an 
 infinitely attenuated quantity to be summed, of its being, per 
 unit area, of lower dimensions than 1/r 2 . The quantity p 
 itself attenuates to nothing, as well as q or g, if they are 
 existent at all at infinity. 
 
 The three up-step formulae may, by inspection, be transformed 
 so as to involve entirely different operations. Thus, in (241), 
 put p for 1/471T. Then we have 
 
 B 2 =-pot(C 2 /r) (256) 
 
 Do the same in (240), and we obtain 
 
 C 2 = pot(D 2 /r). (257) 
 
216 ELECTROMAGNETIC THEORY. CH. III. 
 
 Finally, (229) gives 
 
 . . . . (258) 
 
 We may sometimes utilise these formulee for purposes of cal- 
 culation, should the integrations to be performed be amenable 
 to practical treatment. But there is a caution to be men- 
 tioned. The quantities whose potentials are calculated by the 
 last three formulae are not functions of position, which are de- 
 finitely distributed in space, and are independent of the position 
 of the point where the potential is reckoned, but are definite only 
 when this point is fixed. As you pass to another point, the 
 potential there is that of a different distribution from that 
 belonging to the first point. These quantities, C 2 /r, &c., are 
 therefore not subject to the various properties of the circuital 
 and divergent vectors already considered. For example, C 2 is 
 purely divergent, but D 2 /r has curl ; or, G 1 is circuital, whilst 
 VDj/r has divergence, and so on. 
 
 Integration "by parts." Energy Equivalences in the 
 Circuital Series. 
 
 140. The transformations (244), (249), (254), or the more 
 general ones containing the omitted surface-summations, ex- 
 pressed by putting the sign of summation before every term in 
 (242), (247), (252), and converting the summation on the left 
 to a surface-summation by the introduction of N, as in (243), 
 (248), and (253), are examples of what is, in the Cartesian 
 mathematics, called integration " by parts." It is usually a 
 very tedious and uninforming process, that is, in Cartesians, 
 with its triple and double /'s, its dS and dx dy dz, and its 
 I, m, n. The vectorial methods go straight to the mark at 
 once, avoid a large amount of quite useless work, and enable 
 you to keep your attention fixed upon the actual magnitudes 
 concerned, and their essential relations, instead of being dis- 
 tracted by a crowd of coordinates and components. 
 
 There are, of course, many other cases which arise of this 
 " by parts " integration. One of the most important in con- 
 nection with the circuital series of vectors, is the following : 
 Substitute for p in the preceding, a vector, say f, then we shall 
 have 
 
 2fcurlg = 2gcurlf, .... (259) 
 
 the summations being throughout all space, or, at any rate 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 217 
 
 through enough of it to include all the quantities summed. 
 The proof is that since 
 
 divVgf=fcurlg-gcurlf, . . . (260) 
 which is (178) again, we have, by space-summation, 
 
 2fcurlg = 2gcurlf+2divVgf, . . (261) 
 
 and the third term may be at once turned into the surface-sum- 
 mation 2 NVgf. If this vanish, as usual, then (259) follows. 
 
 Applying this result to the series of circuital vectors A 15 A 2 , 
 A 3 , &c., each of which is the curl of the preceding, we obtain 
 various equivalences. Thus, 
 
 2D 1 2 = 2C 1 E 1 = 2B 1 F 1 = 2A 1 G 1 , . . (262) 
 
 through all space. Operate on one member by curl and on the 
 other by (curl)- 1 , to pass from one form to the next. Also 
 
 2 D^ = 2 0^ = 2 6^ = 2 A^, . (263) 
 
 in a similar manner ; and so on. 
 
 In any of these summations we may convert either of the 
 circuital vectors involved to a general vector by adding any 
 divergent vector. For we see, by (259) that if f is polar, the 
 right summation vanishes, and therefore so does the left. So 
 2D X 2 is the same as 2DjD, for 20^ = 0. Similarly, 20^, 
 is the same as 2 OF lf or the same as 2 1 F, since 2 OjFg = 0, 
 and 2 CgFj = 0. It will be understood here that the suffix 1 
 refers to the circuital vectors entirely, and the suffix 2 to the 
 divergent vectors, and that = 0! + C 2 , &c. 
 
 But we cannot generalise both vectors at the same time. Take 
 OF for example. We have, 
 
 2 OF = 2 (Cj + 2 ) (F! + F 2 ) = 2 0^ + 2 2 F 2 , . (264) 
 
 so that the summation of the scalar product of the two diver- 
 gent vectors now enters. The series of divergent vectors and 
 their intermediate scalars have properties similar to (262), (263). 
 
 Energy and other Equivalences in the Divergent Series. 
 
 141. These properties of the space-integrals of scalar products 
 in the divergent series are formally obtainable from the corres- 
 ponding ones in the circuital series by changing the suffix from 
 
218 ELECTROMAGNETIC THEORY. CH. III. 
 
 1 to 2 ; at the same time changing the type from clarendon to 
 roman should the quantity typified be a scalar. Thus, analo- 
 gous to (262), (263), we have 
 
 2C 2 2 = 2B 2 D 2 = 2A 2 E 2 =..., . . (265) 
 2D 2 2 = 2C 2 E 2 = 2B 2 F 2 = ..., . . (266) 
 
 starting from the square of a vector in the first set, and from 
 the square of a scalar in the second. These transformations all 
 rest upon (249) ; that is, we pass from any form to dhe next 
 by the exchange of v and div between the factors. For 
 example, 
 
 2 D 2 2 = 2 D 2 div C 2 , by definition of D 2 , 
 
 = - 2 C 2 VD 2 , by integration, using (249), 
 = 2 2 E 2 , by definition of E 2 . 
 
 Similarly in all the rest. The preceding equations conveniently 
 summarise them. 
 
 But it will be observed that there is another way of pairing 
 terms in the divergent series, viz., a vector with a scalar. The 
 corresponding transformations do not work so symmetrically as 
 the previous. For instance, 
 
 2 C 2 D 2 = - 2 VB 2 .D 2 , by definition of C 2 , 
 - 2B 2 VD 2 , by (244), 
 = - 2 B 2 E 2 , by definition of E 2 , 
 = - 2 div A 2 .E 2 , by definition of B 2 , 
 = 2A 2 V.E 2 , by integration. . (267) 
 
 Here the symmetry breaks down. The last transformation 
 depends upon 
 
 2(Nf)g = 2(Vf)g = 2(gdivf+f v .g),. . (268) 
 
 which is a case of the theorem (147) or (153). Note that in 
 the second form V has to differentiate both f and g, so that the 
 full expression is in the third form. If the surface-integral 
 vanishes at infinity, we have 
 
 2gdivf= -2fV.g, .... (269) 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 219 
 
 which is the transformation used in getting (267). We have 
 also, by going the other way, 
 
 2C 2 D 2 = 2C 2 divC 2 =-2C 2 V.C 2 , . . (270) 
 
 by using the same transformation (269). Next take D 2 E 2 . 
 Here the vector is the slope of the scalar it is multiplied by, 
 whereas in the former case, the divergence of the vector was 
 the associated scalar. So it now goes quite differently. Thus, 
 
 2 D 2 E 2 = - 2 D 2 VD 2 = - 2 VJD 2 2 = - 2 N|D 2 2 . (271) 
 The result is therefore zero, if the surface-integral at infinity 
 vanishae. 
 
 The Isotropic Elastic Solid. Relation of Displacement to 
 Force through the Potential. 
 
 142. It usually happens that potentials present themselves 
 in physical mathematics as auxiliary functions introduced to 
 facilitate calculations relating to other quantities. But in the 
 theory of the elastic solid, the potential function presents itself 
 in a very direct and neat manner; besides that, the elastic 
 theory presents excellent illustrations of the above transforma- 
 tions and the general theory of v. If the solid be homo- 
 geneous and isotropic, there are but two elastic constants, the 
 rigidity n and the coefficient of resistance Tc to compression or 
 expansion. Let also m = k + Jra, in Thomson and Tait's nota- 
 tion ; then the equation of motion is 
 
 f + 7iV 2 G + wVdivG = /oG, . . . (272) 
 where G is the displacement, and f the impressed force per 
 unit volume, whilst p is the density. Whatever limitations 
 may need to be put upon the values of m and n in treating of 
 solids as we find them, in speculations relating to ethers they 
 may have any values not making the stored energy negative. 
 Thus m may, as we shall see presently, go down to the nega- 
 tive value - n, keeping n positive. 
 
 Split f into f x + f 2 , and G into G x + G 2 , where f x and G T are 
 circuital, and the others divergent. Then (272) may be split 
 into two equations, one circuital, the other divergent. Thus, 
 remembering (193), 
 
 pGy . . . (273) 
 
 /oG 2 . . . . (274) 
 
220 ELECTROMAGNETIC THEORY. CH. III. 
 
 In the circuital equation Jc does not appear, so that the propa- 
 gation of circuital disturbances depends only upon the rigidity 
 and density ; speed, (n/p)l. In the divergent equation both n 
 and k appear, and the speed of disturbances of this class is 
 {(n + m)/p}l. In another form, if we take the curl of (272), the 
 divergence goes out, so that the curl of G, which is twice the 
 rotation, has the curl of f for source, and is propagated inde- 
 pendently of compressibility. 
 
 And if we take the divergence of (272), we see that the 
 divergence of G, or the expansion, has the divergence of f for 
 source, and the rate of its propagation depends upon both 
 rigidity and compressibility. But when m = 0, there is but one 
 speed of propagation, and a complete amalgamation of the two 
 kinds of disturbances. 
 
 In equilibrium, when equilibrium is possible, the right 
 members of the last three equations vanish, since they repre- 
 sent "rate of acceleration of momentum" (a long-winded 
 expression). We therefore have 
 
 (275) 
 . . . (276) 
 
 The solutions are visible by inspection. We see that nGr l is 
 the potential of f lt and (n + m)G 2 the potential of f 2 . That is, 
 the displacement produced by circuital impressed force is its 
 potential divided by n ; and the displacement produced by 
 divergent impressed force is its potential divided by n + m ; 
 whilst, in the general case, we must split the impressed force 
 into circuital and divergent parts, and then re-unite them in 
 different proportions to obtain the resultant displacement. 
 
 Symbolically, 
 
 (277) 
 
 (278) 
 (279) 
 
 In the case of incompressibility Jc, and therefore m, is infinite, 
 so that G 2 is zero, and the displacement is pot fjn, whatever 
 f 2 may be. For f 2 is balanced by difference of pressure, which 
 is set up instantly. The corresponding speed is infinite, but 
 there is no displacement. 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 221 
 
 If, again, m = 0, and there is but one speed, the displacement 
 is the potential (divided by n) of the impressed force, whatever 
 be its type. 
 
 On the other hand, if ra + w = 0, we see by (274) that there 
 is no steady state possible due to divergent force, as f 2 is then 
 employed simply in accelerating momentum on the spot. The 
 corresponding speed is zero. The circuital disturbances are 
 propagated as before. This is the case of Sir W. Thomson's 
 contractile ether, in which the wave of normal disturbance is 
 abolished by making the speed zero. 
 
 The Stored Energy and the Stress in the Elastic Solid. The 
 
 Forceless and Torqueless Stress. 
 
 143. We may also find expressions for the stored energy 
 from the equation of motion. The work done and stored by 
 the impressed force f is JfG, though, if f be put on suddenly, 
 an equal amount is dissipated, since, G being the final displace- 
 ment, fG is the work then done by f, per unit volume. Now 
 
 by (264). Calling the first part U^ and the second part Ug, 
 we have, by (273), (274), 
 
 U 1= -2|wG 1 V 2 G 1 = 2Jn(curlG)2 .... (280) 
 U 2 = - 2 i(m + w)G 2 V 2 G 2 = 2 \(m + n) (div G) 2 . (281) 
 
 The transformations here used are (259) for Uj and (249) for 
 U 2 . We see that the energy stored is expressed in terms of 
 the squares of the rotation and of the expansion. In the 
 contractile ether U 2 is zero, and the stored energy is U r 
 We do not correctly localise the energy by the above formulae, 
 but only express the total amounts. For correct localisation 
 we need to know the stress and the distortion. Now the 
 distortion is a function of the variation of displacement, so is 
 known in terms of G. Can we, however, find the stress itself 
 from the equation of motion ? If not, we can come very close 
 to it. Thus, in a state of equilibrium, F + f=0, if F is the 
 force arising from the stress. So, by (275), (276), 
 
 ..... (282) 
 (283) 
 and F =/tV 2 G + wV 2 G 2 ..... (284) 
 
ELECTROMAGNETIC THEORY. CH. III. 
 
 Now let P N be the stress on the plane whose normal is N. 
 We have, by consideration of the equilibrium of a unit cube, 
 
 FN = divP N , (285) 
 
 to express the relation between an irrotational stress and the 
 force arising from it. Therefore, applying this to (284), 
 
 divP N = 7iV 2 GN + wV 2 G 2 N. . . . (286) 
 
 Here the divergence of P N is given ; find P N itself. The 
 immediate answer is, by (201), or (214), 
 
 PN= -VpotdivP N , 
 or P N = ttVGN + mVG 2 N, . . . (287) 
 
 provided P N is divergent. That is, we have constructed a 
 stress-vector giving the proper force required. But, without 
 interfering with this essential property, we might add on to 
 the right side of (287) any circuital vector. That we must do 
 so now, we may see by remembering that the stress must be 
 irrotational, or produce no torque, and then by finding that 
 (287) does give a torque. To show this, consider the first 
 part only of the stress (287), say with m = 0. Then 
 
 NP M - MP N = 7i(NV.MG - MV.NG). 
 Here take M=j and N = k; then 
 
 kPj - jP k = tt(V 3 G 2 - V 2 G 3 ) = - ni curl G, 
 
 by (149). This gives the i component of the torque, which is 
 therefore - n curl G. But G 2 has no curl, therefore the second 
 part of (287) produces no torque. We require, therefore, to 
 add to P N in (287) a stress giving no force, but a torque n curl G. 
 Such a stress is 
 
 X N = ncurlVGN (288) 
 
 as may be tested in the above manner. 
 This brings us from (287) to 
 
 P N = W (VGN + curl VGN) + mVG 2 N, . . (289) 
 
 which is a stress-vector giving the correct force and no torque. 
 [It should be noted here that since (285) applies to an irro- 
 tational stress, the process employed is only justified when we 
 finally get rid of the torque, as in (289), (290). If the stress is 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 223 
 
 rotational, the divergence of P N is really the N component of 
 the force due to the conjugate stress. Thus the force due to 
 nv.GN is nVdivG, and the torque ncurlGr. The force due 
 to ?ftV.G 2 N is mvdivG, with no torque. The force due to 
 n curl VGN is ncur! 2 G, and the torque n curl G. Adding 
 together the three stresses and the three forces we obtain the 
 stress (289), with no torque and the correct force.] 
 
 But on comparison with the real stress deduced from the 
 elastic properties of a solid, which is 
 
 P N = tt(V-GN + NV.G) + N(m-rc)divG . . (290) 
 
 we see that they do not agree. Yet they can only differ by 
 a stress which produces neither force nor torque. And we 
 know already, by (288), that if G = G 2 , the modified stress will 
 give no force or torque. In fact, on comparing (289), (290), 
 we find that their difference is of this nature, (290) being 
 equivalent to 
 
 P N = rc(VGN + curl VGN) + m(VG 2 N - curl VG.N), (291) 
 
 where the first and third terms on the right are sufficient to 
 give the correct force, but with a torque, which is, however, 
 cancelled by the second term ; whilst the fourth term is 
 apparently (so far as force and torque are concerned) like a 
 fifth wheel to the coach, off the ground. If we inquire under 
 what circumstances the real stress can assume this singular 
 form, we shall find that m + n = and curl G = will do it. 
 With these conditions, only the circuital part of P N is now left, 
 and (291) reduces to 
 
 P N = 2curlVG 2 N (292) 
 
 It is the case of irrotational displacement in the contractile 
 ether, previously referred to, and is entirely remote from the 
 real elastic solid. The noteworthy thing is that we cannot 
 apparently conclude what the stress is, even when the force 
 and torque to correspond are everywhere given, owing to the 
 forceless and torqueless stress coming into the formulae. The 
 form (292) is convenient for showing at sight that there is no 
 force. It may, by (181), remembering the constancy of N, be 
 expanded to 
 
 P N = 2w(NV.G 2 -NdivG 2 ), , , . (293) 
 
224 ELECTROMAGNETIC THEORY. CH. III. 
 
 where put N = i, j, k in turns to get the three stresses on the 
 planes having these normals. One-third of the sum of the 
 normal tractions on these planes is - (4^/3) div G 2 , or, which is 
 the same, + Jc div G. It is the negative of the pressure. But 
 in the real stress (291) itself, the fourth part, from its having 
 the negative sign prefixed, correctly associates pressure and 
 compression. But this forceless and torqueless stress does not 
 contribute anything to the total energy. The amount of 
 stored energy is 
 
 = - J2 (F^ + F 2 G 2 + F 3 G 3 ) 
 
 div P 2 + G s div p s) 
 P 3 VG 3 ), .... (294) 
 
 where the first line needs no remark, the second is got by 
 (285), and the third by the common transformation (249), P p 
 etc., being the stresses on the i, j, k planes. The final form in 
 (294) shows the correct distribution of the energy in terms of 
 the stress and the distortion. Now, if in the stress P N be in- 
 cluded any terms giving rise to no force, the above transforma- 
 tion shows that they contribute nothing on the whole to the 
 energy of distortion. This is the case with the fourth term in 
 (293), and would also be the case with the second term, only 
 that we have no right to ignore it, on account of the torque 
 thereby brought in. If the solid be not unbounded, it is 
 sufficient for its boundary to be at rest for the same principles 
 to apply. 
 
 [The practical meaning is that in the contractile ether the 
 energy of distortion due to any irrotational displacement is 
 zero on the whole, the sum of the positive amounts in certain 
 parts being equal to the sum of the negative amounts in the 
 rest.] 
 
 Other Forms for the Displacement in terms of the Applied 
 
 Forcive. 
 
 144. The simple solution (279) may, of course, receive 
 many other forms. If it be desired to find the displacement 
 due to an explicitly given impressed forcive, it is a matter of 
 some importance to select a method of obtaining it which shall 
 not be unnecessarily difficult in execution j for different pro- 
 
ELEMENTS OP VECTORIAL ALGEBRA. AND ANALYSIS. 225 
 
 cesses leading to the same result may vary greatly in readiness 
 of application. Now, if the impressed forcive be either circuital 
 or divergent, we do not need to modify. But if of a mixed 
 type, then it may be desirable. Put t l = t- f 2 , then an alter- 
 native form is 
 
 * & J* ^ 
 
 ; . . , (295) 
 
 and now, when f is given, it is only the part pot f 2 that needs 
 development. We have 
 
 . . (296) 
 as before explained. Now, if we notice that 
 
 W = 2/r, ..... (297) 
 
 which may be proved by differentiating, we can further con- 
 clude that 
 
 . . (298) 
 
 a very curious result. The potential of a radial vector following 
 the inverse-square law of intensity is a radial vector with a 
 constant tensor in all space. Using this result, we have 
 
 .. (299) 
 so that (295) becomes 
 
 . . (300) 
 
 n( 
 
 wherein f alone appears on the right side. 
 Another form is got by using 
 
 f 2 = V2fV ? 7, /. potf 2 = Vpot2fVp. . (301) 
 But a better one is 
 
 potf 2 = 2/v^/87r, .... (302) 
 as 
 
 where s is length measured along f, and / is the tensor of the 
 latter. This makes 
 
 O-Ipotf 5/?^Z/v* . . . (303) 
 
 n n 
 
 which is generally suitable for practical calculation. 
 
 Q 
 
226 ELECTROMAGNETIC THEORY. CH. III. 
 
 A considerably more complex form is given in Thomson and 
 Tait. It may be obtained from (303) by means of the identity 
 
 where s l is unit s, and therefore parallel to f, making 
 
 . 
 
 ds r 
 
 the same as Thomson and Tait's formulae when expanded in 
 cartesians. But this is a gratuitous complication, as (303) is 
 simpler in expression and in application. Of course (300) is 
 simpler still in expression, and the practical choice may lie 
 between it and (303), or (295). 
 
 Notice that an impressed forcive of the divergent type with 
 a single point-source produces not merely uniform radial dis- 
 placement, as per (298), but an infinite discontinuity, in fact, a 
 disruption, at the source itself. It is a very extreme case, of 
 course. 
 
 The Elastic Solid generalised to include Elastic, Dissipative, 
 and Inertial Resistance to Translation, Rotation, Ex- 
 pansion, and Distortion. 
 
 145. The elastic solid with two elastic constants (k and n) 
 has not been found sufficiently elastic to supply a thoroughly 
 satisfactory analogy with Maxwell's ether, though partial 
 analogies may readily be found. Other kinds of elasticity 
 than resistance to compression and distortion, and other kinds 
 of resistance than elastic resistance, present themselves to the 
 consideration of searchers for analogies between the propa- 
 gation of disturbances in Maxwell's ether and in the brutally 
 simple elastic solid of theory, which is, however, known to 
 fairly represent real solids within a certain range. 
 
 There are four distinct ideas involved in the displacement 
 of a small portion of matter, viz., the translation as a whole, 
 the rotation as a whole, the change of size, and the change of 
 shape. These separate themselves from one another naturally. 
 As regards the mere translational motion, if it be only iner- 
 tially resisted, we have the equation of motion 
 
ELEMENTS OP VECTOBIAL ALGEBRA AND ANALYSIS. 227 
 
 (306) 
 
 where p is the density, q. the velocity, f impressed force (per 
 unit volume), F tfae force arising from the stress associated 
 with the strain (including rotation, distortion, and expansion), 
 and 8/St the time-differentiator for moving matter. This equa- 
 tion is constructed on simple Newtonian principles. We may, 
 however, wish to have elastic and frictional resistance to trans- 
 lation, as well as inertial resistance. Then generalize the 
 above to 
 
 f +F- ( ft + ft * +*)<, . . . (307) 
 
 where G is the displacement, when small departures from 
 equilibrium are concerned. Here p 2 is the p of (306), p l is 
 the frictionality (Lord Kelvin's word for coefficient of friction), 
 and p Q is an elastic constant. The displacement from equi- 
 librium calls into action a back force p G proportional to the 
 displacement, with storage of potential energy ; a force /a 1 G- 
 proportional to the velocity, with waste of energy ; and a force 
 /o 2 G proportional to the acceleration, with associated kinetic 
 energy. The potential energy is J/o G 2 , the rate of waste ftG 2 , 
 and the kinetic energy Jp 2 G 2 . 
 
 As regards the force F, this is given by (284), when the stress 
 is irrotational, and the elastic constants are n and k. But, in 
 general, we may proceed thus. First separate the rotation 
 from the strain vector. We have 
 
 NV.G = J(NV.O- + V.NG) + J(NV.G - V.NG) 
 
 = p N -JVNcurlG (308) 
 
 Here Ny.G means the variation of displacement per unit dis- 
 tance along N, which is any unit vector, so that by giving N 
 all directions (practically only three) the complete state of 
 strain is known. This strain vector is above analysed into the 
 strain p N without rotation, and the second part depending upon 
 rotation. But p N includes the expansion, as well as the dis- 
 tortion, or mere change of shape. To exhibit the distortion 
 without expansion, one-third of the expansion (vectorised) must 
 be deducted. Thus 
 
 NV.G = (p N - IN div G) + iN div G - JVN curl G (309) 
 
 Q2 
 
228 ELECTROMAGNETIC THEORY. CH. III. 
 
 shows the separation of the strain into a distortion (without 
 change of size), an expansion, and a rotation, which are 
 naturally independent, 
 
 If distortion, expansion, and rotation are all elastically 
 resisted, three independent elastic constants (in an isotropic 
 medium) intervene between the above strain and the corre- 
 sponding stress P N upon the plane whose normal is N. To 
 obtain P N , multiply the distortional part by 2n, the expansional 
 part by 3&, and the rotational part by 2v, and add the results. 
 Thus, 
 
 PN = 2/t(p N - JN div G) + Nfc div G - vVN curl G. (310) 
 
 Compare with (290). Here n and k are as before, whilst v is a 
 new elastic constant connected with the rotation, and which 
 was previously assumed to have the value zero. That is, there 
 was assumed to be no resistance to rotation. The corres- 
 ponding torque is 
 
 S = 2vcurlG ...... (311) 
 
 With the rotational part of the stress is also associated trans- 
 lational force, given by 
 
 - curl JS= -v curl 2 G. . . . (312) 
 
 Adding this on to the former expression for the force (or 
 deriving the force from (310) directly), we find that 
 
 F = w(V 2 G + JV div G) + &V div G - v curl 2 G (313) 
 
 represents the translational force to be used in the equation of 
 motion (307). We should notice here that the quantity 
 k div G in (310) represents a uniform tension. It equals one- 
 third of the sum of the normal tractions on any three mutually 
 perpendicular planes. Its negative represents the pressure, or 
 p = - k div G. Now when there is incompressibility, k is 
 infinite, and div G is zero. But their product usually remains 
 finite. So in any case we may replace the k term in (310) by 
 - Np, and the k term in (313) by - Vp. 
 The energy of the strain is 
 
 . . (314) 
 
 or one-half the sum of the scalar products of the stress vector 
 and the strain vector for three perpendicular planes, P v P 2 , P 3 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 229 
 
 being the stresses on the i, j, k planes respectively. On 
 reckoning up, by (310), (309), we find the energies of expan- 
 sion, rotation, and distortion are all independent, and that U 
 is their sum, given by 
 
 U = n[ Pl 2 + p 2 2 + p 3 2 _ l(div G)2] + J&(div G)2 + 1 v(curl G)', (315) 
 
 where the v term is the energy of rotation ( = \ torque x 
 rotation), the Jc term is the energy of expansion, and the n 
 term the energy of distortion. 
 
 Now the stress (310) is derived from the strain (309) by 
 the introduction of elastic resistances only. There is, however, 
 no reason why we should limit the nature of the resistance in 
 this way. Consider the n term only for example, relating to 
 distortion. There may also be dissipative or frictional resist- 
 ance to distortion. To exhibit it, change n to n Q + n^d/dt) in 
 (310), (313). Then n Q will be the rigidity and n the viscosity, 
 or coefficient of frictional resistance to distortion. For instance, 
 if we abolish n and v, and retain n v and k, we have the stress 
 in a real viscous fluid according to Stokes's theory. It may be 
 that there is not a complete disappearance of the rigidity in 
 a fluid ; if so, then retain both n and n v When the rigidity 
 is marked, as in a solid, n Q is the important part of n ; whilst 
 in a fluid it is the other part. It is, however, not at all to be 
 expected that the expression of the viscosity of solids by n v if 
 true at all, would extend beyond the small range of approxi- 
 mately perfect elasticity. Lord Kelvin's experiments on the 
 subsidence of the oscillations of wires tended to show that a 
 different law was followed than that corresponding to the 
 viscosity of fluids, and that elastic fatigue was concerned in 
 the matter. 
 
 There may also be, conceivably, inertial resistance to dis- 
 tortion, and not only elastic, but also frictional and inertial 
 resistance to rotation, and the same with respect to expansion. 
 These will be brought in by generalising the three elastic 
 constants k, n, and v of (310) thus : 
 
 . . (316) 
 
230 ELECTROMAGNETIC THEORY. CH. III. 
 
 where of the nine coefficients on the right side; the first three 
 are elastic constants, the second three dissipative constants, the 
 last three inertial constants ; the elastic constants involving 
 potential energy, the dissipative constants- waste of energy, and 
 the inertial constants kinetic energy, similarly to (307) as 
 regards translation. Out of these twelve constants only four 
 are in use, or seven, if v and v,, and p-^ be included, which 
 additional have been speculatively employed. This does not 
 represent finality by any means ; but sufficient has been said 
 on the subject of generalising the elastic constants to emphasize 
 the fact that there is plenty of scope for investigation in the 
 theory of the motion of media of unknown internal constitution, 
 which, externally viewed, involve the four elements translation, 
 rotation, expansion and distortion. 
 
 The activity of the stress vector P N is P N q, and the flux of 
 energy is (see 68, 72 also), if q be the speed, or tensor of the 
 velocity, 
 
 Qtf, ..... (317) 
 
 where U and T are the complete potential and kinetic energies, 
 per unit volume, and Q N is the stress conjugate to P N , to 
 obtain which change the sign of the last (rotational) term in 
 (310). The convergence of the energy flux represents the 
 work done and stored or wasted in the unit volume. Consider- 
 ing the term - Q g ^ only (the rest being the convective flux 
 of energy), its convergence is 
 
 div (Q l2l + Q 2 ? 2 + Q 3?3 ) = VjtoQ) + V 2 (P 2 q) + V 3 (P 3 q) 
 
 - & div Qi + & div Q 2 + q 3 div Q 3 + Q x Vft + Q 2 V? 2 + Q 3 V<? 3 
 
 . . (318) 
 
 where P is the force as in (313). Fq. is the. translational 
 activity, which may, by (307), be employed in increasing kinetic 
 and potential energy or be wasted. The rest represents the 
 rate of increase of the sum of the three potential energies when 
 there is merely elastic resistance but if we use (316) we 
 obtain also the sum of the rates of waste of energy, through 
 &P n v v v and also the rate of increase of the sum of the kinetic 
 energies, through & 2 , n 2 , v 2 . 
 
ELEMENTS OP VECTOBIAL ALGEBRA AND ANALYSIS. 231 
 
 Thus we have, by (310), 
 1 V 1 + P 2 V 2 + P s V 3 )q 
 
 = ^ ( 2n(p, - Ji div G) + M di v G - vVi curl G ) 
 dx\ / 
 
 - Jj div G) + Jcj div G - vVj curl G 
 
 + 2(p s - Jk div G) + k div G - vVk curl G 
 dz\ / 
 
 = 2w(p, VjCt + P 2 V 2 q + p 3 V 3 q) - ^ div G div a 
 
 + & div G div q + v curl G curl q . . . (319) 
 
 Take for illustration, the rotational term v curl G curl q. This 
 is simply (dldt) {}>v (curl G) 2 }, or the rate of increase of the 
 rotational energy, when v is a constant, but when we use the 
 last of (316) it becomes 
 
 ) + v^curl a) + ^(i" 2 (rl a)*), (320) 
 
 showing the rates of increase of potential and kinetic energy, 
 and the rate of waste (the middle term) due to rotational 
 friction. Similarly as regards the other terms in (319). The 
 expansion terms give 
 
 ), . (321) 
 
 where the middle term is the rate of waste, and the others 
 rates of increase of potential and kinetic energies. Finally, 
 the distortional terms of (319) give 
 
 (p, 2 + pV + p' 3 2 - 
 
 (322) 
 
 where the first line is the rate of increase of the potential 
 energy, the second line the frictional rate of waste, and the 
 third line the rate of increase of kinetic energy. We thus 
 complete the energy relations of the stress with the generalised 
 
232 ELECTROMAGNETIC THEORY. CH. III. 
 
 elastic constants, and so far as small motions are concerned, 
 the equations are manageable. 
 
 Electromagnetic and Elastic Solid Comparisons. First 
 Example : Magnetic Force compared with Velocity in 
 an incompressible Solid with Distortional Elasticity. 
 
 146. Comparisons between the propagation of electromag- 
 netic disturbances in Maxwell's ether, or in a homogeneous 
 isotropic dielectric, whether conducting or not, and the propa- 
 gation of motion in an elastic solid, either simple or generalised, 
 may be made in a variety of ways. They all break down 
 sooner or later, but are nevertheless useful as far as they go. 
 A few cases will be now considered, based on the preceding. 
 First take the case of a non-conducting dielectric at rest, and 
 compare it with the regular elastic solid made incompressible 
 by infinite resistance to compression. The incompressibility 
 includes inexpansibility. 
 
 In the dielectric we have, if p stands for d/dt, for convenience, 
 
 curl(H-h) = cpE, curl (e - E) = /*pH, . (323) 
 
 the circuital equations connecting E and H ( 38, 24, 66), 
 where e and h are impressed, and //., c are the inductivity and 
 permittivity. Now, the equation of motion in an incompressible 
 solid is, by (273), 
 
 . . (324) 
 
 where p and n are the density and rigidity, G the displacement, 
 q. the velocity, and f x the circuital part of the impressed forcive, 
 the divergent part being inoperative. Or we may take the 
 impressed forcive to be circuital to begin with. 
 
 Now, let h = 0, and e be finite. If it has no curl, it does 
 nothing, as before shown ( 89). The source of disturbance is 
 curl e. This being the case in the dielectric, and f x being the 
 source of motion in the solid, it will be convenient to take f x 
 and curl e as corresponding. Let the latter be (temporarily) f. 
 Then (323) gives (remembering that div H = 0), 
 
 f = upB. + curl E = /^H + curl 2 H/cp, 
 
 f=w - H = ^ 2 -- ' ' (325) 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 233 
 
 Comparing this with the second of (324), and assuming the 
 equality of f and f 1? we see that H is velocity, /* density, c~ l 
 rigidity ; and so on. IL/p means the time-integral of H. Or, 
 more explicitly, let Z be the time-integral of H, so that H=>Z; 
 then 
 
 f=(^2-c-iV 2 )Z (326) 
 
 which compares with the first of (324). Thus 
 
 Z = H/p corresponds to G, (spacial displ.) 
 (mag. force) H = pZ q, (velocity), 
 
 (el. current) curl H curl q, (2 x spin), 
 
 (el. displ.) D = cE curl G, (2 x rotation), 
 
 (inductivity) /* p, (density), 
 
 (permittivity) c ,, l/n, (compliancy), 
 
 (mag. source) f= curie f p (impd. force), 
 
 (mag. energy) J//H 2 |pq 2 , (kin. energy). 
 
 This is, as far as it goes, an excellent analogy, particularly on 
 account of its directness and ease of application, and on account 
 of the similarity of sources. The disturbances of H generated 
 by f in the dielectric and propagated away at speed (/*c)~* are 
 precisely represented by the velocity generated by similarly 
 distributed impressed Newtonian force, which is propagated in 
 the same manner through the solid at the equivalent speed 
 (n/p)%. Of course the correspondence at similar moments also 
 applies to the other quantities compared. 
 
 But it is imperative (in general) that the motions in the 
 solid should be small. The disturbances should therefore be 
 of the fluctuating or alternating character, so that Z, the 
 time-integral of H, does not mount up. For, correspondingly, 
 if this were allowed, then G would mount up, and the elastic 
 solid get too much out of shape to allow us to treat d/dt at a 
 fixed point and 8/8t for a moving particle as identical. 
 
 Observe also, that the electric energy -JcE 2 is not matched 
 by the potential energy of distortion ; although their total 
 amounts are equal, their distributions are entirely different. 
 This is a fatal failure in detail. Nor have we any right to 
 expect that a scheme like Maxwell's, depending upon rotations, 
 can be perfectly matched by one which depends on shears. 
 
234 ELECTROMAGNETIC THEORY. CH. III. 
 
 Second Example: Same as last, but Electric Force 
 compared with Velocity. 
 
 147. Another analogy of the same class is got by making 
 the displacement in the solid represent the time-integral of E 
 in the dielectric. Thus let e = G in (323), and - curl h = + g. 
 It is now g that is the source. Eliminate H between the two 
 equations (323). We get 
 
 epE + g = curl (- curl E)//*p. . . . (327) 
 Or, since divE = (when there is no electrification) 
 
 g-(q,-2!Wg*-2_V . . (328) 
 
 \ ftp/ \ [A/ 
 
 if A = E/p, making E = pA. 
 
 Comparing (327) with (324) for the solid, we see that we 
 have E representing velocity, ft compliancy, c density, and so 
 on, being a general turning over of all the quantities. Thus 
 
 A = E/p corresponds to G (displacement), 
 
 c p, 
 
 (el. force) E q, (velocity), 
 
 H racurlG, 
 
 curl h f x (circuital impd. force). 
 
 The conclusions are similar as regards the generation and 
 propagation of disturbances of the two kinds compared. There 
 is a similar failure to before as regards the energy of dis- 
 tortion. It ought to be the magnetic energy J/*H 2 now, but 
 it is not. This want of correspondence as regards one of the 
 energies will be removed (and can only be removed) by doing 
 away with the rigidity and substituting something else, which 
 must, however, be equivalent in its results as regards the 
 propagation of disturbances from place to place. 
 
 Third Example : A Conducting Dielectric compared with a 
 
 Viscous Solid. Failure. 
 
 148. As we see from the preceding that the propagation 
 of electromagnetic disturbances through a non-conductor can 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSTS. 235 
 
 be imitated in an incompressible solid, possessing the usual dis- 
 tortional rigidity, let us next introduce electric conductivity 
 on the one hand, and examine what changes are needed in the 
 analogies of 146, 147 to keep them working, if it be possible. 
 Since there is waste of energy associated with the conduction 
 current, it is clear that some sort of frictional force requires to 
 be introduced into the elastic solid. 
 
 First try distortional friction. The equation of motion of 
 an incompressible viscous solid is 
 
 fi-^ 2 -(o + Mv 2 ]Gi ' (329) 
 
 where f x is the impressed force per unit volume, G the displace- 
 ment, p the density, n Q the rigidity, and n^ the associated fric- 
 tionality. Also, p stands for d/dt, for practical convenience in 
 the operations, whether direct or inverse. Thus, p~\ or qjp 
 means the time-integral of q, which would in the ordinary 
 
 notation of the integral calculus be expressed by / qcft, a nota- 
 
 J o 
 
 tion which is not convenient for manipulative purposes in in- 
 vestigations of this class. We may derive (329) from (324) by 
 changing n to n^ + n^ ; or by (306), (313). [See also p. 229.] 
 
 Now the circuital equations of E and H in a conducting 
 dielectric (homogeneous, isotropic, stationary), are 
 
 curl (H - h) = (k + cp) E, curl(e-E)=/xpH, . (330) 
 
 comparing which with (323) we see that the effect of introducing 
 conductivity is to change cp to k + cp, so that the equation of 
 H is got by making this change in (325), giving 
 
 < < 331 > 
 
 where Z = H/p. But on comparing this equation with (329), 
 we see that the viscosity in the one case and the conductivity 
 in the other enter into the equations in different ways. We 
 can only produce a proper correspondence when the coefficients 
 of y 2 are equal; or, in another form, when the operators 
 p(k + cp)~ ! and (n Q + n } p) are equivalent in effect. This is not 
 generally possible. But there are two extreme cases of agree- 
 ment. One we know already, viz., when there is no conduc- 
 
236 ELECTEOMAGNBTIC THEORY. CH. Ill 
 
 tivity and no viscosity, as in 146. The other is when there is 
 no permittivity and no rigidity, which is sufficiently important 
 to be separately considered. 
 
 Fourth Example: A Pure Conductor compared with a 
 Viscous Liquid. Useful Analogy. 
 
 149. This is the extreme case of a pure conductor, or a 
 conductor which cannot support elastic displacement. It dissi- 
 pates energy, but does not store it electrically ; though, on the 
 other hand, its magnetic storage capacity is retained. It 
 is compared with an incompressible viscous solid with the 
 rigidity abolished; that is to say, with an incompressible 
 viscous liquid. Put c = in (331), and we have the equation 
 
 f=(/^-ArV)H; . . (332) 
 
 and putting n Q = in (329), with q the velocity substituted for 
 p&, gives us the equation 
 
 (333) 
 
 in the liquid. These admit of immediate comparison. Observe, 
 however, the curious fact that whereas in 146 the permit- 
 tivity was the reciprocal of the rigidity, so that the vanishing 
 of one means the infinitude of the other, yet now both the 
 permittivity and the rigidity vanish together, as if they were 
 equivalent. This emphasises the incompatibility of (329), (331). 
 Our present analogy must stand by itself, all idea of permit- 
 tivity and rigidity being thrown away. 
 
 We compare magnetic force H in a pure conductor with the 
 velocity q in a viscous liquid ; the inductivity ft with the 
 density of the liquid, so that the magnetic energy and the 
 kinetic energy are compared, and the current is twice the 
 spin. Also the source f = curie in the case of the conductor is 
 compared with circuital impressed force in the liquid. So far is 
 the same as in 146. But now, in addition, we have the electric 
 resistivity k~ l represented by the liquid viscosity. We conclude 
 that the disturbances q generated by f,_ in the liquid are similar 
 to the disturbances H in the conductor generated by f, and 
 that the propagation of electrical disturbances in a conductor 
 is like that of motion iii a viscous incompressible liquid. It 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 237 
 
 takes place by the process called diffusion. It is the limiting 
 case of elastic wave-propagation, with distortion and dissipation 
 of energy by friction. 
 
 We are usually limited to small motions in this analogy, so 
 that the impressed forces should not act in one sense continu- 
 ously, but should fluctuate about the mean value zero, for the 
 reason mentioned before. But we are not always limited to 
 small motions. In certain symmetrical distributions of mag- 
 netic force and current we may remove this restriction 
 altogether. In the liquid case (333) p strictly means 8/8t, 
 whilst in (332) it means d/dt. In the former a moving particle 
 is followed ; in the latter we keep to jane place. But in the 
 cases of laminar flow referred to, S/St reduces to d/dt readily 
 enough, so that the analogy is carried on to cases of steadily 
 acting impressed forces. 
 
 The analogy is an important one, owing to the readiness with 
 which the setting in motion of water by sliding friction can be 
 followed, as when a wind blows over its surface. Two interest- 
 ing cases are the penetration of magnetic induction and (with 
 it) electric current into a core enveloped by a solenoidal coil, 
 in whose circuit an impressed force, variable or steady, acts ; and 
 the penetration of magnetic induction and electric current into a 
 straight wire when an impressed force acts in its circuit. Consider- 
 ing the latter we may state the analogy thus. Understanding, 
 in the first place, that the circumstances should be such that 
 the influence of electric displacement outside the wire is negli- 
 gible, we may replace the wire by a similar tube of water, and 
 then replace the impressed voltage in the electric circuit by a 
 uniform tangential drag upon the surface of the water in the 
 direction of the length of the pipe. The current of water 
 in the pipe and the electric current in the wire, will 
 vary similarly under the action of similarly varying impressed 
 forces. Thus, a steadily applied force on the water will first 
 pull the outermost layer into motion; this, by the viscosity, 
 will pull the next layer, and so on up to the axis. The initial 
 current is purely superficial ; a little later we have a central 
 core in which the water is practically motionless, surrounded 
 by a tubular portion whose outermost layer is moving 
 rapidly, and innermost very slowly ; later still, the whole mass 
 is in motion, though less rapidly at the axis than at the boundary; 
 
238 ELECTROMAGNETIC THEORY. OH. III. 
 
 finally, the whole mass of water acquires a state of uniform 
 motion. Substituting electric current for current of water, we 
 obtain a representation of the way the electric current is set 
 up in a wire, passing through the various stages from the initial 
 surface current to the final uniformly distributed current. If 
 the impressed force be rapidly oscillatory, we stop the penetra- 
 tion as above described in its early stage, so that if the frequency 
 be great enough, there is a practical confinement of the current 
 (in sensible amount) to the skin of the wire (or of the water 
 respectively). But if the mean value of the oscillatory force 
 be not zero, it is the same as having two impressed forces, one 
 steady, the other periodic, without any bias one way or the 
 other. Then we have finally a steady current throughout the 
 wire, plus the oscillatory current with superficial concentration. 
 When a straight core is magnetised in a solenoid, it is the 
 magnetic induction which is longitudinal, or parallel to the axis. 
 Comparing this with the current of water in the pipe, we have 
 the same state of things as before described, as regards the 
 penetration of magnetic induction into the core. 
 
 One effect is to increase the resistance of a wire when the 
 frequency is great enough. It will be remembered that Prof. 
 Hughes (in 1886) brought forward evidence of this increase, 
 and, therefore, in the opinion of others, of the truth of the 
 theory of surface conduction along wires under certain circum- 
 stances which was advanced by me a year previous. There has 
 since been plenty of confirmatory evidence of a more complete 
 nature ; that is, not merely of an approximation towards, but 
 of an almost complete attainment of surface-conduction. 
 
 This action of conductors is sometimes referred to as magnetic 
 screening. It should, however, be noted that the screening is 
 done by the conductor itself, not by the currents " induced" in 
 the outer part of the wire or core, which are, as it were, merely 
 a sign that the screening is taking place. The screening action 
 often seems to be merely superficial ; but this is accidental, 
 from the disturbances being communicated to the conductor 
 from without. In reality, any and every part of a conductor is 
 screened from the rest by the portion immediately surrounding it, 
 that is, by its skin, and by that only. It is primarily the con- 
 ductivity that causes the screening, so that it is rather conduc- 
 tive screening than magnetic screening. The result is that dis- 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 239 
 
 turbances can only pass through a conductor by the (relatively) 
 very slow process of diffusion, so unlike that by which they are 
 transferred through a non-conductor. But the property of con- 
 ductivity acts conjointly with the indue tivity, so that although 
 copper would be the best screener on account of its high conduc- 
 tivity, the great inductivity of iron usually far more than com- 
 pensates for its inferior conductivity, and makes iron take the 
 first place as a screener. 
 
 Fifth Example : Modification of the Second and Fourth. 
 
 150. As we were not able to make an analogy with c and 
 k both finite, likewise n and n lt when H was taken to be 
 velocity, let us try the effect of taking E to be velocity, as in 
 147. Change the source from f to g= -curlh; then we 
 have merely to alter cp in (328) to k + cp to obtain the required 
 modification. Thus, 
 
 (334) 
 
 if A = E/p. Here, again, on comparison with (329), we see 
 that distortional friction will not furnish a proper analogy in 
 the general case. There seem to be but two special cases 
 possible; first, with the conductivity and frictionality both 
 zero (not the resistivity and frictionality), which reproduces 
 the case in 147 ; and, secondly, the permittivity and the 
 rigidity both zero (not permittivity and compliancy), which 
 brings us to the viscous liquid again. Thus, putting c = in 
 (334) brings us to 
 
 (335) 
 
 which compares with (333). We now compare conductivity 
 with density, and inductivity with the reciprocal of the 
 viscosity. This is by no means so useful as the previous form 
 of the viscous fluid analogy, although it is fundamentally 
 of the same nature, involving propagation by diffusion. 
 
 As in 146, 147, we had a failure of representation of one 
 of the energies by the energy of distortion, so now, in the 
 viscous fluid analogies, we fail to represent the dissipation ol 
 energy properly. The latter depends, as shown before, on the 
 velocity of distortion, so we might expect a similar failure. 
 
240 ELECTROMAGNETIC THEORY. CH. III. 
 
 Sixth Example : A Conducting Dielectric compared with an 
 Elastic Solid with Translational Friction. 
 
 151. But we can get a working analogy to suit the con- 
 ducting dielectric in another way. Thus, still keeping the 
 distortional elasticity or rigidity, give up the viscosity or dis- 
 tortional friction, and substitute translational friction. For 
 the equation of motion of an incompressible solid with the usual 
 constants, rigidity n and density /o 2 , and with translational 
 frictionality p 1 is 
 
 t^(p lP + P^-n^)Q,. . . . (336) 
 
 where f a is circuital impressed force. Now operate on (331) 
 by 1 + k/cp, producing the equation 
 
 - < 337 > 
 
 Comparing this with (336) we see that there is a proper cor- 
 respondence of form on the right sides, though not on the left. 
 The circuital impressed force fj_ is replaced, not by f, but by 
 f + M/cp. This is awkward for the analogy as regards the gene- 
 ration of disturbances. But away from the sources, there is a 
 useful correspondence. As in 146, Z corresponds to spacial 
 displacement in the solid, H to velocity, ju- to density, c to com- 
 pliancy (distortional) ; whilst now, in addition, the translational 
 frictionality in the solid has the complex representative pk/c in 
 the dielectric ; or &(/xr) 2 , if v is the speed of propagation of 
 disturbances. So disturbances of H are propagated in the same 
 way through a conducting dielectric, as are disturbances in an 
 incompressible solid with the usual rigidity, with frictional 
 resistance to displacement superadded. But we do not cor- 
 rectly localise the energy dissipated. The rate of waste per 
 unit volume is &E 2 in the dielectric, whilst it is /^q 2 in the solid, 
 which corresponds to &(/xv) 2 H 2 in the dielectric. This being 
 entirely against Joule's law renders this form of analogy 
 unsatisfactory. But we may easily make a substantial improve- 
 ment by taking E to be velocity in the solid. 
 
 Seventh Example : Improvement of the Sixth. 
 152. Thus, let g= - curl h be the source of disturbance in 
 the dielectric, so that we have the equations (334) to consider. 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. "J41 
 
 Take the second form and compare it with (336), and we see at 
 once that there is a considerably more simple correspondence 
 than in the case last treated. Thus, the source f x of circuital 
 displacement G- corresponds to the source g of circuital A (the 
 time-integral of E) ; the electric force now means velocity ; /A 
 is compliancy ; c is density ; k itself is the frictionality ; the 
 electric energy JcE 2 is kinetic energy ; and the waste (Joulean) 
 is properly localised in the solid by the frictional waste of energy 
 Pi<l 2 per second. The induction ^H is twice the rotation. In 
 fact, this case is the same as in 147, with translational fric- 
 tion added without destroying the analogy. It is clear that the 
 correct localisation of the waste, and the correspondence of 
 sources, makes the analogy be more readily followed. Of course, 
 there is the same failure as regards the distortional energy as 
 in 147. 
 
 Eighth Example: A Dielectric with Duplex Conductivity 
 compared with an Elastic Solid with Translational 
 Elasticity and Friction. The singular Distortionless 
 Case. 
 
 153. As a last example in which use is made of distortional 
 elasticity, let us generalise the non-conducting dielectric by the 
 introduction of magnetic conductivity. We have 
 
 curl(H-h) = (& + cp)E, . . . (338) 
 curl (e-E) = for + flp)H, . . . (339) 
 
 where g is the magnetic conductivity. So, if curl h = 0, and 
 we eliminate E, we have 
 
 . . (340) 
 
 .__ 
 Or, if curl e = and we eliminate H, we have 
 
 . . (341) 
 
 We see that we cannot, when k and g are both finite, get a 
 satisfactory analogy as regards the generation of disturbances 
 by f p circuital impressed force in a solid. Consider, therefore, 
 only the free propagation, subject to the equation 
 
 , . . (342) 
 
242 ELECTROMAGNETIC THEORY. OH. III. 
 
 where for Z we may substitute H or E or other electrical quan- 
 tities. Since, in (342), we have a term proportional to Z itself 
 (or its substitute), we must introduce elastic resistance to dis- 
 placement in the elastic solid to make an analogy. Thus, 
 
 . . (343) 
 
 is the equation of motion in the incompressible solid if there be 
 elastic and frictional as well as inertial resistance to translation, 
 by (307), (313), 145. 
 
 Now expand (342) and divide by c. We get 
 
 which is suitable for comparison with (343). When Z is the 
 spacial displacement, H is velocity, c compliancy, /* density, as 
 in 146 ; but in other respects the correspondence is complex, 
 for the elastic resistance to translation p Q is kg/c, whilst the 
 frictionality p l is g + kp/c. 
 
 If we take E to be velocity there will be another set of corre- 
 spondences, c being density and p compliancy. 
 
 However unsatisfactory the analogy maybe in details, we have 
 still the result that disturbances in a dielectric with duplex 
 conductivity (electric and magnetic) are propagated similarly to 
 motions in an elastic solid constrained in the manner above de- 
 cribed. The effect of the two kinds of distortion and waste of 
 energy involved in the existence of the two conductivities are 
 therefore imitated by one kind of frictionality in the elastic solid, 
 assisted by elastic resistance to translation. 
 
 The singular distortionless case ccmes about when 
 
 4/W>2 = />i 2 in the solid, 
 
 k/c=g/{j, in the dielectric, 
 
 In either case the effect of the frictional resistance on the dis- 
 turbances is made merely attenuative. Disregarding the atten- 
 uation with the time, the propagation takes place in the di- 
 electric in the same way as if it were non-conducting, and in the 
 solid in the same way as if it were devoid of frictional and 
 elastic resistance to translation. In the electromagnetic case 
 we may follow this into detail with ease in a symmetrical 
 manner. It is, however, rather troublesome in the elastic 
 
 } ^ 
 j ' 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 243 
 
 solid, on account of the want of correspondence in detail, due 
 to the employment of distortional elasticity, and for other 
 reasons. 
 
 The Rotational Ether, Compressible or Incompressible. 
 
 153 A. All the above analogies, however good or bad they 
 may be in other respects, are deficient in one vital property, 
 inasmuch as they involve the elasticity of distortion, and with 
 it, the energy of distortion. But the electromagnetic equations, 
 especially when put in the duplex form symmetrical with 
 respect to the electric and magnetic sides, have nothing in 
 them suggestive of distortional forces, nor can we represent 
 either the electric or the magnetic energy as the energy of 
 distortion. On the other hand, the equations are fully 
 suggestive of rotation. If, then, the elastic solid has still 
 to do duty for purposes of analogy, and yet not fail upon the 
 point mentioned, it is clear that the rigidity must be done 
 away with, at least as an electromagnetically active influence. 
 Its place must then be taken by elastic resistance to rotation, 
 by using a rotational instead of a shearing stress. This 
 brings us to the medium invented by Lord Kelvin, called by 
 him simply Ether, and contrasted with Jelly, which means 
 an incompressible elastic solid with ordinary rigidity. This 
 rotational ether has mass of course, for one thing, which 
 brings in kinetic energy (of translation) ; and, on the other 
 hand, it possesses, by means of internal arrangements with 
 which we are not immediately concerned, the property of 
 elastically resisting rotation, and consequently stores up 
 energy of the potential kind, to balance the kinetic. If this 
 were all, there would be little more to say. But Lord Kelvin 
 found that his Ether enabled him to complete a long-delayed 
 work, viz., to produce a satisfactory elastic-solid analogy to 
 suit the problem of magnetic induction (" Mathematical and 
 ?hysical Papers," Vol. I., art. 27, and Vol. III., art. 49) ; and 
 I >ave pointed out that this rotational ether enables us to 
 construct good all-round analogies for the propagation of dis- 
 turbances in a 'stationary dielectric (Appendix to chap. II., 
 ante), with correct localisation of both energies and of the 
 flux of energy. As these are the only all-round elastic-solid 
 analogies yet known, we may here consider them again, 
 
244 ELECTROMAGNETIC THEORY. CH. III. 
 
 especially in relation to the previously given analogies, and 
 to show where the changes come in. 
 
 Go back to equation (313), giving the translational force P 
 due to the stress in an elastic solid when there is rotational 
 elasticity as well as rigidity. We have 
 
 F = n (V 2 G + JV div G) + JfcV div G - v curl 2 G, (346) 
 
 where G is the displacement, n the rigidity, k the compressive 
 resistivity, and v the new elastic constant connected with the 
 rotation. Now, we have curl 2 = V div - V 2 , so we may rearrange 
 (346) thus, 
 
 F = ( + v)V 2 G + (& + Jtt-v)VdivG, . (347) 
 
 and this may be split up into circuital and divergent parts, viz., 
 
 (circuital), . (348) 
 2 , (divergent). (349) 
 
 Now observe that in the circuital equation, n and v occur 
 additively, so that equal parts of each have the same effect, 
 and either may be replaced by an equal amount of the other. 
 This is irrespective of the compressibility, which is not con- 
 cerned in circuital disturbances. We see, then, that the 
 rigidity may be done away with altogether and equal v substi- 
 tuted, without affecting the propagation of circuital disturb- 
 ances. This is a very remarkable property. 
 
 Observe, also, that in the divergent equation v does not 
 enter at all, although n does, and in the same way as in the 
 ordinary compressible rigid solid. In fact, the coefficient k + ^n 
 is the same as the former m + n, of 142 and later. So the 
 propagation of divergent disturbances is the same as if the 
 rotational elasticity were done away with. 
 
 Now, we have already noticed the case in which the speed 
 of the divergent wave is brought down to zero the contractile 
 ether of 142, 143. In an ordinary rigid solid, which was 
 then referred to, this requires negative compressibility. But 
 now, if n be abolished and v substituted, so that the circuital 
 propagation is unaffected, the divergent equation will contain 
 only k t so that to make the speed of the divergent wave zero 
 we need only abolish the resistance to compression, or make 
 & = 0. The medium does not now resist either change or 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 245 
 
 shape or of size, but resists rotation only, and the potential 
 energy is the energy of rotation. This is one extreme form of 
 the rotational ether, made by combining it with the contractile, 
 by the evanescence of the rigidity and of the resistance to com- 
 pression. The medium is quite neutral as regards expansion 
 and compression. 
 
 Now, it is clear, from equation (348) and matter in previous 
 paragraphs, that we may construct analogies to suit the non- 
 conducting dielectric by abolishing n and using v instead, 
 without concerning ourselves with k at all, which may have any 
 value from zero to infinity. But Lord Kelvin's ether is got by 
 going to the other extreme from the neutral case just men- 
 tioned. 
 
 First Rotational Analogy : Magnetic Force compared with 
 
 Velocity. 
 
 154. Take & = oo (with n = 0) making the medium incom- 
 pressible, and therefore making disturbances in an unbounded 
 medium be necessarily circuital, whether the impressed forcive 
 be circuital or not. We now have 
 
 ...... (350) 
 
 by (348), whilst the other equation (349) will merely assert 
 that F 2 is balanced by difference of pressure. This we are not 
 concerned with, since our impressed forcive should be circuital 
 in the electromagnetic comparisons. The stress (310) is now 
 
 P N = -vVNcurlG, .... (351) 
 
 where we ignore the uniform pressure for the reason mentioned. 
 The torque accompanying this stress is, by (311), 
 
 S = 2vcurlG, ..... (352) 
 and the translational force is 
 
 F= -curljS=i>v 2 G, (353) 
 agreeing with (350). So the equation of motion is, by (306), 
 
 fi = (^ 2 'vV2)G = (^-^V 2 )q, . . (354) 
 
 where fj is the circuital impressed force, and q. is the velocity. 
 As before, we keep to small motions in general. 
 
246 ELECTROMAGNETIC THEORY. CH. III. 
 
 Equation (354) may be at once compared with the electro- 
 magnetic 
 
 f=(^-V 2 \ H = (^_Vf\ z> . . (355) 
 \ cpJ \ c/ 
 
 as in (325), (326), our first example, and we therefore deduce 
 the following correspondences : 
 
 Z = H/p stands for G, (spacial displ.) 
 
 (mag. force) H ,, q, (velocity), 
 
 (el. current) curl H curl q, (2 x spin), 
 
 (el. displ. ) D ,, curlG, (2 x rotation), 
 
 (inductivity) /x, ,, ,, /a, (density), 
 
 (permittivity) c ,, ,, 1/v, (rotl. compliancy), 
 
 (induction) B ,, /oq, (momentum), 
 
 (el. force) E ,, JS, (J x torque), 
 
 (mag. energy) J/xE 2 /oq 2 , (kin. energy), 
 
 (el. energy) JcE 2 ,, Jv(curlG) 2 , (rotl. energy), 
 
 (source of H) f= curie ,, f lt (impd. force), 
 
 (energy flux) VEH VJSq, (energy flux) 
 
 Referring to the similar list in 146, it will be seen that the 
 correspondences there given are repeated here, except that the 
 elastic constant changes its meaning, and that there are now 
 additional correspondences. The electric energy is correctly 
 localised by the potential energy of the rotation. Since the 
 kinetic energy is also correctly localised, we need not be 
 surprised to find that the energy-flux has the same distribution. 
 The activity of the stress P N is P N q per unit area, which is 
 the same as g-NP^, if P q be the stress on the plane whose 
 normal is q, provided the stress is irrotational. But if ro- 
 tational, as at present, it is ^NQ q , where Q is conjugate to P. 
 This activity means energy transferred from the side of the 
 plane when the stress vector is reckoned, to the. other side, or 
 against the motion. The vector expressing the energy flux is 
 therefore -^Q q , not counting the convective flux. In our 
 present case, by (351), Q is the negative of P, because it is 
 purely a rotational stress, so W = qP q is the vector flux of 
 energy ; or, by (351), 
 
 W - - vqV^ curl G - - vVq curl G, . (356) 
 
ELEMENTS OF YECTORIAL ALGEBRA AND ANALYSIS. 24? 
 
 which is, by the above table, the same as VEH, the electro- 
 magnetic flux of energy. (In the above, q and ^ are the tensor 
 and unit vector of q.) 
 
 The above reasoning is applied directly to the unit element 
 of volume. But it may be easier to follow by taking am- 
 volume into consideration. If N is the normal outwards from 
 the surface enclosing it, then P N is the pull, per unit area, of 
 the matter outside, on the matter inside the surface, across the 
 unit area of the interface ; and P N q. is its activity. The total 
 activity of P N all over the surface is therefore 2P N q, which is 
 the same as - 2 N^P^, and expresses the work done per second 
 by the matter outside on the matter inside the surface, or the 
 rate of transfer of energy from the outside to the inside ; and 
 its equivalent is the rate of increase of the stored energy, 
 potential and kinetic, within the surface. The convergence of 
 the vector qP^ expresses the same property for the unit volume, 
 so that <?P q is the flux of energy (per unit area) itself. (The 
 small convective flux of energy is ignored here.) 
 
 Circuital Indeterminateness of the Flux of Energy in general. 
 
 155. That any circuital flux of energy may be siiperadded, 
 without making any difference in the transformations of 
 energy, is a fact which is of importance as evidence against the 
 objectivity of energy, but is of no moment whatever in the 
 practical use of the idea of a flux of energy for purposes of 
 reasoning. We should only introduce an auxiliary circuital 
 flux when some useful purpose is served thereby. 
 
 I may here remark that (speaking from memory) when Pro- 
 fessor J. J. Thomson first objected to the VEH formula as 
 representative of energy-flux, by reason of the circuital inde- 
 terminateness, he added that a knowledge of the real flux 
 could only be determined from an actual knowledge of the 
 real dynamical connections involved, that is to say, by a know- 
 ledge of the mechanism. But this is surely an inconclusive 
 argument, because the circuital indeterminateness applies even 
 then. In fact, it is universal in its application, as energy is at 
 present understood. For there to be an absolutely definite flux 
 of energy through space seems to require energy to have objec- 
 tivity in the same sense as matter, which is a very difficult notion 
 to grasp, and still more difficult to accept. But even if it be ac- 
 
248 ELECTROMAGNETIC THEORY. CII. Ill 
 
 cepted, the argument of circuital indeterminateness remains in 
 action, when it is desired to find what is the flux in a given case. 
 
 In a recent number of the Phil. Mag., Mr. Macaulay, by the 
 addition of a certain circuital flux to VEH, brings out the 
 result that in stationary states the modified flux reduces to 
 pG, where p is potential and current-density. I am, how- 
 ever (unless I forget much of what I have learnt in the last 
 15 years), unable to see that the auxiliary flux proposed serves 
 any useful purpose. In the first place, it greatly complicates 
 the flux of energy in general, and is entirely against the more 
 simple ideas to which we are naturally led in the study of 
 electromagnetic waves, whether in dielectrics or Conductors. 
 Besides that, it is implied that the function p, or electric 
 potential, is a determinate quantity specifying some definite 
 state of the medium. This seems to me to be an idea which 
 has no place at all in Maxwell's theory. It may have place in 
 other theories, but that is not to the point. To exemplify, 
 put a closed circuit (with battery) supporting a steady current 
 inside a metal box, and electrify the latter from outside. 
 According to the pG formula, the flux of energy in the wire 
 suffers a very remarkable change, by reason of the raising of 
 the potential of the box and its interior. But, according to the 
 interpretation of Maxwell's scheme which I expound, there is 
 no change whatever in the electrical state of the interior of the 
 box in the steady state (though there may and must be a 
 transient disturbance in the act of charging it, which is a 
 separate question), and the energy distribution and its flux are 
 the same as before, because E and H, which settle the state of 
 affairs, electric and magnetic, are unchanged. There are, 
 however, I believe, some electricians who will be much gratified 
 with the pG flux in steady states. 
 
 The total flux of energy through a wire will be pC, where by 
 C we mean the total current, obtained from pG by integration 
 over the cross-section. Now this is somewhat suggestive of 
 my expression VC for the total flux of energy along a circuit of 
 two conductors. But there are radical differences. For VC 
 is the integral form of VEH in the dielectric. Although C in 
 VC means the same as in pC, being the circuitation of H, or 
 the gaussage, V has no connection with p the potential, for it 
 (V) is the line integral of E across the circuit from on& wire 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 249 
 
 to the other i.e., the transverse voltage. Again, the VC 
 formula holds good in variable as well as in steady states, 
 whilst the proposed pG holds for steady states only. Even in 
 steady states, when V specializes itself and becomes difference 
 of potential, it remains different from p. 
 
 But although I cannot see the utility of the proposed change, 
 which seems to be a retrograde step, I think that Mr. 
 Macaulay's mathematics, which is of a strong kind, may be of 
 value in the electromagnetic field in other ways, especially 
 when cleared of useless and treacherous potentials. 
 
 Second Rotational Analogy : Induction Compared with 
 Velocity. 
 
 156. Returning to the suppositional Ether, the conception 
 is of an incompressible medium possessing mass, which involves 
 translational inertia, and therefore kinetic energy, and which 
 elastically resists rotation, and so stores potential energy ; and 
 we have supposed that the velocity of the medium means 
 magnetic force, and that its density means /x, so that the 
 kinetic energy and the magnetic energy are compared. But 
 we may equally well have this comparison of the energies com- 
 bined with a different interpretation of /A. Take, for instance, 
 the induction to mean velocity. Then, since the magnetic energy 
 is J/^B 2 , we see that it is now, not /x, but fjr 1 that is the density, 
 whilst H is momentum. Also, since /xcv 2 = 1 = pv~ l v 2 , we see 
 that the new interpretation of c is vy~ 4 . That is, the change of 
 [j, from density to its reciprocal involves the change of c from 
 compliancy to its reciprocal multiplied by v~^ making p 2 /v or p/v' 1 , 
 But it is better to write them out side by side, thus : 
 
 (mag. energy) J/*~ 1 B 2 stands for 
 
 (induction) 
 (reluctivity) 
 (mag. force) 
 (el. current) 
 (el. displ.) 
 (permittivity) 
 (el. force) 
 (el. energy) 
 
 B 
 
 H 
 
 curlH 
 D 
 c 
 E 
 
 (kin. energy), 
 (velocity), 
 (density), 
 (momentum). 
 
 p curl q, 
 p curl G, 
 
 vp~ l curl G, 
 
 (potl. energy). 
 
250 ELECTROMAGNETIC THEORY. CH. III. 
 
 The potential energy still correctly localising the electric 
 energy, in spite of the other changes, we may expect the flux 
 of energy to be correct. We have 
 
 VEH = vp~ l V curl G . />q = - vVq. curl G, 
 
 as before, equation (356). 
 
 Similarly, we may assume that p is not the density, nor its 
 reciprocal, but an unstated function of the density, say /A =f(p), 
 and work out the various correspondences that this necessitates. 
 We shall, for example, by the formulae for magnetic and kinetic 
 energy, require 
 
 Of course, in thus comparing magnetic with kinetic energy, we 
 shall always have H compared with q, or with some multiple 
 of q, which may change its " dimensions," as, for example, in 
 the above detailed cases, where H is changed from velocity to 
 momentum. 
 
 Probability of the Kinetic Nature of Magnetic Energy. 
 
 157. If it be asked why, in the previous analogies, a pre- 
 ference has usually been shown for the representation of magnetic 
 force (or a constant multiple thereof) by velocity, the answer 
 would be, substantially, that it has been done in order to make 
 the magnetic energy be kinetic. Now, it is true that in a per- 
 fectly abstract electromagnetic scheme, arranged in duplex 
 form in which every electric magnitude has its magnetic re- 
 presentative and, therefore, including a magnetic conduction- 
 current with waste of energy, there would be a perfect 
 balance of evidence as regards the kinetic nature of eiiher 
 the electric or the magnetic energy, when the other is to be 
 potential energy, as of a state of strain in an elastic medium. 
 For, if it were argued from a certain set of relations that 
 the magnetic energy was kinetic, the force of the argument 
 could be at once destroyed by picking out an analogous set of 
 relations tending to show (in the same way as before) that 
 the electric energy was kinetic. 
 
 But, as matters actually stand, with an imperfect corre- 
 spondence between the electric and magnetic sides of the elec- 
 tromagnetic scheme, there seems to be a considerable pre- 
 ponderance of evidence in favour of the kinetic nature of the 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 251 
 
 magnetic energy. We may refer, in particular, to the laws of 
 linear electric circuits, which were shown by Maxwell to be 
 simply deducible by the ordinary equations of motion (gene- 
 ralised) of a dynamical system, on the assumption that 
 magnetic energy is kinetic; there being one degree of 
 freedom for every circuit, and the variables being such 
 that every linear electric current is a (generalised) velocity. 
 In this theory, the dielectric in which the conducting circuits 
 are immersed is regarded as unyielding, so that electric dis- 
 placement cannot occur in it, and the currents are confined 
 entirely to the conductors. 
 
 Now, we could construct a precisely similar theory of con- 
 ductive magnetic circuits immersed in a medium permitting 
 displacement, but destitute of magnetic inductivity. The 
 circuital flux of magnetic induction in the former case would 
 now be replaced by a similar circuital flux of electric dis- 
 placement; the former electric currents becoming magnetic 
 currents, and the magnetic energy becoming electric energy. 
 But this electric energy, when expressed in terms of the linear 
 magnetic currents, would possess the property of allowing us 
 to deduce from it the laws of the magnetic circuits, by using 
 the generalised equations of motion, on the assumption that 
 electric energy is kinetic, in a manner resembling Maxwell's 
 deduction of the laws of electric circuits. Therefore, sup- 
 posing the state of things mentioned to really exist, we might 
 become impressed with the idea that electric energy is kinetic ; 
 just as, at present, it seems hardly possible to avoid entertain- 
 ing the idea of the kinetic nature of the magnetic energy. 
 
 We do not, however, need to go to generalised dynamics to 
 arrive at this probable conclusion. It is sufficient to start 
 with a sound general knowledge of dynamical facts and prin- 
 ciples, not necessarily mathematical, but such as may be ac- 
 quired in practical experience by an intelligent and thoughtful 
 mind, involving clear ideas about inertia, momentum, force, 
 and work, and how they are practically connected (the Act of 
 Parliament notwithstanding). On then proceeding to the expe- 
 rimental study of electrokinetics, including the phenomena of 
 self-induction in particular, the dynamical ideas will be found to 
 come in quite naturally. Lastly, the generalised theoretical 
 dynamics will serve to clinch the matter. Although adding 
 
252 ELECTROMAGNETIC THEORY. CH. III. 
 
 little that is novel, it will corroborate former conclusions, and 
 co-ordinate the facts in a compact and systematic manner, suit- 
 able to a dynamical science. 
 
 On the other hand, there is little that is suggestive of 
 kinetic ideas in electrostatics, whilst there is much that is 
 suggestive of the potential energy of a strained state. This 
 fact, combined with the kinetic suggestiveness of the facts (and 
 the equations embodying them) in which magnetic induction 
 and electric currents are concerned, explains why the associa- 
 tion of magnetic with kinetic, and electric with potential 
 energy becomes natural. It is, therefore, somewhat a matter 
 of surprise, as well as rather vexing, to find that in order to 
 extend the last-considered rotational analogy to a conducting 
 dielectric, we must, if we wish to do it simply, give up the com- 
 parison of magnetic force with velocity and electric force with 
 torque, and adopt the converse system, making the electric 
 energy be kinetic, and the magnetic energy the potential 
 energy of the rotation. 
 
 Unintelligibility of the Rotational Analogue for a Conduc- 
 ting Dielectric when Magnetic Energy is Kinetic. 
 
 158. We can, perhaps, most easily see that this plan should be 
 adopted by writing down the two circuital equations of electro- 
 magnetism, and then, immediately under them, the correspond- 
 ing proposed circuital equations of the rotational ether. Thus, 
 taking H to be velocity, as before, we have, to express the first 
 circuital law in the dielectric and its mechanical companion, 
 
 (357) 
 . . . (358) 
 
 Here the terms in vertical are to be compared. Magnetic 
 force H becoming velocity q, the electric force E becomes |S, 
 or half the torque ; c the permittivity is the compliancy v~ l , 
 and gj is a newly-introduced companion to g, which is the source 
 - curl h. It will be interpreted later. 
 
 The other circuital equations are 
 
 .... (359) 
 . . (360) 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 253 
 
 We now have the additional analogues of inductivity and 
 density, and f, which is the source curl e, is compared with t v 
 which is impressed translational force per unit volume. 
 
 So far relating to a non-conducting dielectric, and giving an 
 intelligible dynamical analogy, if we wish to extend it to a con- 
 ducting dielectric, according to Maxwell's scheme (including, of 
 course, a pure conductor which has, or is assumed to have, no 
 permittivity), we require to change cp in the first circuital law 
 (357) to k + cp, whilst the second circuital law needs no change. 
 But the right member of (358) needs to be changed to match 
 the modified (357). Thus, 
 
 g + curl H = (Jc + cp)E, .... (361) 
 g 1 + curU = ft + v-^)(iS), . . (362) 
 
 where ^ is the new coefficient, to match k. But what is its 
 interpretation in the rotational ether, and how is the latter to 
 be modified to make & x as intelligible as the other constants ? 
 
 Now, since we use rotational elasticity to obtain the po- 
 tential energy, and we associate the rotation with the electric 
 displacement, it is suggested that rotational friction should be 
 introduced to cause the waste of energy analogous to that of 
 Joule. But if there were frictional resistance proportional to 
 the spin, we should have 
 
 S = 2(v + Vjjp)ourlG, .... (363) 
 
 instead of (352), connecting the torque -with the rotation. 
 But this will not harmonise at all with (362). We cannot do 
 what we want by rotational friction, but require some special 
 arrangement, whose nature does not appear, in order to interpret 
 (362) intelligibly. 
 
 The Rotational Analogy, with Electric Energy Kinetic, 
 extended to a Conducting Dielectric by means of Trans- 
 lational Friction. 
 
 159. But, by changing the form of the analogy, choosing 
 the electric energy to be kinetic, the extension to a conducting 
 dielectric can be made in a sufficiently obvious manner. Put 
 (360) under (361), thus 
 
 g + curl H = (k + cp)E, .... (364) 
 . . . (365) 
 
254 
 
 ELECTROMAGNETIC THEORY. 
 
 OH. in. 
 
 where we Introduce p l to match k. 
 to get the other pair, thus 
 
 Also put (358) under (359) 
 
 (366) 
 (367) 
 
 We have now a fit, with an intelligible meaning to be given to 
 the new coefficient that brings in waste of energy. For (365), 
 which is compared with the first circuital law, is the trans- 
 lational equation of motion in the rotational ether when there 
 is frictional resistance to translation, expressed by p 1 q, so that 
 P 1 is the frictionality, and /a^ 2 the rate of waste. 
 The correspondences in detail are as follows : 
 
 (el. force) 
 (permittivity) 
 (el. displ.) 
 (el. energy) 
 (mag. force) 
 (inductivity) 
 (induction) 
 (mag. energy) 
 (conductivity) 
 (cond. current) 
 (Joule-heat) 
 (true current) 
 
 stands for G, 
 ,< Q, 
 
 Ptt, 
 
 i/xA 
 
 -js, 
 
 (spacial displ.). 
 
 (velocity). 
 
 (density). 
 
 (momentum). 
 
 (kin. energy). 
 
 ( - | torque). 
 
 (compliancy). 
 
 B 
 
 Jv(curlG-) 2 , (rotl. energy). 
 Pa ' (frictionality). 
 
 (rate of waste). 
 
 These should be studied in connection with the circuital 
 equations (364), (366), and their matches underneath them, if 
 it be desired to obtain " an intelligent comprehension " of the 
 true nature of the analogy, and to correct any errors that may 
 have crept in. 
 
 If we keep away from the sources of energy, there is little 
 difficulty in understanding the analogy in a broad manner. 
 But the sources require special attention before the electro- 
 magnetic and rotational analogues are intelligible. We know 
 that activity is the product of two factors, a " force " and a 
 " velocity." Newton knew that. In modern dynamics, too, 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 255 
 
 where the velocity is not a primitive velocity, but has a 
 generalised meaning the time-rate of change of some variable 
 the corresponding force generalised (not a primitive force) is 
 still such that the product of " force " and " velocity " is 
 activity, or activity per unit volume, &c., according to con- 
 venience. In our electromagnetic equations, for instance, EC 
 and HB and ED are activities (per unit volume), and we call 
 E and H the " forces " (intensities), and the other factors the 
 fluxes, the corresponding " velocities." Similarly, eO, eD, and 
 hB are activities (per unit volume understood), where e and li 
 are intrinsic, communicating energy to the system. 
 
 Now, since the " variables " may be variously chosen in a 
 dynamical system, we need not be surprised if it should some- 
 times happen that the (generalised) force turns out to be a 
 (primitive) velocity, and the (generalised) velocity to be a (primi- 
 tive) force. Here is food for the scoffer, for one thing. At 
 any rate, we should be careful not to confound distinct ideas, 
 and remember the meaning of the activity product. Our pre- 
 sent rotational analogy furnishes an illustration. We have the 
 equation of activity, 
 
 . (368) 
 where W = V(E - e) (H - h). . . (369) 
 
 W is the energy-flux, Q the rate of waste, U the electric, and 
 T the magnetic energy per unit volume, whilst the left member 
 represents the rate of supply of energy by the intrinsic sources 
 e and h, being the sum of their activities. 
 
 Now what are the analogues of e and h ? Remember that 
 only their curls appear in the circuital equations, and that f and 
 g are the sources of disturbances. By inspection of (365), 
 (367), we see that f x is the curl of a torque, and g l the curl 
 of a velocity, say 
 
 2f x = curl S , g l = - curl q . . . (370) 
 
 Then, just as the analogue of H is - JS, the analogue of h is 
 -|S . Impressed magnetic force is, therefore, represented 
 by an impressed torque per unit volume, and its activity is 
 
 (-4S )v-X-iS), . . . (371) 
 
256 ELECTROMAGNETIC THEORY. CH. Ill 
 
 which is, of course, plain enough. But the other activity is 
 
 Qo (ft + /*>), ( 372 ) 
 
 where q is the analogue of e. Here the supposed " force " has 
 become a velocity, and the " velocity " a force. For the factor 
 of q in (372) is the analogue of electric current- density, and 
 means the (primitive) force per unit volume in the rotational 
 ether, partly employed in increasing momentum, partly work- 
 ing against friction. Of course this perversion is rather 
 extreme. 
 
 The rotational analogues of (368), (369) may be readily 
 written down by proper translations in accordance with the above, 
 remembering (370). 
 
 Mr. W. Williams, who has recently (Physical Society, 1892) 
 published a very close study of the theory of the " dimensions " 
 of physical magnitudes, on applying his views to the electro- 
 magnetic equations and the rotational ether, has arrived at the 
 conclusion that the representation of either E or H by velocity 
 leads to the only two systems that are dynamically intelligible. 
 It should be remembered, however, that his conclusion is 
 subject to certain limitations (mentioned in his paper) regarding 
 dimensions, for otherwise the conclusion might seem to be of 
 too absolute a nature, as if one of E or H must be velocity. In 
 any case, I cannot go further myself at present than regard the 
 rotational ether as furnishing a good analogy, which may lead 
 later to something better and more comprehensive. The pre- 
 sent limitation to small motions (in general) is a serious one, 
 and there are other difficulties. 
 
 Symmetrical Linear Operators, direct and inverse, referred 
 to the principal Axes. 
 
 160. Linear vector functions of a vector play a very 
 important part in vector-algebra and analysis, just as simple 
 equations do in common algebra. Thus, in the theory of 
 elasticity the strain vector is a linear function of the direction 
 vector, and the stress vector is another linear function of the 
 direction vector. In electromagnetism the electric displace- 
 ment is a linear function of the electric force, and so is the 
 conduction current-density ; whilst the magnetic induction is a 
 linear function of the magnetic force, when its range is small 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 257 
 
 enough. There are, therefore, at least five examples of linear 
 vector functions to be considered in electromagnetism. In 
 other sciences too, the linear function often turns up, and 
 frequently in pure geometry, when treated algebraically. 
 Finite rotations may be treated by the method, and in the 
 algebra of surfaces of the second order the linear connection 
 between two vectors is prominent. It would be inexcusable 
 not to give some account of linear operators in this chapter 
 on vector-analysis and its application to electromagnetism. 
 The subject, however, is such a large one that it is only 
 possible to deal with its more elementary parts in a somewhat 
 brief manner not, however, so brief as to be useless. 
 
 The simplest kind of linear connection is that of Ohm's law 
 in isotropic conductors. We have C = E, where E is the 
 intensity of electric force and C the current-density, whilst k 
 is a constant, the conductivity. No specification of direction 
 is here made. Bub when vectorised, we have the equation 
 C = E instead, Tc being as before. We now assert that E 
 and are parallel, whilst their tensors are in a constant ratio, 
 for all directions in space. 
 
 Suppose, however, we alter the conductivity of the body in 
 a certain direction, say that of i (e.g., by means of compression 
 parallel to i), making it k v whilst its conductivity is k 2 in all 
 directions transverse to i. We now have Cj_ = /^E^ and also 
 O x = &}}, when the electric force is parallel to i. But if it be 
 transverse to i, then we have C = & 2 E. There is, in both cases, 
 parallelism of E and 0, but the ratio of the tensors changes 
 from &L to Jc 2 . What, then, is the current when E is neither 
 parallel nor transverse to i ? The answer is to be obtained by 
 decomposing E into two components, one parallel to i, the other 
 transverse ; then reckoning the currents to match by the above, 
 and finally combining them by addition. Thus 
 
 C = i. 1 E 1 +j. 2 E 2 + k.& 2 E 3 , ... (1) 
 where E I} E 2 , E 3 are the i, j, k scalar components of E. We 
 have here made use of the linear principle. It is that the sum 
 of the currents due to any two electric forces is the current 
 due to the sum of the electric forces. This principle (suitably 
 expressed) applies in all cases of linear connection between two 
 vectors, and is the ultimate source of the simplicity of treat- 
 ment that arises. 
 
258 ELECTROMAGNETIC THEORY. CH. HI. 
 
 More generally, let the conductivity be different in the 
 three co-perpendicular directions of i, j, k, being k lt k y k z re- 
 spectively. Then we have 
 
 = *E-U 1 E 1 +J.fc 8 E 2 + kJfc 8 E s . ... (2) 
 
 The current is now coincident with the electric force in three 
 directions only, namely, along the principal axes of conductivity. 
 When E has any other direction than that of a principal axis, 
 we have no longer parallelism of and E, and their relation 
 is expressed by equation (2). It is the general type of all 
 linear relations of the symmetrical kind when referred to the 
 principal axes, if we remove the restriction which obtains in 
 the electrical example that the &'s must be all positive. Given, 
 then, that = JcE, with the understanding that is a symme- 
 trical linear function of E, so that k must represent the linear 
 operator connecting them (the conductivity operator in the 
 above example) the full answer to the question, What is the 
 corresponding to a given E? is obtainable in the above 
 manner, viz., by first finding the axes of parallelism of E and 
 and the values of the principal &'s, and next, by adding together 
 the three C's belonging to the three component E's parallel to 
 the axes. 
 
 The inverse question, Given E, find 0, is similarly answerable. 
 For if C = />E, where p is the operator inverse to k (or the 
 resistivity operator in the case of conduction-current), we know 
 that ^ft = 1, if p l is the constant resistivity parallel to i, and 
 similarly k 2 p 2 = 1 and & 3 /o 3 = 1 for the j and k axes, so that in 
 full, 
 
 ... (3) 
 
 where C v C 2 , C 3 are the scalar components of current. 
 
 Precisely similar remarks apply to the permittivity operator 
 c in D = cE, connecting the displacement D with the electric 
 force E, and to the inductivity operator /A in B = /*H, connect- 
 ing the induction B with the magnetic force H, when the con- 
 nection is a linear one. It is not without importance to con- 
 stantly bear in mind the above process of passing from E to C 
 through &, and the process of inverting k to /o, because when 
 treated in a general manner, with reference to any axes, the 
 relations become much more complicated, whilst their intrinsic 
 and resultant meaning is identically the same. 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 259 
 
 Geometrical Illustrations. The Sphere and Ellipsoid. 
 Inverse Perpendiculars and Maccullagh's Theorem. 
 
 161. Next let us obtain some geometrical illustrations of 
 the pre tdous. The transition from to is like that from a 
 sphere to an ellipsoid. If a solid be uniformly subjected to 
 what is termed a homogeneous strain, any initially spherical 
 portion of it becomes an ellipsoid. If this be done without 
 rotation, the state is such that all lines parallel to a certain 
 axis, say that of i, in the unstrained solid, are lengthened or 
 shortened in a certain ratio in passing to the strained state, 
 without change of direction. The same is true as regards lines 
 parallel to two other axes, say j and k, perpendicular to each 
 other and to the first axis, with different values given to the 
 ratio of lengthening or shortening in the three directions of 
 preservation of parallelism. 
 
 The equation of the ellipsoidal surface itself is 
 
 when referred to the centre and the principal axes. Here a, 6, c, 
 are the lengths of the principal semi-axes, whilst x, y, z, are 
 the scalar components of r, the radius vector from the centre 
 to any point of the surface. It may be written 
 
 = N 2 say, . . (5) 
 
 from which we see that when r belongs to the ellipsoid, N is 
 a unit vector, the vector radius of a certain sphere from which 
 the ellipsoid r may be obtained by homogeneous strain, the 
 three ratios of elongation being a, &, c (that is, the co-ordinate 
 x/a in the sphere becomes x in the ellipsoid, and so on). 
 Now, if we square (3) and divide by E 2 , we may write it 
 
 where the C's are the components of 0, and E is the tensor of 
 E. Now suppose E is constant, and let E take all directions 
 in succession. Its extremity will range over a spherical 
 surface, whilst the end of the corresponding or &E will 
 range over an ellipsoidal surface. For we may take ^E, # 2 E, 
 fe 3 E in (6) to be a, b, c in (4) ; when G I} C 2 , C 3 will simulta- 
 neously be x, y, z. The semi-axes of this current-ellipsoid are 
 
 s2 
 
260 ELECTROMAGNETIC THEORY. CH. III. 
 
 the principal currents in the directions of parallelism of electric 
 force and current. 
 
 Similarly to (6), we have, by squaring (2), 
 
 ( 7 ) 
 
 from which we see that when C is constant, so that the vector 
 ranges over a spherical surface, the corresponding E simulta- 
 neously ranges over the surface of an ellipsoid whose principal 
 semi-axes are the principal E's. 
 
 There are other ways of illustration than the above. Thus 
 we may see from the form of the right member of (4) that it is 
 the scalar product of r and another vector s, thus 
 
 rs = l, ....... (8) 
 
 where r and s are given by 
 
 ..... (9) 
 
 Here we see that s is a linear function of r, and such that if 
 N = <r, then s = <N = <<r = < 2 r. See equation (5). The in- 
 terpretation of the new vector s is easily to be found. It is the 
 reciprocal of the vector perpendicular from the centre upon the 
 tangent plane to the ellipsoid at the extremity of r. For if p 
 be this perpendicular, the projection of r upon p is evidently 
 p itself, because the three vectors r, p, and the line in the 
 tangent plane joining their extremities form a right-angled 
 triangle whose longest side is r. Therefore 
 
 rp=P 2 , ....... (11) 
 
 or, dividing by p 2 , 
 
 rp- 1 -!, ....... (12) 
 
 comparing which with (8), which is true for all r's of the 
 ellipsoid, we see that s = p" 1 , as stated 
 Now we have 
 
 = -l=^il+JV- + JL (13) 
 
 EC ftEC /> 2 EC /o 3 EC' 
 
 from which we see that if we construct the ellipsoid whose 
 principal semi-axes are (p 1 EO) i , and so on, the radius vector 
 will be E and the corresponding reciprocal perpendicular will 
 be C/EO. Here EC is understood to be constant. 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 
 
 Similarly, we have 
 
 i 2 2 2 
 = 
 
 so that in the ellipsoid with principal semi-axes (^EC)*, &c., the 
 radius vector is C and the reciprocal perpendicular is E/EC. In 
 both these methods of representation neither E nor has con- 
 stant tensor, but their product, the activity, is constant. 
 
 In the case (6), where it is E that is considered constant, the 
 reciprocal of the perpendicular on the tangent plane to the 
 ellipsoid of is 
 
 . _ ii . = 
 * E 2 E 2 ' 
 
 This is also an ellipsoid. For E is an ellipsoid, and pE only 
 differs in the principal constants being the reciprocals of those 
 in the former case. 
 
 Similarly, in the case (7), where C is constant, the reciprocal 
 perpendicular is 
 
 = c 
 C 2 ' 
 
 and again the extremity of JcG ranges over an ellipsoidal 
 surface. 
 
 Thus the sphere of E with constant tensor is associated with an 
 ellipsoid &E, whose reciprocal perpendicular is />E/E 2 or ^r~ 1 E/E 2 , 
 another ellipsoid. It follows that the reciprocal perpendicular 
 of the latter gives us the former ellipsoid again. 
 
 This reciprocal relation of two ellipsoids through their in- 
 verse perpendiculars is an example of Maccullagh's extraor- 
 dinary theorem of reciprocal surfaces. It may be stated thus 
 Let 
 
 1^ = 1, and r 2 s 2 = l, .... (17) 
 
 be the equations of two surfaces referred to the same origin, 
 T l and r 2 being the radius vectors, and s 1} S 2 the reciprocals of 
 the vector perpendiculars from the origin on their tangent 
 planes. If, then, we choose r 2 to be B I} we shall simultaneously 
 make S 2 be r r That is, starting with the surface r lt construct 
 the surface whose radius vector r 2 is B V the reciprocal perpen- 
 dicular of the ij surface. Then the reciprocal perpendicular s 2 
 of the second surface (s 1 or r 2 ) will be the radius vector r x of 
 the first surface. 
 
262 ELECTROMAGNETIC THEORY. CH. III. 
 
 This admits of simple vectorial proof. For if ijSj = 1 be the 
 equation of a surface according to the above notation, we 
 obtain, by differentiation, 
 
 r^Sj + s^ = 0, (18) 
 
 if di l and efSj are simultaneous variations of r x and s r But di l 
 is in the tangent plane to r 15 and therefore at right angles to 
 BH so that s^ij = 0. This leaves T l ds l = also. But ds 1 or 
 c?r 2 equivalently is in the tangent plane of the surface whose 
 vector radius is S 1} or r 2 ; therefore r x is parallel to the perpen- 
 dicular on the same, or parallel to S 2 , say, ij = xs 2 , from which 
 r 1 s 1 = a;s 1 s 2 or #r 2 s 2 . But 1^ = 1 and r 2 s 2 = l, so ==!, and 
 ij = s 2 , which completes the connections. 
 
 In terms of the perpendiculars, let p l and p 2 be their lengths 
 and r v r 2 the corresponding radius vectors, then 
 
 *i'- r*=Pi'- Pv ( 19 ) 
 
 and P! is parallel to p 2 , and r 2 to p r 
 
 Internal Structure of Linear Operators. Manipulation of 
 several when Principal Axes are Parallel. 
 
 162. Leaving now the geometrical illustrations connected 
 with the ellipsoid, consider the symmetrical linear operator by 
 itself. If D = cE, where c is the linear operator, we know by 
 the above exactly how it operates through the principal o's. 
 We can, however, write c in such a manner that it shall state 
 explicitly its meaning. Thus, if c v c 2 , c 3 be the principal c's 
 belonging to the axes of i, j, k, we have 
 
 cE = i.c 1 E 1 +j.c 2 E 2 + k.c 3 E 3 , . . . (20) 
 by 160. Now put the E's in terms of E, producing 
 
 cE = i.c 1 iE+j.c 2 jE + k.c 3 kE. . . . (21) 
 
 In this form, the operand may be separated from .the operator, 
 thus 
 
 cE = (i.c 1 i+j.c 2 j+k.c a k)E, . , . (22) 
 
 so that the expression for c itself is 
 
 c = i.c 1 i+j,c 2 j + k.c 3 k (23) 
 
 In this form, with a vector to operate upon implied, the 
 nature of c is fully exhibited. But the operand need not 
 
ELEMENTS OP VECTORtAL ALGEBRA AND ANALYSIS. 263 
 
 follow the operator. If it precedes it the result is just the 
 same. Thus by (23), 
 
 EC = Eucji + Ej.Cgj + Ek.c 3 k 
 
 = c 1 E 1 .i + c 2 E 2 .j + c 3 E 3 .k = cE. . . (24) 
 
 When, however, the operator is unsymmetrical, the vectors 
 cE and EC are not identical. They are said to be conjugate to 
 one another, so that in the symmetrical case of identity, c is 
 also called a self-conjugate operator. The distinction between 
 symmetrical and skew operators will appear later. 
 
 There are six vectors concerned in c, viz., i, j, k, and ^i, c 2 j, 
 c 3 k, which call I, J, K. Thus 
 
 c = i.I+j.J + k,K ..... (25) 
 
 is the type of a symmetrical operator referred to the principal 
 axes. Now the general form of c referred to any axes and with 
 the symmetrical restriction removed is obtained by turning 
 these six vectors to any six others; thus, 
 
 < = a.l + b.m + c.n, ...... (26) 
 
 so that <E = a.lE + b.mE + c.nE J ..... (27) 
 
 E( = Ea.l + Eb.m + Ec.n, ..... (28) 
 
 show the linear function and its conjugate explicitly. This is 
 (with a changed notation, however) Prof. Gibbs's way of re- 
 garding linear operators. The arrangement of vectors in (26) 
 he terms a dyadic, each term of two paired vectors (a . 1, &c.,) 
 being, a dyad. Prof. Gibbs has considerably developed the 
 theory of dyadics. 
 
 Returning to the simpler form (25) or (22), we may note the 
 effect of the performance of the operation symbolised by c a 
 number of times, directly or inversely. Thus by (22) or (21), 
 or (20), we have 
 
 2 
 
 and so on. Thus c n means the operator whose principals are 
 the nth powers of those of the primitive c, with the same 
 axes. We therefore get a succession of ellipsoids with similarly 
 
264 ELECTROMAGNETIC THEORY. CH. IIL 
 
 directed principal axes, only altering their lengths, starting 
 from initial E with constant tensor. (That is, provided the 
 principal c's are all positive. If some be negative, we have 
 other surfaces of the second order to consider.) 
 
 We may also manipulate in the same simple manner any 
 number of operators, a, b, c, &c., which have the same principal 
 axes, though they may differ in other respects. Thus, using 
 the same kind of notation, 
 
 aE = i . a 1 E l + j . a 2 E 2 + k . a 3 E 3 , \ 
 
 bdE = abE = i . a^Ej + j . a 2 6 2 E 2 + k . a 3 6 3 E 3 , I . (30) 
 
 c&aE = a&cE = i . a^c^ + j . a 2 6 2 c 2 E 2 + k . a^c^, J 
 
 and so on. That is, the successive action of any number of 
 symmetrical operators with common principal axes is equiva- 
 lent to the action of a single operator of the same kind, whose 
 principals are the products of the similar principals of the set 
 of operators. The resulting ellipsoids have their axes parallel 
 throughout. 
 
 Next, multiply the equation (20) by any other vector F, 
 producing 
 
 FcE = c 1 E 1 F 1 + c 2 E 2 F 2 + c 3 E 3 F 3 . . . . (31) 
 
 By symmetry we see that this is identically the same as 
 EcF. We might also conclude this from the fact that cE and 
 EC are the same vector ; or, thirdly, by forming cF, and then 
 multiplying it by E. We may therefore regard EcF as the 
 scalar product of E and cF (or Fc), or, as the equal scalar pro- 
 duct of EC (or cE) and F. This reciprocity is the general cha- 
 racteristic of symmetrical operators, by which they can be 
 distinguished from the skew operators. The electrical mean- 
 ing in terms of displacement and electric force is that the com- 
 ponent displacement in any direction N, due to an electric force 
 acting in any other direction M, equals the component dis- 
 placement along M, due to an electric force with the same 
 tensor acting along N. 
 
 Theory of Displacement in an Eolotropic Dielectric. The 
 
 Solution for a Point-Source. 
 
 163. We see from the preceding that when it is the electric 
 force that is given in a dielectric there is no difficulty in find- 
 ing the displacement ; and conversely, when the displacement 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 265 
 
 is given, the electric force similarly becomes known through 
 the values of the principal permittivities. Suppose, however, 
 that the data do not include a knowledge of either the electric 
 force or the displacement, both of which have to be found to 
 suit other data. It is then sufficient to find either, the linear 
 connection settling the other. Thus, to take an explicit case 
 which has, for a reason which will -appear, a special interest, 
 suppose the dielectric medium to have uniform permittivity 
 transverse to the axis of k, but to be differently permittive to 
 displacement parallel to this axis. Further, put a point-charge 
 q at the origin, and enquire what is the equilibrium distribu- 
 tion of displacement. 
 
 The statement that there is a charge q at the origin means 
 nothing more than that displacement diverges or emanates from 
 that place, to the integral amount q. We also understand, of 
 course, that the displacement has no divergence anywhere else. 
 But the manner of emanation is, so far, left quite arbitrary. 
 Certain general results may, however, be readily arrived at. To 
 begin with, if the medium is isotropic, the emanation of the 
 displacement must not favour one direction more than another, 
 from which it follows that the displacement at distance r from 
 the source is spread uniformly over the area 47rr 2 , so that q/lirr 2 
 expresses its density. Now if we alter the permittivity to dis- 
 placement parallel to k only, we favour displacement in that 
 direction if the permittivity is increased, at the expense of the 
 transverse displacement, because the total amount, which 
 measures the strength of the source, is the same. Conversely, 
 if we reduce the permittivity parallel to k, we favour the 
 transverse displacement. The lines of displacement, originally 
 spread equably, therefore separate themselves about the k axis 
 (both ways), and concentrate themselves transversely, or about 
 the equatorial plane, the plane passing through the charge 
 which is cut by the k axis perpendicularly. 
 
 If we carry this process so far as to reduce the permittivity 
 along k to zero, there can be no displacement at all parallel 
 to k, so that the displacement must be entirely confined to 
 the equatorial plane itself. In this plane it spreads equably 
 from the source, because of the transverse isotropy assumed at 
 the beginning. The amount q therefore spreads uniformly over 
 the circle of circumference 2?rr in the displacement sheet, so 
 
266 ELECTROMAGNETIC THEORY. CH. III. 
 
 that the density (linear) is now q/2irr. The law of the inverse 
 square of the distance has been replaced by the law of the 
 inverse distance, with confinement to a single plane, however. 
 
 Next, if we introducs planar eolotropy, say by reducing the 
 permittivity in direction j, we cause the displacement in the 
 sheet to concentrate itself about the i axis, at the expense of 
 the displacement parallel to j. Finally, if we abolish 
 altogether the permittivity parallel to j, the whole of the 
 displacement will be confined to the i axis, half going one 
 way and half the other. 
 
 We may go further by introducing heterogeneity. Thus, 
 reduce the permittivity on one side of the equatorial plane. 
 The effect will be to favour displacement on the other side j 
 and, in the limit, when the permittivity is altogether abolished 
 on the first side, the whole of the displacement goes unilaterally 
 along the k axis on the other side. 
 
 In the case of the planar distribution of displacement, the 
 surface-density is infinite, but the linear-density is finite, except 
 at the source. We may, however, spread out the source along 
 a finite straight line (part of the k axis) when the displacement 
 will simultaneously spread out in parallel sheets, giving a finite 
 surface-density. Similarly, in the case of the linear distribu- 
 tion of displacement along the i axis, we may spread out the 
 source upon a portion of the j, k plane, when the former line 
 of displacement will become a tube, with finite surface-density 
 of displacement inside it. 
 
 In the above manner, therefore, we obtain a general know- 
 ledge of results without mathematics. Except, however, in the 
 extreme cases of spherical isotropy, of planar isotropy with zero 
 permittivity perpendicular to the plane, and of purely linear 
 displacement due to the vanishing of the transverse permit- 
 tivity, and other extreme cases that may be named, we do not 
 obtain an exact knowledge of the results, except in one respect, 
 that the total displacement must always be the same. Return, 
 therefore, to the case of initial isotropy upset by changed per- 
 mittivity parallel to the k axis, or, if need be, with three differ- 
 ent principal permittivities, so that the displacement is 
 
 cF, . . . (32) 
 if F be the electric force. We know that in the state of equi- 
 
ELEMENTS OF VECTORIAL ALGEBKA AND ANALYSIS. 267 
 
 librium, there must be no voltage in any circuit, or the curl of 
 P must be zero, so that F must be the slope - VP of a scalar 
 P, the potential. Furthermore, the divergence of D is the 
 electrification- density p. Uniting, then, these results, we obtain 
 
 div(-cVP) = />, ..... (33) 
 
 or (c 1 V 1 2 + c 2 V 2 2 + c 3 V 3 2 )P=- /0 , . . . (34) 
 
 which is the characteristic of P when referred to the principal 
 axes. Now when there is a point-source at the origin, and the 
 three c's are equal, we know that the potential is q/^irrc at 
 distance r. From this, remembering that r 2 is the sum of the 
 squares of the components of r, it is easy to see that the 
 potential in the case of eolotropy is 
 
 where f is some constant, because this expression (35) satisfies 
 (34) with p = 0, that is to say, away from the source. 
 
 The constant / may be evaluated by calculating the displace- 
 ment passing through the surface of any sphere r = constant, 
 according to (35), and equating it to q. The result is /= (c^Cg)"*. 
 The complete potential in the eolotropic medium due to the 
 point-source q at the origin is therefore 
 
 g/47T ( } 
 
 -- 
 
 The equipotential surfaces are therefore ellipsoids centred at 
 the charge, the equation of any one being 
 
 . . . (37) 
 
 so that the lengths of the principal axes are proportional to the 
 square roots of the principal permittivities. The lines of elec- 
 tric force cut through the equipotential surfaces at right angles. 
 But the lines of displacement do not do so, on account of the 
 eolotropy. To find their nature, first derive the electric force 
 from the potential by differentiation. Then (36) gives 
 
 -' 
 
 and the displacement is obtained by operating by c on this, 
 which cancels the c~\ and gives 
 
268 ELECTROMAGNETIC THEORY. OH. III. 
 
 -i 
 
 The displacement is therefore radial, or parallel to r itself. 
 That is, the lines of displacement remain straight, only altering 
 their distribution as the medium is made eolotropic. 
 
 Observe, in passing, that the scalar product FD varies as 
 P 4 ; that is, the density of the energy varies as the fourth 
 power of the potential. Also note that Fr is proportional to P ; 
 that is, the radial component of F, or the component parallel 
 to the displacement, varies as P/r. 
 
 If we select any pair of equipotential surfaces between which 
 the electric force and displacement are given by (38), (39), we 
 may, if we please, do away with the rest of the electric field. 
 That is, we may let the displacement and electric force 
 terminate abruptly upon the two ellipsoidal surfaces. What 
 is left will still be in equilibrium. For the voltage remains 
 zero in every circuit possible. This is obviously true between 
 the surfaces, because no change has been made there. It is also 
 true in any circuit beyond the surfaces, because of the absence 
 of electric force. And lastly, it is true for any circuit partly 
 within and partly beyond the region of electric force, because 
 the electric force is perpendicular to the surface. So the electric 
 field is self-contained. The electrification is on the surfaces, 
 where the normal component of the displacement measures the 
 surface-density. That is, there is a total electrification q on 
 the inner and - q on the outer surface. The elastance of the 
 condenser, or " leyden," to use Lord Rayleigh's word, formed 
 by the two surfaces, is the ratio of the voltage between them 
 to q. That is, the coefficient of q in (36) is the elastance be- 
 tween any equipotential surface and the one at infinity. 
 
 In thus abolishing the electric field beyond the limited 
 region selected, we may at the same time change the nature of 
 the medium, making it isotropic, for example,- or conducting, 
 &c. Any change is permissible that does not introduce 
 sources that will disturb the equilibrium of the electric field 
 between the equipotential surfaces. 
 
 Starting from isotropy, when the equipotential surface is a 
 sphere, if we keep Cj = c 2 constant, and reduce c 3 , the sphere 
 will become an oblate spheroid, like the earth (and other 
 bodies) flattened at the poles. Let the common value of c t 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 269 
 
 and c 2 be denoted by c , then the potential due to the central 
 charge q is 
 
 P s __ Qfa _ , (40) 
 
 and the displacement and electric force are given by 
 
 As c 3 is continuously reduced the oblate spheroid of equili- 
 brium becomes flattened more and more, and is finally, when 
 c 3 = 0, reduced to a circular disc. The displacement is now 
 entirely in the equatorial plane, so that if we terminate the 
 displacement at distance r from the centre, we obtain a 
 circular line of electrification (that is, on the edge of the disc). 
 We may have the displacement going from the central charge 
 to an equal negative charge spread over the circle, or from one 
 circle to another in the same plane ; and so on. 
 
 Similarly, when c x and c 2 are unequal, but c 3 = 0, the circular 
 disc is replaced by an elliptical disc. But, without introducing 
 transverse eolotropy, we see that the abolition of the permittiv- 
 ity in a certain direction has the effect of converting the usual 
 tridimensional solutions relating to an isotropic medium to 
 bidimensional solutions, in which the displacement due to 
 sources situated in any plane perpendicular to the axis of 
 symmetry is also in that plane. We may have any number of 
 such planar distributions side by side, with the axis of symmetry 
 running through them. But they are quite independent of 
 one another. If we restore the permittivity parallel to the 
 axis of symmetry, the displacement will usually spread out to 
 suit tridimensional isotropy. The exception is when every 
 plane perpendicular to the axis has identically similar sources, 
 similarly situated. Then the restoration of the permittivity 
 will produce no change. 
 
 Theory of the relative Motion of Electrification and the 
 
 Medium. The Solution for a Point- Source in steady 
 
 rectilinear motion. The Equilibrium Surfaces in General. 
 
 164. Let us now pass to what is, at first sight, an entirely 
 
 different problem, but which involves essentially the same 
 
 mathematics. As before, let there be a point-source of displace- 
 
270 ELECTROMAGNETIC THEORY. CH. TIT. 
 
 ment at the origin, but let the dielectric medium be homo- 
 geneous and fully isotropic. The displacement will be radially 
 distributed, without bias one way or another. Now, keeping 
 the charge fixed in space, imagine the medium to move steadily 
 from right to left past the charge. How will this affect the 
 displacement ? 
 
 If we desire the solution expressing how the displacement 
 changes from its initial distribution, as the medium is brought 
 from rest into steady motion, we can obtain it by going the 
 right way to work. But it is complex, and will, therefore, not 
 be given here, especially as it is not intelligible without close 
 study. But if we only wish to know the finally-assumed dis- 
 tribution of displacement when the initial irregularities have 
 subsided, we may obtain and express the result in a compara- 
 tively simple manner. Thus, to begin from the foundation, we 
 have the two fundamental circuital laws, 
 
 curl(H-h) = cpE + /ou, .... (42) 
 curl(e-E) = fipH, (43) 
 
 where E and H are the electric and magnetic forces, /t and c 
 the inductivity and permittivity, p the density of electrification, 
 supposed to have velocity u ; and e, h are the motional electric 
 and magnetic forces given by 
 
 e = VwB, h = VDw, . . . (44) 
 
 B and D being the induction and displacement, and w the 
 velocity of the medium. (See equations (1), (2), or (3), (4), 
 66, and equations (5), (6), 44, for the motional forces. Also 
 33 to 70 generally.) 
 
 Now in our present case, E and H are steady, and the elec- 
 trification is at rest. The right sides of (42), (43) are 
 therefore ze/o, which is a great simplification. By (42) and 
 the second o~ (44) we obtain 
 
 curl H = curl h = curl VDw. . . . (45) 
 But B is circuital, and therefore so is H. It follows that 
 
 H = VuD, (46) 
 
 if we substitute - u for w, for future convenience. This gives 
 H explicitly in terms of the displacement, when that is known, 
 so we have no further trouble with the magnetic force. 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 271 
 
 To find the displacement, (43) gives, along with the first of 
 
 curl(E + VuB) = 0, .... (47) 
 or say, curlf=0, if f=E + VuB. . , , (48) 
 
 Now put the value of B in f by (46), thus, 
 
 . . (49) 
 
 since /zc 2 = l, v being the speed of propagation of distur- 
 bances, and u the tensor of u, supposed to be parallel to k. 
 
 Expanding by the fundamental formula (52), 114, we 
 obtain 
 
 . . (50) 
 
 Now let f = (1 - tt 2 /v 2 )F. Insert in (50), and divide by 
 (1 - tt 2 /^ 2 ). The result is 
 
 ' 
 
 This vector F must also have no curl. Therefore, if P is the 
 potential whose slope is F, we have 
 
 d? dP n u? } dP 
 %=-_-, J^ 2 = - .b = -(!-) 
 
 ax ay v* dz 
 
 and the components of displacement are given by 
 
 *& %' *&' < 53 > 
 
 if Ci = C2 = Cj an( j c 3 = c(l-u 2 /v 2 ). . . (54) 
 
 Now observe that (53) express the displacement in terms of the 
 electric force in an eolotropic medium at rest, whose principal 
 permittivities are given by (54), when P represents the electro- 
 static potential. Our present problem of a moving isotropic 
 medium is therefore reduced to one relating to a stationary 
 eolotropic medium. Thus, to state the comparison fairly, when 
 the isotropic medium moves bodily past a stationary charge, 
 the displacement distribution becomes identically the same as if 
 the medium were at rest, but had its permittivity in the direc- 
 
272 ELECTROMAGNETIC THEORY. CIL III. 
 
 tion of motion reduced from c to c(l - w 2 /v 2 ). The solution is 
 therefore given by (40), (41) in 163, by taking c = c, and c 3 
 as in (54). The displacement concentrates itself about the 
 equatorial plane, or plane through the charge perpendicular to 
 the axis of motion in the one problem, or axis of reduced per- 
 mittivity in the other. In the limit, when the speed of motion 
 reaches v, the speed of light, the displacement is wholly planar, 
 as in the eolotropic case when the permittivity c 3 vanishes. 
 
 Observe, in (52), that the square of u occurs. It does not 
 matter, therefore, which way the motion takes place, so far as 
 the displacement is concerned. But it makes a great difference 
 in the magnetic force which accompanies the lateral concen- 
 tration of the displacement. Being given by (46), we see that 
 H reverses itself when the direction of motion is reversed. 
 
 It should be carefully noted that P, which is the electrostatic 
 potential in the eolotropic stationary medium, is not the electro- 
 static potential in the moving medium, although it is the same 
 function precisely. In the eolotropic medium its slope is the 
 electric force F. In the moving medium its slope is the same 
 force F, but this is no longer the electric force, which is E 
 instead, obtained from F by (51) or (52). Again, whilst the 
 displacement and electric force are not parallel in the stationary 
 eolotropic medium ; on the other hand, in the moving medium 
 they are parallel, because the medium is isotropic. To com- 
 plete the differences, there is no magnetic force in the eolotropic 
 case, but there is in the moving medium, and it is its existence 
 which allows the parallel to exist as regards the identical 
 distributions of displacement. 
 
 Instead of moving the medium past a fixed charge, we may 
 fix the medium, and move the charge through t the reverse 
 way at the same speed. This is why we put w = - u. The 
 above results apply when the charge q moves with velocity u 
 through the stationary medium. But it is necessary now to 
 travel with the charge in order to see the same results, because 
 the origin is taken at the charge, and it is now in motion. The 
 preliminary mathematics is therefore less easy. 
 
 The above theory of convection- currents and the eolotropic 
 comparison I have given before.* I have now to add a some- 
 what important correction. For the opportunity of making 
 * "Electrical Papers," Vol. II., pp. 492-499, 504-516. 
 
ELEMENTS OP VECTOUIAL ALGEBRA AND ANALYSIS. 273 
 
 this correction I am indebted to Mr. G. F. C. Searle, of Cam- 
 bridge, who has been at the trouble of working his way through 
 my former calculations. Whilst fully confirming my results 
 for a point- charge, and therefore all results obtained there- 
 from by integration, he has recently (in a private communica- 
 tion) cast doubt upon the validity of the extension of the 
 solution to the case of a moving charged conducting sphere> 
 and has asked the plain question (in effect) : What justification 
 is there for taking distributions of displacement calculated in 
 the above manner, and cutting them short by surfaces to which 
 the displacement is perpendicular, which surfaces are then 
 made equilibrium surfaces? For example, in the solution for a 
 point- charge, I assumed that the solution was the same for a 
 charged sphere, because the lines of electric force met it per- 
 pendicularly. On examination, however, I find that there is no 
 justification for this process. The electric force should not be 
 normal to a surface of equilibrium, exceptions excepted. 
 
 The true boundary condition for equilibrium, however, needs 
 no fresh investigation, being implicitly contained in the above. 
 In the stationary case of eolotropy, the vector F must have no 
 curl, and since it is the electric force, it is the same as saying 
 that there must be no voltage in any circuit, so that F must 
 be normal to an equilibrium surface. Now, in the problem 
 of a moving medium with charge fixed the same condition for- 
 mally obtains, viz., that F shall have no curl. But as it is not 
 the electric force E, it is clear that the displacement (which is 
 parallel to E) cannot be perpendicular to an equilibrium sur- 
 face. The voltage calculated by the fictitious electric force F 
 must come to zero in every circuit. The boundary condition 
 is, therefore, that F is perpendicular to an equilibrium surface. 
 That is, P = constant is the equation to a surface of equilibrium 
 (no longer equipotential in the electrostatic sense), where P is, 
 however, the same function as in the eolotropic stationary 
 problem of electrostatics. 
 
 Therefore, by 163, we see that the sphere of equilibrium 
 when the medium (isotropic) is stationary, becomes an oblate 
 spheroid when the medium is set into a state of steady motion, 
 the shorter axis being parallel to the line of motion. In the 
 limit, when the speed is v, the spheroid is flattened to a 
 circular disc, on whose edge only is the electrification. 
 
 T 
 
274 'ELECTROMAGNETIC THEORY. CH. III. 
 
 This applies when the whole dielectric medium is moving 
 past a fixed spheroidal surface of electrification. It equally 
 applies when the electrification moves the other way through a 
 fixed medium. There will be no electric force in its interior. 
 We may therefore fill it up with conducting matter, provided 
 we do not interfere with the free motion of the external 
 medium through it in the one case, or with the rest of the 
 external medium when the conductor moves through it. But 
 in the limiting case of motion at the speed of light, when only 
 a single circular line of electrification is concerned, it would 
 seem to be immaterial whether the conductor be a flat disc or 
 a sphere, subject to the reservation of the last sentence. 
 
 Whether it is possible to move matter through the ether 
 without disturbing it forms an entirely different question, to 
 which no definite answer can be given at present. We can, in 
 any case, fall back upon the more abstract theory of electrifi- 
 cation moving through the ether, without having conductors 
 to interfere. 
 
 Theory of the relative Motion of Magnetification and the 
 
 Medium. 
 
 165. From the preceding, relating to the magnetic effects 
 produced by moving the medium bodily past stationary elec- 
 trification ; or equivalently, by moving the electrification the 
 other way through the medium ; or, more generally, by rela- 
 tive motion of the sources of displacement and the medium 
 supporting it, we may readily deduce the corresponding results 
 when the sources are of the flux induction, instead of displace- 
 ment. Thus, in (42), (43), do away with the convective electric 
 current /ou, and substitute convective magnetic current cm, 
 where o- is volume-density of magnetification (suppositional). 
 Our circuital equations are then 
 
 curl(H-h) = cpE, . .... (55) 
 curl (e-E) = /*pH + <ni, . . . (56) 
 
 with the auxiliary equations of motional electric and magnetic 
 force 
 
 e = VwB, h = VDw, .... (57) 
 
 as before, where w is the velocity of the medium, and with the 
 
ELEMENTS OP VECTOIUAL ALGEBRA AND ANALYSIS. 275 
 
 further auxiliary conditions that the divergence of the displace- 
 ment is zero, whilst that of the induction is or. Now we really 
 do not need to employ the formal mathematics. Knowing the 
 results relating to the motion of electrification, we may infer 
 the corresponding ones concerning the motion of magnetifica- 
 tion, by making use of the analogies between the electric and 
 magnetic sides of electromagnetism, translating results in the 
 appropriate manner. But as our object is to illustrate the 
 working of vectors as well as electromagnetic principles, it 
 will not be desirable to leave out the mathematics entirely, 
 especially as the safe carrying out of the analogies requires 
 that they should be thoroughly understood first, which requires 
 some practice. 
 
 First, if we convert the problem to one of stationary waves, 
 by keeping the sources at rest, and letting only the medium 
 move bodily past them, we have, when the stationary state of 
 induction and displacement is reached, disappearance of the 
 right members of (55) and (56). Then (56) gives, when united 
 with the first of (57), 
 
 curl E = curl e = curl VwB. . . . (58) 
 
 In the result for a moving point-charge, curl E is perpendicular 
 to w, so now we may write 
 
 ..... (59) 
 
 where w = - u, because, similarly, curl H will be perpendicular 
 to w, and so make VBu circuital. Here u is the equivalent 
 velocity of the sources, when the medium is stationary. Thus 
 E, and therefore D, are fully known in terms of B. Comparing 
 with (46), we see that the electric force set up when a magnetic 
 charge moves is related to the induction from the charge in 
 the same way as the magnetic force set up when an electric 
 charge moves is related to the displacement from it, with, 
 however, a change of sign, or reversal of direction. 
 
 Using the second of (57) in the other circuital equation (55) 
 produces 
 
 curl(H-VuD) = = curlf, say, . . (60) 
 
 where, by (59), 
 
 . . (61) 
 
 T2 
 
276 ELECTROMAGNETIC THEORY, CH. III. 
 
 or, if u is parallel to k, 
 
 f = H + VkVkH = H + (kH 8 - H) 
 
 . . (62) 
 
 from which we see that the vector F derived from f by 
 f=(l -u 2 /v 2 )T has no curl, or is derived from a scalar poten- 
 tial fl thus, F = - vQ, and that the induction is given by 
 
 *"' B *=-"S' B 3 =-,d-wf. (63; 
 
 That is, the induction distributes itself in the same way as if 
 the medium were at rest, but had its inductivity reduced from 
 p to /*(! -u?/v z ) in the line of motion. In fact, the theory of 
 the effects produced by relative motion of magnetic sources 
 and the medium is essentially the same as that of moving 
 electric sources ; translating displacement in the latter case to 
 induction in the former, permittivity to inductivity, electric 
 potential (real or fictitious, as the case may be) to magnetic 
 potential, and the magnetic induction which accompanies 
 moving electrification to electric displacement set up by 
 moving magnetification, remembering, however, the change 
 of sign. 
 
 We know, therefore, the result of the steady rectilinear 
 motion of any distribution of magnetic sources. If the same 
 motion is common to all the elementary sources, we reduce the 
 problem to that of magnetic eolotropy without relative motion 
 of the medium and sources, and there is a definite system of sur- 
 faces of equilibrium (ft = constant) where we may, if we please, 
 terminate the electric and magnetic fields, thereby substituting 
 new arrangements of magnetic sources for the old. In this 
 case it is most convenient to keep the sources at rest, and 
 move the medium alone. 
 
 But should the relative motion of source and medium not be 
 the same for every source, then, since we cannot move the 
 medium bodily more than one way at a time, we may imagine 
 it to be stationary, and obtain the effect due to the motion of 
 the magnetification by superimposing the separate effects of 
 the elementary sources, which are known by the above. 
 
ELEMENTS OF VECTOKIAL ALGEBRA AND ANALYSIS. 277 
 
 Theory of the relative Motion of Magnetisation and the 
 Medium. Increased Induction as well as Eolotropic 
 Disturbance. 
 
 166. There is, however, no such thing as magnetifi cation, 
 the magnetic analogue of electrification. The induction is 
 always circuital. If it were not circuital, we should have 
 unipolar magnets. We must, therefore, somewhat modify the 
 conditions assumed to prevail in the above, in order to come 
 closer to reality. Consider, therefore, instead of magnetifica- 
 tion, a distribution of intrinsic magnetisation. This quantity 
 is I = /xh , where h is the equivalent intrinsic magnetic force. 
 Abolish o- in (56), and introduce h in (55). The circuital 
 equations are now 
 
 curl (H - VDw - h ) = cpE, . . . (64) 
 
 curl (VwB - E) = /^H, .... (65) 
 
 where we have introduced the motional forces (57). In the 
 steady state, the right members vanish as before. Now assume 
 that 
 
 E = VwB, (66) 
 
 so that the theory is unchanged so far, the displacement 
 depending on the induction and velocity of the medium in the 
 same way as when the source was magnetification. We shall 
 see the limitation of application later. Next put (66) in (64) 
 and we find, if the fraction 1 u 2 /v 2 be denoted by s, 
 
 curl f= curl h , (67) 
 
 curl F = curl h /s, .... (68) 
 
 where f and F are the same vectors (in terms of H) as in tho 
 last case, such that sF = f. We see that the vector F has now 
 definite curl, and that the induction is derived from F by 
 
 B^/iFp B 2 = ^F 2 , B 3 = MsF 3 ; . (69) 
 
 that is to say B = AF, where A is the inductivity operator whose 
 scalar principals are /x, p, and ps. We therefore have to find 
 the steady state from these complete connections, 
 
 divB = 0, B = AF, curl F = curl h /s . (70) 
 
 That is, the induction is the same as in a stationary eolotropic 
 medium (according to A), provided the intensity of the intrinsic 
 
278 ELECTROMAGNETIC THEORY. CH. III. 
 
 magnetic force be increased from h to h /s. This is an 
 important point. 
 
 We know that there are certain respects in which the theory 
 of induction due to intrinsic magnetic force is identical with 
 that due to magnetification. For instance, the steady induction 
 outside a magnet due to h is the same as that due to a dis- 
 tribution of cr, measured by the convergence of the intrinsic 
 magnetisation. From this we might hastily conclude that the 
 disturbance from the steady distribution produced by moving 
 the medium would be the same for h as for the equivalent o-, 
 provided we keep outside the region of magnetisation. But 
 the above investigation only partly confirms this conclusion. It 
 shows a likeness and a difference. The likeness is in the 
 eolotropic peculiarity brought in by the motion. In both cases 
 we may do away with the motion provided we simultaneously 
 reduce the inductivity parallel to the (abolished) motion from 
 fj. to s/x, so as to cause the induction to retreat from this direc- 
 tion and concentrate itself transversely. The difference is in 
 the reckoning of the strength of sources. In the case of 
 magnetification (as of electrification) we do not alter the 
 strength of the source. But in the case of magnetisation we 
 do, or, at any rate, produce an equivalent result. For, along 
 with the reduction of inductivity in the direction of motion, 
 we require to increase the intensity of the intrinsic magnetic 
 force from h to h /s. Remember that s is a proper fraction, 
 going from unity to zero as the speed of motion increases fron. 
 to v t the speed of propagation of disturbances. 
 
 To exemplify this, put 
 
 ..... (71) 
 
 as we see we may do, by the third of (70). Using this in the 
 second and first of (70) we obtain 
 
 .... (72) 
 the equation of the potential 12. Or, in terms of Cartesians, 
 
 XV + v 2 2 + v 8 8 )fl -/*-Woi + v 2 A 02 + *VA S ) . (73) 
 
 The equivalent magnetification is the convergence of Ah /s, 
 not of fihQ the real intensity of magnetisation, nor yet of /*h /s, 
 the same increased in a constant ratio. 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 27 T 
 
 There is, therefore, a remarkable difference between the two 
 cases of motion of the medium parallel to and transverse to 
 the lines of real magnetisation. Thus, first let h be parallel 
 to w or k. Then the s" 1 outside the brackets in (73) cancels 
 the s inside, so that the right member becomes simply /xV 3 ^ 03 , 
 where A 03 is now the tensor of h . That is, the effective 
 magnetification is the convergence of the real magnetisation, 
 just as when the medium is at rest. But if h is perpendicular 
 to k, say parallel to i, the right member of (73) becomes 
 /xs-'V^ou or t* 16 effective magnetification is s~ l times the con 
 vergence of the magnetisation, and is therefore increased. 
 
 Suppose, for example, our magnet is a straight filamentary 
 magnet. It has two poles, of strength m and - m say, so that 
 induction to the amount m diverges from one and converges to 
 the other pole, whilst continuity is made between them in the 
 filament itself. Now let the medium move past the filament, 
 the direction of motion being parallel to it. Then the strength 
 of the poles is unchanged, but the induction outside is diverted 
 laterally by the effective reduced inductivity parallel to the 
 filament. This case, then, resembles that of a pair of oppositely 
 signed point-charges, if we keep outside the filament. In the 
 limit, therefore, when the speed is raised to v, we have a pair 
 of parallel plane induction-sheets whose cores are joined by the 
 straight filament of induction, to make continuity. 
 
 For (66) to be valid, B should be such as to make 
 circuital. This requires that w and curl B should be perpen- 
 dicular to one another. They are perpendicular in the example 
 just mentioned, on account of the symmetry of B with respect 
 to the axis or line of motion. But if the same filament be held 
 transversely across the line of motion the property stated is no 
 longer true, so that (66) is not true, and this will, to an 
 unknown extent, upset the later equations. We can, however, 
 still employ them provisionally to obtain a first approximation 
 to the results. On this understanding, then, the strength of 
 the poles (effective) is made m/s and - m/s, with unimpaired 
 effective inductivity parallel to the filament, whilst that per- 
 pendicular to the filament is reduced, as before. So there is 
 now an increased total induction as well as concentration 
 about the plane through the filament perpendicular to the 
 motion. 
 
280 ELECTROMAGNETIC THEORY. CH. III. 
 
 We may, however, arrange matters in such a way that the 
 equation (66) shall be still valid when the line of motion is 
 perpendicular to the magnetisation, and so obtain an exact 
 solution showing the increased induction, that is, exact in the 
 absence of working errors. Let the region of magnetisation be 
 confined between two infinite parallel planes, and the magneti- 
 sation I = fih be uniformly distributed, parallel to the 
 boundaries. We thus do away with the poles. Now when the 
 medium is at rest the induction is B = I within the plate 
 (which may be of any thickness), and zero outside, whilst 
 there is no displacement. The intrinsic magnetic force h pro- 
 duces the greatest effect that it can produce unaided. But if 
 the medium be made to move steadily straight across the 
 magnetised region, then, after certain transient effects have 
 passed away (which may be readily calculated, because they 
 form simply plane electromagnetic waves), although there will 
 still be no induction (or displacement) outside the plate, that 
 within it will be increased (without change of direction) from 
 I to I/s. That is, the motional magnetic force comes in to 
 assist the intrinsic magnetic force. Here we have the explana- 
 tion of the previous result relating to the increased effective 
 strength of poles. 
 
 Along with this increased induction, there will be electric 
 force, according to (66). It is entirely within the magnetised 
 region. As the speed increases up to v t the induction and the 
 accompanying displacement in the plate go up to infinity. 
 
 The theory of an electrised plate is similar. The displace- 
 ment due to intrinsic electrisation J = ce , where e is the 
 equivalent electric force, the electrisation being uniform and 
 parallel to the sides of the plate, will be increased from D = J 
 to D = J/s by setting the medium moving straight through it, 
 it being now the motional electric force that assists the in- 
 trinsic. The accompanying magnetic force is accordingly given 
 
 (74) 
 Outside the plate there is no disturbance in the steady state. 
 
 It will, of course, be understood that if, instead of supposi- 
 tional impressed or intrinsic forces in a uniform medium, we 
 employ actual material plates, magnetised or electrised as the 
 case may be, they must, in the first place, be non-conductors, 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 281 
 
 and next, they must not interfere with the supposed uniform 
 motion of the medium. In short, the reservation is similar to 
 that mentioned in connection with moving charged conductors. 
 Of course, in our present case, the plates may be moved, 
 whilst the medium is supposed to be at rest. 
 
 It will be readily seen that the full investigation of the 
 effects of moving practicable electromagnetic arrangements of 
 conductors presents considerable difficulties. We can, how- 
 ever, get some information relating to the motion of a linear 
 circuit. 
 
 Theory of the Relative Motion of Electric Currents and 
 the Medium. 
 
 167. Thus, let us first abolish the intrinsic magnetisation 
 of 166, and substitute equivalent electric current. Here, 
 by equivalent electric current, we mean a distribution of 
 electric current which produces the same induction as the 
 intrinsic magnetisation ; so that if both were to exist together, 
 the induction would be everywhere doubled ; and then, if either 
 of them were negatived, the resulting induction would be nil. 
 That this is possible, whatever may be the distribution of 
 inductivity, is a very remarkable property. The induction due 
 to magnetisation is conditioned solely by the curl of the intrin- 
 sic magnetic force, and the " poles " are essentially quite a 
 secondary matter. That is, we may vary the poles as we like, 
 provided we do not alter the curl of the intrinsic force, without 
 affecting the induction or the associated energy. 
 
 The question then presents itself whether this equivalence 
 continues to hold good when the medium is set in motion past 
 the stationary magnetisation or electric current respectively. In 
 the circuital equation (64), put h = 0, and introduce a term 
 on the right side, producing 
 
 curl(H-VDw) = C + cpE, . . . (75) 
 
 expressing the first circuital law ; whilst (65), or 
 
 curl(VwB-E) = /ijpE .... (76) 
 
 expresses the second. Now (75) only differs from (64) in the 
 substitution of C here, for curlh there. That is, the new 
 C and the old curlh are equivalent. We may, therefore, 
 
282 ELECTROMAGNETIC THEORY. CH. III. 
 
 dismiss the idea of magnetisation, and let the source of the 
 induction be any distribution of intrinsic electric current C . 
 With it is associated a certain distribution of induction, iden- 
 tical with that due to any distribution of intrinsic magnetisation 
 for which we have curlh = C , and the equivalence persists 
 when the medium is moving, or when the sources are moving. 
 Corresponding to (70) we shall have 
 
 divB = 0, B = AF, curlF = C /s, . (77) 
 for the determination of the induction in the eolotropic manner 
 previously pursued, when the steady state of affairs is reached j 
 subject also to the previous reservation that w and curl B are 
 perpendicular. That is to say, the induction is affected by the 
 motion in the same way as if the medium had its inductivity 
 decreased from /x to fts in the direction of motion, without any 
 change transversely, whilst at the same time the source C is 
 effectively increased to C /s. 
 
 Equations (77) are suitable when the electric currents are in 
 planes perpendicular to the direction of motion. For instance, 
 let there be (to take a very easy example) two parallel plane 
 sheets of electric current of surface-density C and - C respec- 
 tively. The induction between them, which is B=//,C when 
 at rest, becomes ftC /s when they are in motion with velocity u 
 perpendicular to their planes. Here s is the fraction (1 - u 2 /v*) 
 as before, and the electric force accompanying -the changed in 
 duction is simply the motional electric force. 
 
 There is another way of making the eolotropic comparison, 
 namely, by employing the vector f instead of F. Then, instead 
 of (77) we shall have 
 
 divB = 0, B = (A/)f, curlf=C . . (78) 
 In this way of looking at the matter we regard as suffering 
 no change of effective strength, whilst the eolotropic operator 
 X/s is such as to indicate increased inductivity (from p to /A/*) 
 transverse to the motion, and unchanged inductivity parallel 
 to it. The vector f is the excess of the magnetic force H of 
 the flux over the motional magnetic force. Inasmuch as this 
 way emphasises the fact that the intrinsic sources are really 
 constant, it is not without its advantages. But the idea of 
 increased transverse inductivity is rather an unmanageable one 
 in general, especially in the case of moving electrification. 
 
ELEMENTS OP VEOTORIAL ALGEBRA AND ANALYSIS. 283 
 
 Jf the sources and the medium have a common uniform 
 translational motion, it may be seen from the general circuital 
 equations that the steady distribution of the fluxes with respect 
 to their sources is unaffected by the motion. That is, if we 
 travel with the medium there is no change observable. This 
 applies in the case of circuital sources (as curlh above), as 
 well as divergent sources (as of electrification). The natural- 
 ness of the result is obvious, when the relativity of motion is 
 remembered. 
 
 The General Linear -Operator. 
 
 168. It is now necessary to leave these special cases of 
 eolotropy for fear of being carried away too far from the main 
 subject, which is, the nature of linear vector operators, in ter- 
 mination of this chapter on vector analysis. Up to the present 
 only the symmetrical operator has been under consideration. 
 Three rectangular axes are concerned, each of which is identi- 
 fied with parallelism of the force and flux, or of a vector and a 
 linear function thereof. Now, if we associate the algebraical 
 ratio of the force to the flux with the three directions of 
 parallelism, we see that the symmetrical linear operator depends 
 upon three vectors. If they were arbitrary, this would involve 
 nine scalar specifications ; but, being coperpendicular, there 
 are really only six independent specifications ; and, moreover, 
 if we transform to any other system of axes (independent or 
 non-coplanar) there can still be no more than six independent 
 data. 
 
 But it is easy to see that the general linear vector operator 
 must involve nine scalar specifications, viz., three for each of 
 the three independent axes of reference that may be chosen. 
 Thus, let D be any linear function of E. We must then have, 
 in terms of the i, j, k components, the following set of equa- 
 tions : 
 
 ..... (79) 
 
 = C 31 E 1 + C 32 E 2 + C 33 E 3>. 
 
 where the nine c's may have any values we please. The axes 
 of reference are perpendicular only for convenience ; any set of 
 three independent axes may be used, with nine properly deter- 
 
284 ELECTROMAGNETIC THEORY. CH. IIL 
 
 mined c's to match. The relation between the vectors D and 
 E, which is fully exhibited in (79), may be conveniently 
 symbolised by 
 
 D = cE, ...... (80) 
 
 where c is the general linear operator. 
 
 If we exchange c 12 and c 21 , &c., in the set (79) we shall 
 usually change D, of course. Let it become D'. It is then 
 fully given by the set 
 
 (81) 
 
 and these relations between D' and E may be symbolised, like 
 (80), by the single equation 
 
 D' = c'E ....... (82) 
 
 The manipulation of sets of equations like (79) and (81) is 
 lengthy and laborious. By the use of the linear operators, 
 however, with proper attention to the laws governing them, 
 the vectors may be manipulated with facility. Thus, to give 
 the first and easiest example that presents itself, we may 
 instantly turn AB to Ace^B. This is obvious enough, inas- 
 much as the operation ii dicated by c" 1 will be precisely can- 
 celled by the operation c, so that cc -1 B = B. But there is much 
 more in it than that. For ActT^B is not merely the scalar pro- 
 duct of the vectors A and cc~ 1 B, that is, of A and B, but is 
 the scalar product of the vectors Ac and c~ 1 B, which are quite 
 different. The vector Ac is the same as the vector c'A, whilst 
 c -1 B is the vector which, when operated upon by c, gives the 
 vector B. 
 
 The constituents of the inverse operator c" 1 may be found 
 by solution of (79). Since D is a linear function of E, it follows 
 that E is a linear function of D ; or the E's may be expressed 
 in terms of the D's by means of a set of equations like (79), 
 with new coefficients instead of the c's there. Similarly as 
 regards the inverse operator c'" 1 belonging to (81) and (82). 
 
 The vector formed by taking half the sum of the vectors D 
 and D' is a symmetrical function of E, say 
 
 AK . . ... (83) 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 285 
 
 The constituents of X with two equal suffixes, namely, c n , 
 C 22 C 33 are tne same as those of c and of c. But the other 
 constituents are half the sum of the corresponding ones of c 
 and c' ; for instance J(c 12 + c 21 ) in place of c 12 and c 21 . From 
 this we see that the operator A is its own conjugate, or is self- 
 con jugate, making AE and A/E identical. 
 
 On the other hand, the vector formed by taking half the 
 difference of D and D' is a simple vector product. By inspec- 
 tion of (79) and (81), and remembering the structure of a 
 vector product, we may see that 
 
 (84) 
 where a is the vector given by 
 
 . (85) 
 
 We see, therefore, that any linear function, if it be not 
 already of the symmetrical kind, may be represented as the 
 sum of a symmetrical function and of a vector product. Thus, 
 by addition and subtraction of (83) and (84) we obtain 
 
 cE = D = AE + VaE, ..... (86) 
 
 c'E = D' = AE-VaE ..... (87) 
 
 That is, in terms of the operators alone, 
 
 c = A + Va, ..... (88) 
 
 c' = A-Va, ..... (89) 
 
 where c (and therefore c') is general, whilst A is symmetrical. 
 
 The vector a is quite intrinsic, and independent of axes of refer- 
 ence. The symmetrical operator A involves six scalars, the vector 
 a three more, thus making up the full nine. 
 
 Notice, by (86), (87), that 
 
 .... (90) 
 
 or the scalar product of the force and the flux does not involve 
 the vector a at all. 
 
 The Dyadical Structure of Linear Operators. 
 
 169. Now go back to the equations (79). Observe that the 
 right members, being the sums of three pairs of simple products, 
 
286 ELECTROMAGNETIC THSORT CH. HI. 
 
 are themselves scalar products. Thus, let v&ree new vectors 
 defined by 
 
 be introduced. Then we see that equations (79) are simply 
 D^^E, D 2 = c 2 E, D 3 = c 3 E, . . (92) 
 
 and, therefore, by combining them, we produce the one vector 
 equation 
 
 D = i.c 1 E+j.c 2 E + k.c 3 E, . . . (93) 
 
 thus proving that every linear operator may be exhibited in 
 the dyadical form 
 
 c = i.c 1 +j.c 2 + k.c 3 , .... (94) 
 
 whether it be symmetrical or not. The conjugate operator is, 
 similarly, 
 
 c' = i.c' 1 +j.c' 2 + k.c' 3 , .... (95) 
 
 where the accented vectors may be obtained from the un- 
 accented given in (91) by changing c 12 to c n , <fec., just as D' 
 was got from D, in fact. 
 
 It is now easily to be proved that 
 
 cE = Ec', Ec = c'B, , . . (96) 
 
 that is, if we wish to remove an operator from before to behind 
 a vector, we may do so by simply turning the operator to its 
 conjugate ; that is, by putting on the accent, or by removing 
 it if already there. It will, of course, be understood that when 
 a, vector E follows c, and we use the dyadical form of c, as in 
 (94), it is only the three second vectors that unite with E \ 
 whereas, if E precedes c, it is the first vectors in c that unite 
 with it. Thus : 
 
 Ec = Ei.c 1 + Ej.c 2 + Ek.c 3 , . -. . (97) 
 
 c'E = i.c' 1 E+j.c' 9 E + k.c' 3 E. . . . (98) 
 
 These are identical vectors always. In the symmetrical case, 
 the accents may be dropped, of course ; then EA, = A.E. It will 
 be seen that the above way of writing and working 
 dyadics fits in precisely with all the previous notation em- 
 ployed in the vector algebra. The dots in (94) are merely 
 
ELEMENTS OP VECTOBTAL ALGEBRA AND ANALYSIS. 287 
 
 separators, not signs of multiplication. When we put on a 
 vector, before or behind, it unites with the vectors it is not 
 separated from by dots ; we therefore obtain the forms (97), 
 (98), in agreement with the principles of notation described in 
 die early part of this chapter. 
 
 We have already pointed out that EAT, where A, is sym- 
 metrical, may be regarded as the scalar product of E and AT, 
 or of EA, (the same as AE), and T. The corresponding general 
 property is the same ; that is, c being general, EcT is the 
 scalar product of EC and F, or of E and cP. But EC is the 
 same as c'E, and cP the same as Fc', so we have 
 
 EcF = c'EF = EFc' = Fc'E = FEc. . . (99) 
 
 The operator c must be united with one or other of the two 
 vectors E and F, but it is immaterial which it is, provided we 
 always change to the conjugate properly, as exemplified. 
 
 The same applies when there are many linear operators. 
 Let there be three, a, 6, c. Then we have, to illustrate how 
 the transformations are made, 
 
 . (100) 
 
 and so on. Here the brackets are introduced to show the 
 association of the vectors and the operators. 
 
 We may also introduce anywhere a new operator, accom- 
 panied, of course, by the reciprocal operator ; thus 
 
 cE[aF] = cB[db- l bf] = eE[a&- 1 (F&')] = &c. . (101) 
 
 The above will be sufficient to show the great power gained 
 by the use of the operators, enabling one to do almost at sight 
 algebraical work which would, in Cartesians, cover pages. 
 
 Hamilton's Theorem. 
 
 170. The following little theorem, due to Hamilton, will 
 serve to illustrate the working of vectors, as well as another 
 purpose which will appear later. 
 
 Let m and n be a pair of vectors, then their vector product 
 Vmn is perpendicular to both, by our definition of a vector pro- 
 duct. That is, 
 
 = mVmn, = nVmn, . . . (102) 
 
288 ELECTROMAGNETIC THEORY. CH. III. 
 
 by our definition of a scalar product, and the parallelepipedal 
 property. Now introduce cc~ l betwen the V's and the vectors 
 preceding them in (102). Thus 
 
 = mcc- 1 Vmn, = ncc- 1 Vmn. . . (103) 
 
 These assert (and we cannot but believe it) that the vector 
 c^Vmn is perpendicular to the vectors me and nc, or c'm and 
 c'n ; it is, therefore, parallel to their vector product. The last 
 statement is expressed by 
 
 c- 1 Vmn = Vc / mc'n, .... (104) 
 
 where x is an unknown (or so far undetermined) scalar. Or, 
 operating on both sides by c, 
 
 a;Vmn = cVc'mc'n. .... (105) 
 
 To find the value of x, we have merely to multiply (105) by a 
 third vector, say 1. For this gives 
 
 zlVmn = IcVc'mc'n = c'lVc'mc'n, (106) 
 
 which gives the value of x explicitly, namely, 
 
 < 107 > 
 
 to be used in (105), which, with x thus settled, is the state- 
 ment of the little (but important) theorem. The quantity x 
 appears to depend upon 1, m, n. These vectors, however, enter 
 into both the numerator and denominator in such a way that 
 they can be wholly eliminated from x. This may be seen by 
 expanding the numerator and denominator. In fact, a; is a 
 pure constant, depending upon the operator c alone. 
 
 In the symmetrical case we may find its value by taking 
 1, m, n to be i, j, k, and the latter to be the principal axes. 
 Thus 
 
 cl = ci = Cji, c'm = cjj, c'n = c 3 k, 
 
 if c lf c 2 , c 3 are the principal scalar c's. So, by (107), 
 
 (108) 
 
 the continued product of the principal c's. 
 In the general case we may put, as we know, 
 
ELEMENTS OF VECTORTAL ALGEBRA AND ANALYSIS. 289 
 
 Using this in (107), on the understanding that c lt c 2 , c 3 are 
 the principals of A now, we shall get, similarly, 
 
 .... (109) 
 
 which it will be a useful exercise to verify. 
 
 Since c is any linear operator, (105) remains true when for c 
 we substitute its conjugate c', producing 
 
 aj'Vmn = c'Vcnicn, . . . . (110) 
 
 where x is got from the expression for x by changing c' to its 
 conjugate c. But c and c only differ in the changed sign of a, 
 as in (88), (89). On the other hand (109) contains a quadrati- 
 cally, so that a reversal of its sign makes no difference. There- 
 fore, x' is the same as x, which might not have been anticipated 
 at the beginning of the evaluation. 
 
 Hamilton's Cubic and the Invariants concerned. 
 
 171. The reader of Prof. Tait's profound treatise on 
 Quaternions will probably stick at three places in particular, 
 to say nothing of the numerous minor sticking-points that 
 present themselves in all mathematical works of any value, 
 and which may be readily overcome by the reader if he be 
 a real student as well. First, there is the fundamental 
 Chapter II., wherein the rules for the multiplication of vectors 
 are made to depend upon the difficult mathematics of spherical 
 conies, combined with versors, quaternions and metaphysics. 
 Next, Chapter IV., where the reader may be puzzled to find out 
 why the usual simple notion of differentials is departed from, 
 although the departure is said to be obligatory. Thirdly, 
 Chapter V., where the reader will be stopped nearly at the 
 beginning by a rather formidable investigation of Hamilton's 
 cubic. Only when the student is well acquainted with the 
 nature of linear operators and how to work them can he tackle 
 such an investigation. He should, therefore, pass on, to obtain 
 the necessary experience. On then returning to the cubic he 
 may find the investigation not so difficult after all, especially if 
 it be simplified by some changes calculated to bring out 
 the main points of the work more plainly. 
 
 The reader is led to think that the object of the investiga- 
 tion is to invert a linear operator that is, given D = cE, to find 
 
 u 
 
290 ELECTROMAGNETIC THEORY. CIL III. 
 
 E = c~ 1 D. But if this were all, it would be a remarkable 
 example of how not to do it. For the inversion of a linear 
 operator can be easily effected by other far simpler and more 
 natural means. The mere inversion is nothing. It is the 
 cubic equation itself that is the real goal. The process of 
 reaching it is simplified by the omission of inverse operations. 
 (It is also simplified by not introducing the auxiliary function 
 called x-) 
 
 The fundamental cubic is derived from equation (105), or 
 the equivalent equation (110) last investigated. I should 
 remark that this equation is frequently useful in advanced 
 vector-analysis as a transformation formula, turning Vcmcn to 
 a function of Vmn. Now use the form (110), or 
 
 #Vmn = c'Vcmcn, ...... (HI) 
 
 where a; is a known function of c, viz., 
 
 IVmn 
 
 Here c is any linear operator. Now if g be a constant, gm is 
 a linear function of m, and therefore (c - g)m is any linear 
 function of m; that is, c-g is a general linear operator. 
 Equation (111) therefore remains true when we substitute 
 c - g for c in it, not forgetting to make the change in x, and also 
 the equivalent change in c', viz., from c' to c' g. Equation 
 (111) then becomes 
 
 x g Vmn = (c' - g)V(e - g)m(c - g)n, (113) 
 
 where x g is what x becomes by the change, that is, by (112), 
 
 x - (c-<7)lV( c 
 
 IVmn 
 
 Remembering that g is a constant, we may readily expand (113) 
 to the form 
 
 = c'Vcmcn - g (Vcmcn + c Vmcn + c'Vcmn) 
 - # 3 Vmn +g' 2 (c'Vmn + Vmcn + Vcmn), . . (115) 
 
 where the cubic function of g on the left side is the expansion 
 of x g . The new coefficients x l and rc >2 are, of course, to be 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 291 
 
 found by expanding (114), but we do not want their values 
 immediately. Since a; is a function of c only, x ff is a function 
 of c and g only, so that x l and X 2 are functions of c only. 
 
 Now since (111) is an identity, so is (115) ; and since g may 
 have any value, the last equation must be identically true for 
 every power of g concerned, taken one at a time. That is, it 
 splits into four identities. ~Now, on comparing the coefficients 
 of g* on the left and right sides, we obtain (111) again. Simi- 
 larly, we observe that the coefficients of g 5 are the same. So 
 far, then, we have nothing new. But the g and g 2 terms give 
 
 . . (116) 
 . . (117) 
 
 which are fresh identities. From these various others may be 
 deduced by elimination or combination. The one we are seek- 
 ing is obtained by operating on the first by c' and on the second 
 by c' 2 , and then subtracting the second result from the first. 
 This eliminates the vector in the brackets and leaves 
 
 - a? 2 c' 2 )Vmn = c'Vcmcn - c' 3 Vmn, . (118) 
 = (a;-c' 3 )Vmn, . . . (119) 
 
 where the transition from (118) to (119) is made by usin& 
 (111) again. Rearranging, we have the final result, 
 
 which is Hamilton's cubic. Note that Vmn may be any vector 
 we please, and may, therefore, be denoted by a single letter. 
 Observe, also, that the cubic function of c' is of the same form 
 as that of g in the expansion of x. 
 
 Since (120) is true for all linear operators, we may write c 
 instead of c', giving 
 
 x-x lC + x 2 c*-c* = ..... (121) 
 
 This form is what we should have arrived at had we started 
 from (105) instead of (110). The value of x is the same in 
 either case, and it may be inferred from this that x l and x 2 do 
 not suffer any change when we pass from a linear operator to 
 its conjugate. 
 
 That the complex function x g is independent of 1, m, n, may 
 be seen by inspecting (114), and observing the parallel epipedal 
 
 u2 
 
292 ELECTROMAGNETIC THEORY. <3H. III. 
 
 form of numerator and denominator. For, suppose we alter 
 1 to 1 + am, where a is any constant. The addition made to the 
 numerator vanishes by the parallelepipedal property, and simi- 
 larly in the denominator. The same invariance obtains when 
 we change 1 to 1 + an. Also, the change of 1 to any scalar 
 multiple of itself alters both the numerator and denominator 
 in the same ratio. But, unless 1, m, n are coplanar, we may 
 turn 1 to any vector by adding to it vectors parallel to 1, m 
 and n of the right size. Therefore, x g is the same whatever 
 vector 1 may be. By the same reasoning m may be any vector, 
 and so may n. But 1, m, n should be independent vectors (that 
 is, not coplanar). This is not because we can suppose that an 
 actual discontinuity occurs when (for example) 1 is brought 
 into the plane m, n, but merely because the expression x 9 
 assumes an indeterminate form in the coplanar case. 
 
 From the invariance of x ff follows that of the three functions 
 it contains, namely, x, x v and x y That of #, of course, may be 
 independently seen by itself, readily enough, but that of the others 
 is less plain, because on expansion they are found to each 
 involve three parallelepipedal products in the numerator instead 
 of only one. Thus, expanding (114) we obtain 
 
 1 Vcmcn 4- m Vend + n Vclcm / 1 o o \ 
 
 - -' ' ' (122) 
 
 cl Vmn -f cmVnl + en Vim / 1 o Q \ 
 
 * 2== lVmn~ 
 
 In these we may, by the above, substitute c' for c, that is, 
 turn the linear operator to its conjugate. This does not mean 
 that x g is independent of the vector a which comes in when 
 the operator is not symmetrical, but that it only involves a 
 quadratically. 
 
 The value of x being 
 
 x = CjC^ + aca, . . . . . (124) 
 
 expressed in terms of the principal constants of the symmetrical 
 operator and the rotation vector, we may obtain the reduced 
 values of x l and X 2 from it directly instead of from their general 
 jxpressions. Thus : turn c to c-g (in 124), making it 
 
 $)* ( 125 ) 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 293 
 
 On expansion, the coefficients of g and # 2 show that 
 
 X l = C 2 C 3 + C 3 C 1 + C 1 C 2 + ft2 > (126) 
 Z 2 = Cj+C 2 + C 3 . 
 
 Thus x z is independent of a, whilst #j involves its square, as 
 we concluded previously. 
 
 When referred to the principal axes, Hamilton's cubic there- 
 fore reduces to 
 
 = foc^ + aca) - (c^ + c 2 c 3 + CgCj + a 2 )c 
 
 *<';*><} (127) 
 
 and in the symmetrical case of vanishing a to 
 
 Here the axes are coperpendicular. But we may disregard 
 the principal axes altogether, and write the general cubic in 
 the form 
 
 where the g'a are the roots of x g = 0. If, then, these roots are 
 all real and different, there are three directions of parallelism 
 of r and cr, or three r's such that 
 
 i=0A <**=9f9 <x z =y z i y (130) 
 
 Now multiply the first by r 2 and the second by T V giving 
 
 W-ftVi. ricr 2 =5' 2 r 1 r 2 . . . (131) 
 
 The left members are not usually equal, but in the sym 
 metrical case they are, and then the right members are 
 equalised. Then 1^ = 0, or r x is perpendicular to r 2 . Simi- 
 larly r 2 is perpendicular to r 3 . This is the case of three 
 mutually perpendicular axes of parallelism of force and flux 
 we started from. 
 
 The Inversion of Linear Operators. 
 
 172. If we write c~ l for c in the cubic (121) we see that 
 c -1 r becomes expressed as a function of r, cr, and c 2 r. This is 
 one way of inverting the operator, but a very clumsy way. 
 The simple way is in terms of dyads, and is fully described by 
 saying that if 
 
 c = a.l + b.m + c.n . ,. . . (132) 
 
294 ELECTROMAGNETIC THEORY. CH. III. 
 
 is any linear operator in dyadical form, then its reciprocal is 
 c- 1 = L.A + M.B + N.C, . . . (133) 
 
 where A, B, C is the set complementary to a, b, c, and L, M, N 
 the set complementary to 1, m, n. We have already used 
 these complementary vectors in the early part of this chapter, 
 equations (54) and (55), 114. We showed by elementary 
 considerations that any vector r could be expressed in terms of 
 any three independent vectors a, b, c by 
 
 r = rA.a + rB.b + rC.c, . . . (134) 
 where the complementary vectors A, B, C are got from a, b, c 
 
 by 
 
 A = I^, B = ^i, 0-U (135) 
 
 aVbc aVbc aVbc v ' 
 
 These equations serve to define the set complementary to a, b, c. 
 We also showed at the same time that the vector r could be 
 expressed in terms of the complementary set by 
 
 r = ra.A + rb.B + rc.O. .;-..' . (136) 
 
 To verify, we have merely to multiply (134) by A, B, C in turn, 
 and (136) by a, b, c in turn. 
 
 These equations may be written 
 
 r = (a.A + b.B + c.C)r, . . . (137) 
 r = (A.a + B.b + C.c)r, . ,, . (138) 
 
 showing that the dyadic in the brackets is of a very peculiar 
 kind, inasmuch as its resultant effect on any vector is to repro- 
 duce the vector. That is, taken as a whole, and disregarding 
 its detailed functions, the dyadic is equivalent to unity. 
 Now, suppose it is given that 
 
 R = (a.l + b.m + c.n)r, . . . (139) 
 so that R is any linear function of r. We know that 
 
 R = (a.A + b.B + c.C)R, . . . (140) 
 
 and that r=(L.l + M.m + N.n)r, . . . (141) 
 
 by the property of the complementary vectors explained 
 Comparing (140) with (139), we see that 
 
 lr = AR, mr = BR, nr = CR. . . (142) 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 295 
 
 Using these in. (141), we convert it to 
 
 r = (L.A + M.B + N.C)R, . . . (143) 
 
 comparing which with (139), we see that the inversion of the 
 dyadic has been effected in a simple and neat manner. 
 
 Professor Gibbs calls the vectors A, B, the reciprocals of 
 a, b, c. They have some of the properties of reciprocals. Thus, 
 
 aA = l, bB = l, cO = l. . . . (144) 
 
 But it seems to me that the use of the word reciprocal in this 
 manner is open to objection. It is in conflict with the obvious 
 meaning of the recipocal a" 1 of a vector a, that it is the vector 
 whose tensor is the reciprocal of that of a with unchanged ort 
 (or with ort reversed in the quaternionic system). It will also 
 be observed that Gibbs's reciprocal of a vector depends not upon 
 that vector alone, but upon two others as well. It would 
 seem desirable, therefore, to choose some other name than re- 
 ciprocal. I have provisionally used the word " complementary " 
 in the above, to avoid confusion with the more natural use of 
 "reciprocal." 
 
 Skew Product of a Vector and a Dyadic. The Differentia- 
 tion of Linear Operators. 
 
 173. In connection with dyadtcs, it should be remarked 
 that we have only employed them in the manner they usually 
 present themselves in physical mathematics, namely, so as to 
 make direct products with the vectors they are associated with. 
 But there is also the skew product to be considered in a com- 
 plete treatment. Thus, 
 
 . . (145) 
 . (U6) 
 
 Observe that, as before with the direct products, the vector 
 r only combines with the vectors nearest to it in the dyadic, 
 and not separated from it by the dots. Notice, too, that 
 whereas the direct product <r of the dyadic </> and a vector 
 r is a vector ; on the other hand, the skew products V<r 
 and Vr< are themselves dyadics, and behave as such in 
 union with vectors. Thus sV<r is a vector, and so is sVr<, 
 the first being got by making scalar products of s with a, b, c, 
 
296 ELECTROMAGNETIC THEORY. CH. Ill 
 
 and the second with Vra, &o. But to go further in this 
 direction would be to go beyond the scope of the present 
 treatment. 
 
 The differentiation of linear operators must, however, be 
 mentioned, because the process is of frequent occurrence in 
 electromagnetic investigations. If we think only of the 
 differential coefficient of a scalar, that it is its rate of increase 
 with some variable, it might seem at first sight that the 
 differential coefficient of a linear operator was nonsense. But 
 a little consideration will show that it is a perfectly natural, 
 and by no means a difficult conception. Thus, from 
 
 D = cE, ...... (147) 
 
 where c is a linear operator, we obtain, by differentiation with 
 respect to a scalar variable, say the time, 
 
 (148) 
 
 Here in the second term on the right we suppose that c is con- 
 stant, and in the first that E is constant. The meaning 
 of cE, then, is the rate of increase of D when c alone varies. It 
 is the linear operator whose constituents are the rates of 
 increase of the constituents of c. That this is so will be evident 
 on differentiating the set of equations (79). Therefore, if 
 
 c=a.l + b.m + c.n, .... (149) 
 we shall have 
 
 . . (150) 
 
 the sum of two dyadics. But it may be that our axes of 
 reference are invariable, as for example when 
 
 c = i.c 1 +j.c 2 + k.c 3 , .... (151) 
 
 i, j, k being a fixed set of rectangular orts. Then we have the 
 one dyadic 
 
 .. (152) 
 
 where, of course, the dots over the i and j do not signify any 
 differentiation. 
 
 In an isotropic dielectric of variable permittivity the electric 
 stress leads to the force - V c (^cE 2 ) per unit volume, where the 
 scalar c alone is differentiated, so that the result is - JE 2 Vc. 
 
ELEMENTS OP VECTORIAL ALGEBliA AND ANALYSIS. 297 
 
 But if the dielectric be eolotropic, the corresponding force is 
 This is equivalent to 
 
 < 153 > 
 
 where dc/dx, &c., are differential coefficients of c as above ex- 
 plained. It means the same as 
 
 -(V D -V E )(iED), .... (154) 
 
 where V D means that D alone, and V E that E alone is 
 differentiated. 
 
 Summary of Method of Vector Analysis. 
 
 174. In the last paragraph I came dangerously near to 
 overstepping the imposed limits of my treatment of vectors, 
 which is meant to present the subject merely in the form it 
 issumes in ordinary physical mathematics. If we were to 
 ignore the physical applications, and treat vector algebra as a 
 branch of pure mathematics, regardless of practical limitations, 
 there would be no bounds to the investigation of the subject. 
 It will now be convenient to wind up with a few remarks on the 
 previous, and on the nature and prospects of vector analysis. 
 
 Since we live in a world of vectors, an algebra or language 
 of vectors is a positive necessity. At the commencement of 
 this chapter, 97 to 102, 1 made some general remarks on the 
 nature of cartesian analysis, vector analysis, and quaternions ; 
 and the reader is recommended to read them again from a more 
 advanced point of view. Then, he was supposed to know next 
 to nothing about vectors. Now, although he need not have 
 absorbed all the special applications of vector algebra that have 
 been given since, it may be presumed that he has acquired a 
 general knowledge of the principles of the subject. There is 
 no longer any question as to the desirability and utility of 
 vectorial analysis. The present question is rather as to the 
 form the vector algebra should take. On this point there is 
 likely to be considerable difference of opinion, according to the 
 point of view assumed, whether with regard to physical appli- 
 cations, or abstract mathematical theory. Let us then, to begin 
 with, summarise the leading points in the algebra given above. 
 
 First, there is the idea of the vector as a distinct entity. 
 Nearly everyone nowadays knows and apprecktes the idea. 
 
298 ELECTROMAGNETIC THEORY, Ctt. Ill 
 
 And it is a noteworthy fact that ignorant men have long been 
 in advance of the learned about vectors. Ignorant people, like 
 Faraday, naturally think in vectors. They may know nothing 
 jf their formal manipulation, but if they think about vectors, 
 ihey think of them as vectors, that is, directed magnitudes. 
 No ignorant man could or would think about the three com- 
 ponents of a vector separately, and disconnected from one 
 another. That is a device of learned mathematicians, to enable 
 them to evade vectors. The device is often useful, especially 
 for calculating purposes, but for general purposes of reasoning 
 the manipulation of the scalar components instead of the 
 vector itself is entirely wrong. 
 
 In order to facilitate the reading of vector work, the vector 
 has its special type, thus E. This Clarendon, or any very similar 
 neat black type (not block letters), is meant to mark the vector, 
 always the vector, and never anything else, with the obvious 
 exception of headlines that speak for themselves. Also, to 
 economise letters, and ease the strain on the memory, the same 
 letter suitably modified serves for the tensor, the ort, and the 
 three scalar components. Thus E is the tensor, or size, and Ej 
 the ort (signifying the orientation) so that E = EE X ; whilst the 
 scalar components referred to rectangular axes are Ej, E 2 , E 3 . 
 The physical dimensions may be most conveniently merged in 
 the tensor E. Every vector has its species. The one we are 
 most familiar with is the space vector, or straight line joining 
 two points, or more strictly the displacement from one point to 
 another. But all kinds of vector magnitudes are formally 
 similar in having size and ort, and therefore, when considered 
 vectorially, obey the same laws. The properties of the space 
 vector therefore supply us with the rules or laws of vector 
 algebra. 
 
 The addition and subtraction of vectors is embodied in the 
 assertion (whose truth is obvious) that the sum of any number 
 of vectors making (when put end to end) a circuit is zero ; or, 
 in another form, all the various paths by which we may pass 
 from one point to another are vectorially equivalent, namely, to 
 the vector straight from point to point. Every vector equation 
 therefore expresses this fact. Every term is a vector ; and if 
 all the vectors be put on one side, with zero on the other, we 
 express the circuital property ; whilst if we have vectors on both 
 
. 
 
 
 ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 299 
 
 sides of the equation, we assert the vectorial equivalence of two 
 paths. 
 
 Any vector A and its reciprocal A" 1 have the same ort, and 
 their tensors are reciprocal to one another. 
 
 Coming next to combinations of vectors of the nature of 
 products, we find they are of two kinds, scalar and vector. The 
 scalar product of and F is denoted by EF, and the vector 
 product by VEF. Their tensors are EFcos0 and EFsinfl 
 respectively, 6 being the included angle. The scalar product 
 is directionless ; the ort of the vector product is that of the 
 normal to the plane of E and F. The tensor and ort of VEF 
 are V EF and VjEF. 
 
 The justification for the treatment of the scalar and vector 
 products as fundamental ideas in vector algebra is to be found 
 in the distributive property they possess, thus, 
 
 ., . (155) 
 
 which hold good for any number of vectors. In the first 
 equation the manipulation is as in common algebra, in all re- 
 spects. In the second, in all respects save one ; for the re- 
 versal of the order of the vectors in a vector product negatives 
 it, thus, Vac = - Vca. 
 
 Here we have very complex relations of geometry brought 
 down to elementary algebra. As Prof. Gibbs has remarked, 
 the scalar and vector product, which consist of the cosine and 
 sine of trigonometry combined with certain other simple notions, 
 are "incomparably more amenable" to algebraical treatment 
 than the sine and cosine themselves. 
 
 When we go on to combinations in threes, fours, &c., we find 
 it is the same thing over again in various forms. Thus c.ab 
 and cVab and VcVab define themselves by the above, being ab 
 times c, and the scalar and vector products of c and Vab re- 
 spectively. 
 
 Similarly with four vectors, as in dc.ab, the product of dc 
 and ab ; dc.Vab, which is dc times Vab ; d.cVab, which is cVab 
 times d ; dVcVab and VdVcVab, which are the scalar and vec- 
 tor products of d and VcVab ; VdcVab arid VVdcVab, which 
 are the scalar and vector products of Vdc and Vab. Com- 
 binations of two vectors are universal, and those of three are 
 
300 ELECTROMAGNETIC THEORY. CH. III. 
 
 pretty frequent, but those of four are exceptional. But it 
 should be noted that the scalar and vector product of two 
 vectors are involved throughout, no new idea being introduced 
 when more than two vectors are concerned. 
 
 When we pass on to analysis, we find just the same vector 
 algebra to be involved. Vectors are differentiated with respect 
 to scalars to make new vectors in the same way as scalars, and 
 the ideas connected with differentials (infinitesimal) and diffe- 
 rential coefficients are essentially the same for vectors as for 
 scalars. We never require to differentiate a vector with respect 
 to a vector, and there is a very good reason for it, because the 
 operation is an indeterminate one. 
 
 The fictitious vector y is omnipresent in tridimensional ana- 
 lysis. It only differs from a vector in being a differentiator as 
 well, so that it follows vector rules combined with other func- 
 tions. With it are associated the ideas of the slope of a scalar, 
 and the divergence and curl of a vector, with corresponding 
 important theorems relating to the transition from line to sur- 
 face, and from surface to volume summations. The theory of 
 potentials in its broad sense is also involved in the mathematics 
 of y, including the relations, direct and inverse, of potential, 
 slope, curl, and divergence. 
 
 As for the linear operator, that is only the scalar and vector 
 product system again in a special form. Professor Gibbs's 
 dyadic is a useful idea, and I have modified his notation to suit 
 the rest. 
 
 It is neither necessary nor desirable that a student should 
 know all that is sketched out above before he makes practical 
 use of it. For vector analysis, like many other things, is best 
 studied in the concrete application. The principles may be 
 very concisely stated, being little more than explanations of 
 the scalar and vector product, with some conventions about 
 notation. A student might learn this by heart, and be little 
 the better for it. He must sit down and work if he wants to 
 assimilate it usefully. He may work with scalar products 
 only to begin with, for quite a large ground is covered thereby. 
 When familiarised with the working of vectors by practice so 
 far as the scalar product goes, the further introduction of the 
 vector product will come comparatively easy. The former 
 knowledge is fully utilised, and also receives a vast extension 
 
ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 301 
 
 of application. For the vector product is a powerful engine, 
 which, with its companion the scalar product, is fully capable 
 of working elaborate mathematical analysis in a concise and 
 systematic manner. The student need not trouble about linear 
 operators at first. He will grow into them. It is far more 
 important that he should understand the operation of y as a 
 vector, and its meaning in the space variation of functions. 
 
 Unsuitability of Quaternions for Physical Needs. Axiom : - 
 Once a Vector, always a Vector. 
 
 175. Now, a few words regarding Quaternions. It is 
 known that Sir W. Rowan Hamilton discovered or invented a 
 remarkable system of mathematics, and that since his death 
 the quaternionic mantle has adorned the shoulders of Prof. 
 Tait, who has repeatedly advocated the claims of Quaternions. 
 Prof. Tait in particular emphasises its great power, simplicity, 
 and perfect naturalness, on the one hand; and on the other 
 tells the physicist that it is exactly what he wants for his phy 
 sical purposes. It is also known that physicists, with great 
 obstinacy, have been careful (generally speaking) to have 
 nothing to do with Quaternions ; and, what is equally remark- 
 able, writers who take up the subject of Vectors are (generally 
 speaking) possessed of the idea that Quaternions is not 
 exactly what they want, and so they go tinkering at it, trying 
 to make it a little more intelligible, very much to the disgust 
 of Prof. Tait, who would preserve the quaternionic stream pure 
 and undefiled. Now, is Prof. Tait right, or are the defilers 
 right? Opinions may differ. My own is that the answer all 
 depends upon the point of view. 
 
 If we put aside practical application to Physics, and look 
 upon Quaternions entirely from the quaternionic point of 
 view, then Prof. Tait is right, thoroughly right, and Quater- 
 nions furnishes a uniquely simple and natural way of treating 
 quaternions. Observe the emphasis. 
 
 For consider what a quaternion is. It is the operator which 
 turns one vector to another. For instance, we have 
 
 B=A.A- I B + B-A- I .AB 
 
302 ELECTROMAGNETIC THEORY. CIT. TIT. 
 
 identically. Or, 
 
 B = (A- 1 B + V.VA~ 1 B)A. . . , (157) 
 
 Here the operator in the brackets, say q, depends upon A and 
 B in such a way that it turns A to B ; thus, B = qA.. It is a 
 quaternion. It has a scalar and a vector part. The general 
 type of a quaternion is 
 
 ? = w, + Va, (158) 
 
 where w is any scalar, and a is any vector. Since we know 
 the laws of scalar and vector products we may readily 
 deduce the laws of quaternions should we desire to do so. 
 But we shall find that the above notation, though so well 
 suited for vectors, is entirely unfitted for displaying the 
 merits of quaternions. To do this we must follow Hamilton, 
 and make the quaternion itself the master, and arrange 
 the notation and conventions to suit it, regardless of the con- 
 venience of the vector and the scalar. The result is the 
 singularly powerful algebra of quaternions. To give a notion 
 of its power and essential simplicity we may remark that the 
 product of any number of quaternions, p, q, r t s, say, in the 
 order named, is independent of the manner of association in 
 that order. That is, 
 
 pqrs =p(qrs) = (pq) (rs) = (pqr)s. (1 59) 
 
 This remarkable property is the foundation of the simplicity 
 of the quaternionic algebra. It is uniquely simple. No algebra 
 of vectors can ever match it, and Prof. Tait is quite right in his 
 laudations from the quaternionic point of view. 
 
 But when Prof. Tait vaunts the perfect fitness and natural- 
 ness of quaternions for use by the physicist in his inquiries, 
 I think that he is quite wrong. For there are some very 
 serious drawbacks connected with quaternions, when applied 
 to vectors. The quaternion is regarded as a. complex of 
 scalar and vector, and as the principles are made to suit the 
 quaternion, the vector itself becomes a degraded quaternion, 
 and behaves as a quaternion. That is, in a given equation, 
 one vector may be a vector, and another be a quaternion. Or 
 the same vector in one and the same equation, may be a 
 vector in one place, and a quaternion (versor, or turner) in 
 another, This amalgamation of the vectorial and quaternionic 
 
ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 303 
 
 functions is very puzzling. You never know how things will 
 turn out. 
 
 Again, the vector having to submit to the quaternion, leads 
 to the extraordinary result that the square of every vector is a 
 negative scalar. This is merely because it is true for quad- 
 rantal versors, and the vector has to follow suit. The reciprocal 
 of a vector, too, goes the wrong way, merely to accommodate 
 versors and quaternions. 
 
 And yet this topsyturvy system is earnestly and seriously 
 recommended to physicists as being precisely what they want. 
 Not a bit of it. They don't want it. They have said so by 
 their silence. Common sense of the fitness of things revolts 
 against the quaternionic doctrines about vectors. Nothing 
 could be more unnatural. 
 
 Are they even convenient 1 Not to the physicist. He is very 
 much concerned with vectors, but not at all, or at any rate 
 scarcely at all, with quaternions. The vector algebra should 
 satisfy his requirements, not those of the quaternion. Let the 
 quaternion stand aside. The physicist wants, above all, to 
 have clear ideas, and to him the double use of vectors, as 
 vectors and versors, combined with unnatural properties of 
 vectors to suit quaternions, is odious. Once a vector, always 
 a vector, should be a cardinal axiom. 
 
 If the usual investigations of physical mathematics involved 
 quaternions, then the physicist would no doubt have to use 
 them. But they do not. If you translate physical investi- 
 gations into vectorial language, you do not get quaternions ; 
 you get vector algebra instead. Even Prof. Tait's treatise 
 teaches the same lesson, for in his physical applications the 
 quaternion is hardly ever concerned. It is vector algebra, 
 although expressed in the quaternionic notation. 
 
 Vectors should be treated vectorially. When this is done, 
 the subject is much simplified, and we are permitted to arrange 
 our notation to suit physical requirements. This is a very 
 important matter. Most calculations are, and always will be, 
 scalar calculations, that is, according to common algebra. The 
 special extensions to tridimensional space involving vectors 
 should therefore be done so as to harmonise with the scalar 
 calculations, and so that only one way of thinking is required, 
 instead of two discrepant ways, and so that mutual conversion 
 
304 ELECTROMAGNETIC THEORY. CH. III. 
 
 of scalar aiid vector algebra is facilitated. The system which 
 I expound I believe to represent what the physicist wants, at 
 least to begin with. It is what I have been expounding, 
 though rather by example than precept, since 1882. There is 
 only one point where I feel inclined to accept a change, viz., to 
 put a prefix before the scalar product, say Sab instead of ab, to 
 balance Vab. Not that the S prefix is of any use in the algebra 
 as I do it, but that mathematical writers on vectors seem to 
 want to put the scalar and vector parts together to make a 
 general or complete product, say Qab = Sab + Vab. The function 
 Qab, however, does not occur in physical applications, which are 
 concerned with the scalar and vector products separately. If I 
 could omit the V as well as the S I would do so ; but some sign 
 is necessary, and the V of Hamilton is not very objectionable. 
 
 Gibbs denotes the scalar and vector products by a./3 and a x ft 
 using Greek letters. In my experience this is not a good way 
 when worked out. It is hard to read, and is in serious conflict 
 with the common use of dots and crosses in algebra, which use 
 has to be given up. I use dots in their common meaning, 
 either as multipliers or separators. 
 
 Prof. Macfarlane, who is the latest vectorial propagandist, 
 denotes the scalar and vector products by cos ab and Sin ab. 
 The trigonometrical origin is obvious. But, on this point, I 
 would refer the reader to the equations (155) (156) above, and 
 to the very trenchant remark of Prof. Gibbs which I have 
 already quoted, as to the incomparably greater amenity of the 
 scalar and vector product to algebraical treatment than the 
 trigonometrical functions themselves. The inference is that 
 vector algebra is far more simple and fundamental than trigo- 
 nometry, and that it is a mistake to base vectorial notation 
 upon that of a special application thereof of a more complicated 
 nature. I should rather prefer sea ab and vec ab. Better still, 
 Sab and Vab, with the understanding that the S may be 
 dropped unless specially wanted. 
 
 Macfarlane has also a peculiar way of treating quaternions, 
 about which I will express no opinion at present, being doubtful 
 whether, if the use of quaternions is wanted, the quater- 
 nionic system should not be used, with, however, a distinction 
 well preserved between the vector and the quaternion, by 
 special type or otherwise. 
 
ELEMENTS OF VECTORTAL ALGEBRA AND ANALYSIS. 305 
 
 I am in hopes that the Chapter which I now finish may serve 
 as a stopgap till regular vectorial treatises come to be written 
 suitable for physicists, based upon the vectorial treatment of 
 vectors. The quaternionists want to throw away the " cartesian 
 trammels," as they call them. This may do for quaternions, 
 but with vectors would be a grave mistake. My system, so far 
 from being inimical to the cartesian system of mathematics, is 
 its very essence. 
 
CHAPTER IV. 
 
 THEORY OF PLANE ELECTROMAGNETIC WAVES 
 
 Action at a Distance versus Intermediate Agency. Contrast 
 of New with. Old Views about Electricity. 
 
 176. It has often been observed that the universe is in an 
 unstable condition. Nothing is still. Nor can we keep motion, 
 once produced, to a particular quantity of matter. It is diffused 
 or otherwise transferred to other matter, either immediately or 
 eventually. The same fact is observed in the moral and intel- 
 lectual worlds as in the material, but this only concerns us so 
 far as to say that it underlies the communication of knowledge 
 to others when the spirit moves, even though the task be of a 
 thankless nature. 
 
 The laws by which motions, or phenomena which ultimately 
 depend upon motion, are transferred, naturally form an import- 
 ant subject of study by physicists. There are two extreme 
 main views concerning the process. There is the theory of 
 instantaneous action at a distance between different bodies 
 without an intervening medium ; and on the other hand there 
 is the theory of propagation in time through and by means of 
 an intervening medium. In the latter case the distant bodies 
 do not really act upon one another, but only .seem to do so. 
 They really act on the medium directly, and between the two 
 are actions between contiguous parts of the medium itself. But 
 in the former case the idea of a medium does not enter at all. 
 We may, however, somewhat modify the view so that action, 
 though seemingly instantaneous and direct, does take place 
 through an intervening medium, the speed of transmission 
 being so great as to be beyond recognition. Thus, the two 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 307 
 
 ideas of direct action, and through a yielding medium, may be 
 somewhat harmonised, by being made extreme cases of one 
 theory. For example, if we know that there is a yielding 
 medium and a finite speed, but that in a certain case under 
 examination the influence of the yielding is insensible, 
 then we may practically assume the speed to be infinite. But 
 although this comes to the same thing as action at a distance, 
 we need not go further and do away with the medium alto- 
 gether. 
 
 There is another way of regarding the matter. We may 
 explain propagation in time through a medium by actions at a 
 distance. But this is useless as an explanation, being at best 
 merely the expression of a mathematical equivalence. 
 
 Now consider the transmission of sound. This consists, 
 physically, of vibratory motions of matter, and is transmitted 
 by waves in the air and in the bodies immersed in it. The 
 speed through air is quite small, so small that it could not even 
 escape the notice of the ancients, who were, on the whole, 
 decidedly unscientific in their mental attitude towards natural 
 phenomena. Suppose, however, that the speed of transmission 
 of sound through air was a large multiple of what it is, so that 
 the idea of a finite speed, or of speed at all, did not present 
 itself. We should then have the main facts before us, that 
 material bodies could vibrate according to certain laws, and 
 that they could, moreover, set distant bodies vibrating. This 
 would be the induction of vibrations, and it might be explained, 
 in the absence of better knowledge, by means of action at a 
 distance of matter upon matter, ultimately resolvable into some 
 form of the inverse square law, not because there is anything 
 essentially acoustical about it, but because of the properties of 
 space. Furthermore, we should naturally be always associating 
 sound with the bodies bounded by the air, but never with the 
 air itself, which would not come into the theory. A deep- 
 minded philosopher, who should explain matters in terms of 
 an intervening medium, might not find his views be readily 
 accepted, even though he pointed out independent evidence for 
 the existence of his medium physiological and mechanical 
 and that there was a harmony of essential properties entailed. 
 Nothing short of actual proof of the finite speed of sound 
 would convince the prejudiced. 
 
 x2 
 
308 ELECTROMAGNETIC THEORY. CH. IV 
 
 Now, although there is undoubtedly great difference In detail 
 between the transmission of sound and of electrical disturb- 
 ances, yet there is sufficient resemblance broadly to make the 
 above suppositional case analogous to what has actually hap- 
 pened in the science of eleetromagnetism. There were con- 
 ductors and non-conductors, or insulators, and since the finite 
 speed of propagation in the non-conducting space outside con- 
 ductors was unknown, attention was almost entirely concen- 
 trated upon the conductors and a suppositional fluid which 
 was supposed to reside upon or in them, and to move 
 about upon or through them. And the influence on dis- 
 tant conductors was attributed to instantaneous action 
 at a distance, ignoring an intermediate agency. Again, 
 a very deep-minded philosopher elaborated a theory to ex- 
 plain these actions by the intermediate agency of a yielding 
 medium transmitting at finite speed. But although in doing 
 so he utilised the same medium for whose existence there was 
 already independent evidence, viz., the luminiferous ether, 
 and pointed out the consistency of the essential properties re- 
 quired in the two cases, and that his electromagnetic theory 
 made even a far better theory of light than the old, yet his views 
 did not spread very rapidly. The old views persisted in spite 
 of the intrinsic probability of the new, and in spite of the large 
 amount of evidence in support of the view that some medium 
 outside conductors, and it may be also inside them as well, but 
 not particularly conducting matter itself, was essentially con- 
 cerned in the electrical phenomena. The value and validity 
 of evidence varies according to the state of mind of the judge. 
 To some who had seriously studied Maxwell's theory the 
 evidence in its favour was overwhelming; others did not believe 
 in it a bit. I am, nevertheless, inclined to think that it would 
 have prevailed before very long, even had no direct evidence of 
 the finite velocity been forthcoming. It was simply a question 
 of time. But the experimental proof of the finite speed of 
 transmission was forthcoming, and the very slow influence of 
 theoretical reasoning on conservative minds was enforced by 
 the common-sense appeal to facts. It is now as much a fact 
 that electromagnetic waves are propagated outside conductors 
 as that sound waves are propagated outside vibrating bodies. 
 It is as legitimate a scientific inference that there is a medium 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 300 
 
 to do it in one case as in the other, and there is inde- 
 pendent evidence in favour of both media, air for sound 
 and ether for electrical disturbances. It is also so exces- 
 sively probable now that light vibrations are themselves 
 nothing more than very rapid electromagnetic vibrations, that 
 I think this view will fully prevail, even if the gap that exists 
 between Hertzian and light vibrations is not filled up ex- 
 perimentally, through want of proper appliances provided 
 some quite new discovery of an unanticipated character is 
 not made that will disprove the possibility of the assumed 
 identity. 
 
 Now, the immediate question here is how to propagate a 
 knowledge of the theory of electromagnetic waves. If we 
 could assume the reader to have had a good mathematical train- 
 ing, that would greatly ease matters. On the other hand, it is 
 hardly any use trying to do it for those who have no mathe- 
 matical knowledge least of all for those anti-mathematical 
 attackers of the theory of the wave propagation of electrical 
 disturbances who show plainly that they do not even know in- 
 telligibly what a wave means, and what is implied by its existence. 
 But, between the two, I think it is possible to do a good deal in 
 the way of propagation by means of a detailed exposition of the 
 theory of plane waves, especially in dielectrics. By the use of 
 plane waves, the mathematical complexity of waves in general 
 in a great measure disappears, and common algebra may be 
 largely substituted for the analysis which occurs in more ad- 
 vanced cases. Most of the essential properties and ideas may 
 be assimilated by a thinking reader who is not advanced in his 
 mathematics, and what is beyond him he can skip. Besides 
 this, the treatment of plane waves is itself the best preliminary 
 to more general cases. 
 
 The problems that present themselves by the artificial limi- 
 tation to plane waves are often of an abstract nature. There 
 are, however, some important exceptions; the most notable 
 being that of propagation along straight wires, of which the 
 theory is essentially that of plane waves, modified by the 
 resistance of the wires. Before, however, proceeding to the 
 details of plane waves, which will form the subject of this 
 chapter (although it will be needless to altogether exclude con- 
 nected matters), it will be desirable to prepare the mind by an 
 
310 ELECTROMAGNETIC THEORY. CH. IV- 
 
 outline of some general notions concerning electromagnetic 
 waves, irrespective of their precise type. 
 
 General Notions about Electromagnetic Waves. Generation 
 
 of Spherical Waves and Steady States. 
 
 \ 
 
 177. Consider a n^n-conducting dielectric, to begin with. 
 The two properties it possesses of supporting electric displace- 
 ment and magnetic induction, which we symbolise by ju- and c, 
 the inductivity and permittivity, are, independently of our actual 
 ignorance of their ultimate nature, so related that the speed of 
 propagation v depends upon them in the way expressed by the 
 equation /xcv 2 = 1, or v = (/AC)-*. Thus, an increase either of the 
 permittivity, or of the inductivity, lowers the speed of trans- 
 mission. In transparent bodies we cannot materially alter the 
 inductivity, but we can very considerably increase the per- 
 mittivity, and so lower the speed. If on the other hand, we 
 wish to have infinite speed, for some practical purpose of calcu- 
 lation, we may get it by assuming either p = or else c = 0. 
 In the latter case, for example, we destroy the power of sup- 
 porting electric displacement, whilst preserving the magnetic 
 induction. This is what is done in magnetic problems (the 
 theory of coils, self and mutual induction) when we ignore the 
 existence of electric displacement. On the other hand, in the 
 theory of condensers connected up by inductionless resistances, 
 and in the electrostatic theory of a submarine cable, it is the 
 magnetic induction that is ignored, in a manner equivalent to 
 supposing that /* = 0. In either case we have infinite speed or 
 apparent instantaneous action. 
 
 Now, consider some of the consequences of the property of 
 propagation with finite speed. Let there be a source of dis- 
 turbance at a point for simplicity. It may be either an electric 
 source or a magnetic source. By an electric source we mean 
 a cause which will, if it continue steadily acting, result in 
 setting up a state of electric displacement, whilst a magnetic 
 source steadily acting would result in a state of magnetic 
 induction. In the former case there will be no magnetic 
 induction along with the displacement, and in the latter case 
 no displacement along with the induction. Now, if the speed 
 of propagation were infinite in the former case of an electric 
 source, by the non existence of p, the steady state would be 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 311 
 
 set up instantly, and consequently all variations of the intensity 
 of the source would be immediately and simultaneously accom- 
 panied by the appropriate corresponding distribution of dis- 
 placement in the dielectric. Similarly, with a magnetic source, 
 if the speed be infinite by the non-existence of c, all variations in 
 the source will be simultaneously accompanied by the distribu- 
 tion of magnetic induction appropriate to the instantaneous 
 strength of the source. 
 
 Now do away with the artificial assumption made, and let 
 both p and c be finite, and v therefore also finite. At the time 
 t after starting a source, the extreme distance reached by the 
 disturbance it produces is vt. That is to say, beyond the sphere 
 of radius vt, whose centre is at the source, there is no disturb- 
 ance, whilst within it there is. The wave-front is thus a spheri- 
 cal surface of radius vt, increasing uniformly with the time. 
 Along with this continuous expansion of the range of action 
 there are some other things to be considered. The mere spread- 
 ing causes attenuation, or weakening of intensity as the disturb- 
 ance travels away from the source. Besides this, the intensity 
 of disturbance is not the same at a given distance in all direc- 
 tions from the source, and since the displacement and induction 
 are vectors, their directions are not everywhere the same. They 
 are distributed in the circuital manner, so that we have expand- 
 ing rings or sheets of displacement or induction. Whether the 
 source be of the electric or the magnetic kind, it produces both 
 fluxes initially and when varying, and generally speaking to an 
 equal degree as regards energy. This is a main characteristic 
 of pure electromagnetic waves, a coexistence of the electric and 
 magnetic fluxes with equal energies ; and if the source be in a 
 state of sufficiently rapid alternating variation this state of 
 things continues. But if the source, after varying in any way, 
 finally become steady, the generation of electromagnetic dis- 
 turbances will speedily cease, and the production of a steady 
 state will begin round the source of displacement if the source 
 be electric, and of induction if the source be magnetic whilst 
 the previously generated electromagnetic disturbances will pass 
 away to a great distance. In the end, therefore, we have 
 simply the steady state of the flux corresponding to the 
 impressed force, whilst the electromagnetic disturbances are 
 out of reach, still spreading out, however t and in doing so, 
 
312 ELECTROMAGNETIC THEORY. CH. IV. 
 
 leaving behind them the outside portion of the residual steady 
 flux due to the source, which steady flux, however, requires an 
 infinite time to become quite fully established. 
 
 We may take a specially simple case in illustration of the 
 above general characteristics. Let the source of energy be 
 contained within a small spherical portion of the dielectric, of 
 radius a, and let it be of the simplest type, viz., a uniform 
 distribution of impressed electric force within the sphere. We 
 know by static considerations alone what the nature of the 
 final displacement due to such a source is. It is a circuital 
 distribution, out from the sphere on one half, and in on the 
 other, connection being made through the sphere itself by a 
 uniform distribution ; being, in fact, of the same nature as the 
 induction due to a spherical body uniformly magnetised in a 
 medium of the same inductivity as its own. Now, we can 
 describe the setting up of the final steady state due to the 
 source, when it is suddenly started and kept constant later, 
 thus : At the first moment an electromagnetic wave is generated 
 on the surface of the sphere, which immediately spreads both 
 ways and becomes a spherical shell, whose outer boundary 
 goes outward at speed v, whilst the inner boundary goes inward. 
 At the time t = a/v, therefore, the disturbance fills the sphere 
 of radius 2a, the centre being just reached. The steady state 
 then begins to form, commencing at the centre and expand- 
 ing outwards thereafter at speed . At the time t = 2a/v, 
 therefore, we have the steady state fully formed within 
 the sphere of radius a, whilst just outside it is an electro- 
 magnetic shell of depth 2a. Up to this moment the im- 
 pressed force has been continuously working not, indeed, 
 in all parts of the sphere it occupies, but in all parts 
 passed by the front of the inward wave from the beginning 
 until the centre was reached, and after that, in the parts not 
 occupied by the already formed steady state. Consequently, at 
 the moment t = 2a/v, when the steady state is fully formed 
 throughout the region occupied by the impressed force, the 
 latter ceases to work. It has wholly done its work. The 
 amount done is twice the energy of the final complete steady 
 state, say 2U. The rest of the work is done by the electro- 
 magnetic wave itself. For the subsequent course of events is 
 that the fully-formed electromagnetic shell of depth 2a runs 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 313 
 
 out to infinity, of course expanding on the way, and in doing 
 so it leaves the steady state behind it. That is to say, it 
 drops a part of its contents as it moves on, so that the steady 
 state is always fully formed right up to the rear of the expanding 
 shell during the whole of its passage to infinity. This shell is 
 not a pure electromagnetic wave, with equal electric and 
 magnetic energies. The magnetic energy is constant, of 
 amount ^U, on the whole journey, but the electric energy 
 is in excess. The excess is employed in forming the steady 
 state, so that when the shell has reached a great distance it 
 becomes appreciably a pure electromagnetic wave, having the 
 amount U of energy, half magnetic, half electric, with a slight 
 excess in the latter, to be later left behind in forming the 
 remainder of the steady state. The energy U of the shell is 
 wholly wasted if the dielectric be unbounded, for there is 
 nothing to stop the transference to an infinite distance. 
 
 If we wish to form the steady state without this waste of 
 energy, we must bring the impressed force into action very 
 gradually infinitely slowly, in fact. In this way the energy 
 wasted in the very weak electromagnetic waves generated will 
 tend to become infinitely small, and the work done by the 
 impressed force will tend to the value U, the energy of the 
 steady state. 
 
 When the sphere of impressed force is not finite, but is 
 infinitely small, so that it may be regarded as a special kind of 
 point-source, we simplify matters considerably in some respects. 
 For, by the above, the result is that the moment the source 
 starts, say at full strength, the steady state also immediately 
 begins to grow, so that at the time t it occupies the sphere of 
 radius vt. Outside it there is no disturbance, but on its surface is 
 an infinitely thin electromagnetic shell, which performs the same 
 functions as the previous finite shell. That is, it lays down 
 the steady state as it expands, and then carries out to infinity 
 in itself as much energy as it leaves behind. 
 
 Notice, however, this peculiarity, that if we had started with 
 what appears at first sight to be the simpler problem, viz., a 
 point-source, we should be quite unable to see and understand 
 the functions of the electromagnetic shell on the extreme verge 
 of the steady field. The reader may compare this case with 
 that of the establishment of the steady state of displacement 
 
314 ELECTROMAGNETIC THEORY. OH. IV. 
 
 due to another kind of point-source, viz., electrification suddenly 
 brought to rest at a point after previous motion at the speed of 
 light, as discussed in 55. There is a perfect similarity as re- 
 spects the uniform growth of the steady state. But the nature 
 of the bounding electromagnetic shell is not the same. In the 
 case of electrification it belongs to the zero degree of spherical 
 harmonics ; in our present problem it is of the first degree. 
 
 If, after keeping on the impressed force at a point for, say, 
 an interval T, we suddenly remove it, this is equivalent to 
 keeping it on, but with the addition of an impressed force which 
 is the negative of the former. From this we see that the re- 
 sult of putting on the impressed force for an interval T only is to 
 generate a shell of depth VT, which runs out to infinity. 
 Within this shell is the steady electric displacement due to the 
 source appropriate to the instantaneous position of the shell. 
 On its outer surface is an electromagnetic wave moving out to 
 infinity, and generating or laying down the steady electric dis- 
 placement as it goes, whilst on the inner surface is another 
 wave running after the first, and undoing its effects. That is, 
 it takes up the displacement laid down by the first wave, so 
 that inside the inner wave (just as outside the outer) there is 
 no disturbance. 
 
 Intermittent Source producing Steady States and Electro- 
 magnetic Sheets. A Train of S.H. Waves. 
 
 178. Using the same kind of point-source, let it act inter- 
 mittently and alternatingly ; say, on positively for an interval 
 a, then off for an interval /2, then on negatively for an interval 
 a, followed by off for an interval ft, and so on recurrently, like 
 positive and negative applications of a battery with dead in- 
 tervals. The result is, by the last case, to divide the spherical 
 space occupied by the disturbances at a given moment into 
 concentric shells, of depths va and v/3 respectively. In the 
 latter is no disturbance, since they correspond to the dead in- 
 tervals. In the former are the steady displacements appro- 
 priate to their distance from the source, consecutive shells 
 being positive or negative according to the state of the source 
 when they started from it. On their boundaries are electro- 
 magnetic sheets generating these steady states continuously on 
 one side, and destroying them on the other. 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 315 
 
 Next, do away with the dead intervals, so that the applied 
 force is like that of a common reversing key, first positive 
 of full strength, and then negative of full strength, but without 
 any interval. The result is to completely fill up the sphere of 
 disturbance, the successive shells of depth va containing steady 
 fields alternately positive and negative being brought into 
 contact, and only separated from one another by the electro- 
 magnetic sheets. This may be considered rather an unexpected 
 result. Our impressed force is periodic, but expresses the 
 extremest form of variation, namely discontinuities, and is only 
 expressible simple-harmonically by an infinite series of simple- 
 harmonic forces of all frequencies from zero to infinity. But 
 only when the impressed force varies is electromagnetic dis- 
 turbance generated, so that when in the above manner we 
 confine its variations to be momentary, the electromagnetic 
 disturbances are also momentary, with the consequent result 
 that we have our dielectric occupied by quite steady states of 
 displacement separated by infinitely-thin electromagnetic shells. 
 The latter travel. The steady states also appear to travel, but 
 do not really do so. They are being continuously generated and 
 destroyed at their boundaries, but are quite steady elsewhere. 
 
 If the impressed force vary simple-harmonically instead of 
 discontinuously, the result is less simple. We find that there 
 is no clean separation into steady states and electromagnetic 
 waves, the two being mixed inextricably. We can see this by 
 substituting for the simply periodic variation of the force a 
 very great number of constant forces of small duration and of 
 different strengths, so as to.roughly imitate the simple- harmonic 
 variation. Every discontinuity in the force will behave in the 
 way described, and the final result of the superimposition of 
 effects, when we proceed to the limit and have continuous 
 simple-harmonic variation of the force, is a train of simple- 
 harmonic waves proceeding from the source. They are not 
 pure electromagnetic waves, and there is no steady state any- 
 where. The wave-length X is given by A = VT, where T is the 
 period of the impressed force, or the reciprocal of the 
 frequency. The electric and magnetic forces are simple-har- 
 monic functions of the time everywhere, right up to the 
 extreme wave-front of the disturbance, where, of course, is 
 an electromagnetic sheet following a different law. But 
 
316 ELECTROMAGNETIC THEORY. CH. IV. 
 
 should the source occupy a finite spherical space, then 
 the bounding sheet becomes a shell, of depth equal to the 
 diameter of the sphere, as before described. Now the steady 
 fields become insensible at a great distance in the discontinuous 
 case ; to correspond with this we have the fact that the simple 
 harmonic waves tend to become pure as they expand. The 
 greatest departure from the pure state is round about the 
 source. Here the frequency becomes a matter of importance. 
 When very low, the wave-length is great, and we should, there- 
 fore, have to go a great distance to find any sign of wave- 
 propagation. Moreover, the waves themselves would be 
 exceedingly weak. Only round the source is there sensible 
 disturbance, and it is practically the steady displacement 
 appropriate to the momentary state of the source, accompanied 
 by a weak state of magnetic force connected with the displace- 
 ment nearly in the instantaneous manner. That is, the time- 
 variation of the displacement is the electric current, and the 
 displacement itself is sensibly that of the static theory. Under 
 these circumstances there is next to no waste of energy. 
 
 But by increasing the frequency we can completely alter this 
 state of things. For we increase the rate of change of the 
 impressed force, and therefore the strength of the electromagnetic 
 waves generated, for one thing. Besides this, we can bring the 
 region of electromagnetic waves nearer to the source, and so 
 contract the region in which the displacement was approxi- 
 mately static as much as we please. In the limit with very great 
 frequency we have nearly pure electromagnetic waves close up 
 to the source. The waste of energy increases very rapidly with 
 the frequency, varying as its fourth power with simple-harmonic 
 waves. The extremest case of waste is that of a discontinuity, 
 already mentioned, as when setting up the steady state we 
 waste half the work done, and then, when the force is removed, 
 waste the other half. 
 
 Self-contained Forced Electromagnetic Vibrations. Contrast 
 with Static Problem. 
 
 179. When the sphere of impressed force is of finite size, 
 another effect comes into view of a somewhat striking character, 
 of which the theory of a point-source gives no information. By 
 increasing the frequency sufficiently, we shall reduce the ampli- 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 317 
 
 tude of the external vibrations, and on reaching a certain fre- 
 quency depending upon the size of the sphere, they will entirely 
 vanish. That is, at a certain frequency, a simple harmonic 
 impressed force, acting uniformly within a spherical space in a 
 dielectric, produces no external effect whatever. The vibrations 
 are stationary or standing, and are purely internal, or confined 
 to the sphere itself. By general experience of statical problems 
 this would seem to be impossible. But we must expect to find 
 strange manners when we go into strange lands. 
 
 This strange behaviour arises from the general electromag- 
 netic property that the true source of disturbances due to im- 
 pressed force (electric or magnetic) is the curl thereof. The 
 impressed force itself, on the other hand, is associated with the 
 supply of energy. (See 87.) In the present case the sur- 
 face of the sphere of impressed force is the seat of its curl, its 
 intensity varying as the cosine of the latitude, if the polar axis 
 be parallel to the impressed force ; and consequently, as already 
 described for the case of setting up the steady state, the elec- 
 tromagnetic waves proceed both ways from the surface at 
 which they originate. The waves going inward contract, then 
 cross at the centre and expand again. We have, therefore, 
 two trains of outward waves, and the external effect may be 
 imagined to be due to their superimposition. That they may 
 sometimes assist and sometimes partly cancel one another, 
 according to relative phase, may be readily conceived, and 
 the theory indicates that at any one of an infinite series of 
 definite frequencies there is complete cancellation externally. 
 This refers to the simple-harmonic state. There is always an 
 external effect initially, at any frequency namely, the initial 
 electromagnetic shell of depth equal to the diameter of the 
 sphere of impressed force, because it consists entirely of the 
 beginning part of the first outward wave, the second one only 
 reaching to its rear ; but on the inner side of this shell there 
 may be no disturbance, except within the sphere of impressed 
 force. The waste of energy is not continuous at the critical fre- 
 quencies, but consists merely of the energy in the initial shell. 
 The impressed force works continuously, but as much positively 
 as negatively within a period, so that it is inactive on the whole. 
 
 Effects of the above described kind are principally remark- 
 able from the contrast they present towards more familiar 
 
318 ELECTROMAGNETIC THEORY. CH. IV. 
 
 effects, especially those of a statical kind. The behaviour of 
 electrification on a sphere may be taken as an illustrative con- 
 trast. We know that if we have a spherical surface in a uni- 
 form dielectric covered with electrification, the displacement is 
 partly internal and partly external. Only when the electri- 
 fication is uniformly spread does the displacement vanish 
 internally, and become wholly external. This was for- 
 merly explained by the inverse-square law. Every element 
 of electrification was supposed to exert force equably in 
 all directions round itself, internally as well as externally, 
 and a mathematical consequence of this is the complete 
 cancellation of the internal force when the electrification 
 is uniformly spread on the surface of a sphere. The same 
 reasoning was applied when the sphere was not of the same 
 nature as the external medium when it was a conductor; to 
 wit. Now the internal force is also zero in this case. More- 
 over, it is zero whatever be the nature of the internal material 
 (without electrical sources). But it is certainly untrue that 
 the electric force due to a point-source of displacement is the 
 same in all directions around it when the medium is not homo- 
 geneous. So the reason for the absence of internal force is 
 entirely wrong, although it seems right in one case, viz., that 
 of homogeneity. The true, and sufficient, and comprehensive 
 reason for the equilibrium, and the absence of internal force is 
 the satisfaction of the second circuital law when the external 
 electric force is perpendicular to the surface. This is independent 
 of the nature of the internal matter, and is obviously to be pre- 
 ferred to a hypothesis that is only valid sometimes. Varying the 
 internal material will make the law of spreading of force be 
 different in any number of ways, and, if we like, different 
 for every element of electrification, by arranging various 
 kinds of heterogeneity. But independently of this, we might 
 vary the law of force in many ways in special cases. 
 Thus, in the case of a uniform spread of electrification 
 on a sphere, we may imagine the displacement from any 
 particle of electrification to radiate in any symmetrical manner 
 that will give zero displacement inside and a uniformly 
 radial displacement outside. The simplest case after the usual 
 uniform spreading in all directions, is a planar spreading. Let 
 every element of electrification on the surface send out its 
 
THEORY OF i'LANE ELECTROMAGNETIC WAVES. 319 
 
 displacement in the tangential plane only, though equally in 
 all directions in that plane. This will give the correct static 
 result. It is not altogether a fantastic example, for we can 
 make an electromagnetic problem of it by letting the electrified 
 surface expand at the speed of light. Then the assumed law 
 of spread is the actual law, for every element of electrification 
 is the core of a plane electromagnetic wave. The resultant mag- 
 netic force due to all the waves, however, is zero, and the resultant 
 electric force is as in the static problem. (See 61 and 164.) 
 
 Now when we have vibrating electromagnetic sources on 
 a spherical surface producing electromagnetic waves simple- 
 harmonically, we also in general have both internal and 
 external effects, for there is an internal as well as an exter- 
 nal train of waves. But whilst they cannot cancel internally, 
 they may do so externally. Here is one contrast with the 
 static problem, and along with it is another, viz., that it is the 
 nature of the external medium that is now indifferent (with a 
 reservation), whereas in the static problem we have independence 
 of the nature of the internal medium. The reservation hinted at 
 is connected with the initial uncancelled electromagnetic shell. 
 The external medium should allow it to escape, or absorb it 
 somehow, so that it will not interfere with the effects under 
 consideration. To regard the zero external effect as being 
 actually due to the coexistence of two trains of waves which 
 cancel one another is merely a mathematical artifice, however, 
 because neither train of waves exists. If they did exist we 
 should have to vary their nature to suit the constitution of the 
 external medium, just as in the problem of static equilibrium 
 we require different laws of force to suit the nature of the 
 internal medium. The comparison of the static with the 
 kinetic problems bhows a complete reversal of relations as 
 regards internal and external. 
 
 What we can do with a single surface electromagnetic source 
 we can repeat with others inside it. We see, therefore, that 
 the whole sphere of impressed force may be filled up with 
 vibratory sources of the most vigorous nature without producing 
 any (except initial) external effect, if their periods be properly 
 chosen in relation to their situation, which problem admits of 
 multiple solutions. This is suggestive as regards the stores o> 
 energy bound up with matter. 
 
320 ELECTROMAGNETIC THEORY. CH IV. 
 
 Relations between E and H in a Pure Wave. Effect of Self- 
 induction. Fatuity of Mr. Preece's " KR law." 
 
 180. In the above very little has been said about the dis- 
 tribution of electric and magnetic force from part to part of 
 an electromagnetic wave. This varies greatly in different 
 kinds of waves, and cannot be explained without the formulae. 
 But there is one very important property of pure electromag- 
 netic waves (which also holds good, more or less approximately, 
 in general) which may be described at present. As already 
 mentioned, the electric and magnetic energies are equal in a 
 pure electromagnetic wave. Now the density of the electric 
 energy is -JcE 2 , if E is the electric force (intensity), and that 
 of the magnetic energy is J/*H 2 . If we equate these, we obtain 
 a relation between E and H. Thus 
 
 (1) 
 
 Also /*cv 2 = l ..... (2) 
 
 Therefore E = jj.vR ....... (3) 
 
 The positive or negative sign depends on which way the wave 
 is going. Disregarding this, the electric and magnetic forces 
 have a constant ratio. They are therefore in the same phase, 
 or keep time together in all their variations without lag or 
 lead. Along with this, they have the property of being per- 
 pendicular to one another (that is, E and H are perpendicular, 
 not E and H), and their plane is in the wave-front, or the 
 direction of motion of the wave is perpendicular to E and to H. 
 It is the direction of the flux of energy. These properties hold 
 good in all parts of a pure electromagnetic wave, although the 
 magnitudes and directions of the electric and magnetic forces 
 may vary greatly from one part of the wave to another. 
 
 The above is also the state of things that obtains, more or 
 less perfectly or imperfectly, in long-distance telephony over 
 copper circuits of low resistance, and by lowering the resistance 
 per mile we may approximate as nearly as we please to the 
 state of pure electromagnetic waves. It is the self-induction 
 that brings about this state of things, showing such a contrast 
 to the more familiar relations between the electric and magnetic 
 forces on circuits in general. Like a kind of fly-wheel, the 
 self-induction imparts inertia and stability, and keeps the waves 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 321 
 
 going. It is the long-distance telephoner's best friend who 
 was, not many years since, spurned with contempt from the 
 door. Some people thought there was a very absurd fuss made 
 about self-induction, and that it was made a sort of fetish of ; 
 so, knowing no better, they poured much cold water on the 
 idol. But, whatever opinions we may hold regarding their 
 competence as judges, there can be no question about the stern 
 logic of facts. Self-induction came to stay, and stayed it has, 
 and will stay, having great staying power. Whatever should 
 we think of engineers who declined to take into account the 
 inertia of their machinery? There was also some consider- 
 able fuss made about a supposed law of the squares, or KR 
 law as it was or is called, according to which you could 
 not telephone further than KR = such or such a number, 
 because the speed of the current varied as the square of 
 the length of the line, or else inversely. But in spite of the 
 repeated attempts made to bolster up the KR law, the critical 
 number has kept on steadily rising ever since. I see now that 
 it has gone up to 32,000.* But it need not stop there. Make 
 your circuits longer, and it will go up a lot more. 
 
 As regards the ether, it is useless to sneer at it at this time 
 of day. What substitute for it are we to have ? Its principal 
 fault is that it is mysterious. That is because we know so 
 little about it. Then we should find out more. That can- 
 not be done by ignoring it. The properties of air, so far 
 as they are known, had to be found out before they became 
 known. 
 
 Wave-Fronts; their Initiation and Progress. 
 
 . 181. Still keeping to a simple dielectric, we may always, 
 by consideration of the fact that the speed of propagation is v, 
 find the form of the wave-front due to any collection of point- 
 sources, and trace the changes of shape and position it under- 
 
 * See The Electrician, December 30, 1892, p. 251, for data, in article by 
 Mr. Jos. Wetzler. The 32,000 is for the New York-Chicago circuit of 1,000 
 miles at 4'12 ohms per mile, or 2'06 ohms per mile of wire. Comparison of 
 Mr. Wetzler's with Mr. Preece's figures is interesting. Good telephony is 
 got by Mr. P. at 10,000, and by Mr. W. at 45,000 ; excellent by Mr. P. at 
 5,000, and by Mr. W. at 31,000 ; and so on. But there is still less of a 
 KR law in the American than in the English cases. 
 
 Y 
 
322 ELECTROMAGNETIC THEORY. CH IV. 
 
 goes as It progresses. For obviously, if a point P be at a dis- 
 tance vt from the nearest source, no disturbance can have 
 reached P from it if it started into action after the moment 
 t = ; and generally, the effect at P at a given moment arising 
 from a particular source depends upon its state at the moment 
 r/v earlier (if r be the distance from the source to P), and upon 
 the previous state of the source, causing residual or cumula- 
 tive action at P. We therefore know the limiting distance of 
 action of the sources for every one of their momentary states. 
 The wave-front belonging to a set of disconnected point-sources 
 consists initially of disconnected spheres. But as they expand, 
 they merge into one another to form a continuous extreme wave- 
 front consisting of portions of spheres. When the point-sources 
 are spread continuously over a surface to form a surf ace-source, we 
 have a continuous wave-front from the first moment ; or rather, 
 two wave-fronts, one on each side of the surface. There is an 
 exception, to be mentioned later, when the sources send waves 
 one way only from the surface. This is not the already 
 described case of the cancelling of two trains of waves at par- 
 ticular frequencies, but is a unilateral action obtained by a 
 special arrangement of surface-sources. Passing over this, 
 observe that when we have once got a wave-front we may ignore 
 the sources which produced it, and make the wave-front itself 
 tell us what its subsequent history will be. For, to trace the 
 course of the wave-front from one moment to the next, we have 
 merely to move any element of the surface in the direction of 
 the normal to that element through a small distance a to 
 obtain the new position at the moment a/v later. This being 
 done for the whole surface, gives us the new position of the 
 wave-front, and by continuing this process we may follow the 
 wave-front in its progress as long as we please. Thus the wave- 
 front coming from the surface of a sphere is a sphere ; from a 
 round cylinder, if infinitely long, also a round cylinder ; but if 
 of finite length, then a cylinder with rounded ends ; from a cube, 
 initially a cube with rounded corners, but becoming more and 
 more spherical as it expands ; and so on. It is easily seen that 
 at a sufficiently great distance from a finite collection of sources 
 of any kind, the wave-front tends to become spherical, or the 
 complex source tends to become equivalent, at a great distance, 
 to a point-source of some complex kind. 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 323 
 
 Effect of a Non-Conducting Obstacle on Waves. Also of a 
 Heterogeneous Medium. 
 
 182. Now consider the effect of an obstacle brought into 
 our medium, say a non-conducting dielectric mass of different 
 inductivity or permittivity. The manner of propagation in it 
 is similar in kind to that in the external medium, bat it varies 
 in detail, and the speed will usually be different. Then, when 
 we have a wave coming from (say) a point- source in the first 
 medium, the presence of the obstacle at first makes no dif- 
 ference. It has no immediate action, and, until the wave 
 reaches it, might as well not be there. But immediately the 
 wave does reach it a change occurs. The interface of the two 
 media becomes the seat of sources of fresh disturbances, or they 
 may be considered fictitious sources, in contrast with the 
 original, for they do not bring in any fresh energy. Thence 
 arises a new effect, viz., reflection. Not the whole, but only a 
 part of the wave disturbance enters the new medium. The rest 
 is thrown back, and forms a reflected wave. In the same way 
 as we may trace the course of primary waves from a source in 
 a uniform medium may we trace the course of the secondary 
 waves set up by the obstacle. The disturbance outside it is 
 then due to the superposition of the primary and secondary 
 waves, and, if there be just one obstacle, this state of things 
 continues. Of course the secondary wave- front may itself be 
 complex, because the disturbances going into the second me- 
 dium and transmitted therein according to its nature, may 
 reach the interface again at other parts, and there suffer re- 
 flection and transmission anew. This complication is done 
 away with by making the obstacle infinitely big, with a plane 
 boundary. Then we have just one wave in the obstacle and 
 two in the medium containing the source, viz., the primary and 
 the first (and only) reflected wave. 
 
 But if there be a second obstacle, not only will it, like the 
 first, give rise to a secondary wave when the primary meets it, 
 but each obstacle will act similarly towards the secondary wave 
 from the other, whence arises a pair of tertiary waves, and so 
 on. Thus, with only two obstacles, we shall have a succession 
 of infinitely numerous waves crossing and recrossing one 
 another, arising out of the action of a point-source, with 
 
 Y2 
 
324 ELECTROMAGNETIC THEORY. CH. IV. 
 
 great complications. But, however complex in detail, we have 
 the important fact that the mere knowledge of the speed of 
 propagation allows us to lay down the whole course of the 
 waves generated, and the position of the wave-fronts belonging 
 to a given epoch at the source. The mathematics of the full 
 treatment may be altogether beyond human power in a reason- 
 able time; nevertheless, we can always predict with confidence 
 that the results must have such and such general properties 
 relating to the course of the waves, besides the properties 
 involved in the persistence of energy. 
 
 In the above we were concerned with discontinuous he- 
 terogeneity, or an abrupt change in the value of one or both 
 of the constants c and /* at an interface. When the medium is 
 continuously heterogeneous, or the "constants" change in 
 value continuously from place to place, then the speed v is 
 made a function of position. The process of partial reflection 
 and partial transmission which occurred at the interface now 
 takes place in general wherever the value of c or p changes. 
 These changes being continuous, so are the results, so that we 
 do not have distinctly separable trains of waves, but rather a 
 continuous distortion. We can, however, follow the course of 
 a wave expanding from a point by communicating to the wave- 
 front the speed proper to its position, such speed being, as 
 above, definitively known. A spherical wave may remain 
 spherical, only varying in its speed; or more likely, it may 
 change its form as it expands. It may become ellipsoidal, for 
 example, and of course must do so if the speed in different 
 parts of the medium vary suitably. 
 
 It is possible for the inductivity and permittivity to change, 
 either abruptly or continuously, without any reflection, or cast- 
 ing behind of the disturbance as a wave progresses, although 
 its speed varies. In such a case it will be found that the ratio 
 IJL/C is constant. The change in p,, or the ratio of induction to 
 magnetic force, then fully compensates the change in c, the 
 ratio of displacement to electric force, either of which changes 
 by itself would cause reflection. 
 
 In connection with a heterogeneous medium (including abrupt 
 changes of nature), we may notice its behaviour as regards 
 steady states, in contrast with that of a thoroughly homo- 
 geneous medium. In the latter case, as we have already 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 325 
 
 described, the introduction of a steady point-source causes 
 the steady state to begin immediately at and spread around it, 
 after which no change occurs. That is, the medium being 
 homogeneous, there is no reflex action. But when it is hetero- 
 geneous the case is quite different. The steady state of dis- 
 placement due to impressed voltage depends upon the per- 
 mittivity in all parts of the medium, and only comes about as 
 the final result of the infinitely numerous reactions between 
 different parts, due to the reflections that take place, which 
 modify the final distribution of displacement as well as (usually) 
 the total amount. There is a similar contrast in the mechanism 
 of the mathematics involved in the calculation of the steady 
 state. When the medium is homogeneous we may find it 
 through the potential of fictitious matter at the source alone. 
 When heterogeneous, we require to have fictitious matter all 
 over the medium, or rather in all parts where it changes its 
 nature, and, owing to the infinitely numerous reactions, we 
 should, in a general or complete solution, require to perform 
 not one space-integration, but an infinite series of successive 
 space-integrations, before we could arrive at the true potential 
 function, allowing for the variations of permittivity every- 
 where. Thus, by electromagnetic considerations we obtain 
 some insight of the true nature of transcendental static prob- 
 lems, involving potential functions and assumed instantaneous 
 action at a distance. 
 
 Effect of Eolotropy. Optical Wave -Surfaces. 
 Electromagnetic versus Elastic Solid Theories. 
 
 183. When the medium is not isotropic as regards either 
 the displacement or the induction, or as regards both, we have 
 very remarkable effects, known in the science of optics as 
 double-refraction. There can be two distinct wave-speeds for a 
 given position of the wave-front, and these speeds change when 
 the direction of the normal to the wave-front changes. Hence 
 double refraction, or the separation of a wave entering an 
 eolotropic medium into two waves travelling independently of 
 one another at different speeds. If the medium is only elec- 
 trically eolotropic, although the displacement and induction 
 are still in a wave-front (of either wave), and are perpendicular 
 to one another, the electric force is inclined to the wave-front, 
 
326 ELECTROMAGNETIC THEORY. CH. IV. 
 
 so that the ray, which indicates the direction of the flux of 
 energy, is inclined to the normal to the wave-front. Similarly, 
 with magnetic eolotropy alone, it is the magnetic force that is 
 inclined to the wave front. When the medium is both elec- 
 trically and magnetically eolotropic, both the electric and the 
 magnetic force are inclined to the wave-front, whilst the dis- 
 placement and induction, though still in the wave-front, are no 
 longer perpendicular to one another. 
 
 By imagining plane electromagnetic sheets to traverse the 
 medium in all possible directions about a point, and comparing 
 their positions with respect to the point at equal times after 
 crossing it, we arrive at the conception of the wave-surface. 
 This is obviously a sphere in an isotropic medium. But when 
 it is eolotropic, the two speeds, and their variation according 
 to the direction of motion of the waves, make the wave-surface 
 become a very singular double surface. With electric eolotropy 
 alone, which is the practical case, it is Fresnel's wave-surface. 
 It is also another Fresnel surface with magnetic eolotropy 
 alone. But when the medium is eolotropic as regards both the 
 induction and the displacement, the wave-surface is of a more 
 general and symmetrical character, including the former two 
 as extreme examples. It is still a double surface, however, 
 except in one case. We have already mentioned that in an 
 isotropic medium there is a peculiar behaviour when the ratio 
 P./C is constant, although fj. and c vary. We might anticipate 
 some peculiarity in the wave-surface when /z/c is constant. 
 This constancy now means that the directional properties of //, 
 are exactly paralleled by those of c. That is, the principal 
 axes of p and those of c are coincident, whilst the value of the 
 ratio of the permittivity to the inductivity is the same for the 
 three axes. The result is to reduce the double wave-surface to 
 a single surface, which is an ellipsoid. 
 
 Assuming light to consist of vibrations of an " elastic solid " 
 medium, single or ordinary refraction is explained by an 
 alteration in the density or the rigidity of the ether. Not 
 only is the theory quite hypothetical in considering the kinetic 
 energy to be the energy of vibrational motions of displacement, 
 but the alteration of density or rigidity assumed is a further 
 hypothesis on the top of the main one, for there is no evidence 
 that there is such a change. Moreover, the explanation will 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 327 
 
 not work properly. On the other hand, the electromagnetic 
 theory says that light consists of electromagnetic vibrations in 
 the ether. This, too, is a hypothesis. But the auxiliary part, 
 that refraction is caused by change of permittivity from one 
 medium to another, is not a hypothesis, but a fact. More- 
 over, the theory works. 
 
 Similarly, double refraction in elastic solid theories of light 
 is explained by eolotropy as regards elasticity, or by something 
 similar relating to the density. This is also hypothetical, and 
 not without its troubles. But, on the other hand, Maxwell 
 declared that double refraction occurs because the doubly 
 refracting medium is electrically eolotropic. Now this is a 
 fact too, and the theory is a clear one. 
 
 These remarks will serve to illustrate what I mean by the 
 far greater intrinsic probability of the electromagnetic theory, 
 apart from the experimental work of late years. It is much 
 less hypothetical than elastic solid theories. We know nothing 
 about the density or the rigidity of the ether, or how they vary 
 in different bodies, or' if they vary. But we do know a good 
 deal about electric permittivity and magnetic inductivity. In- 
 stead of dealing with possibilities, we are dealing with actual facts. 
 
 The old objection that a mechanical theory of light was 
 surely to be preferred to an abstract electromagnetic theory 
 was very misleading. The electromagnetic theory is mechan- 
 ical, without, however, a precise specification of the mechanism. 
 An elastic solid theory is merely a special mechanical theory. 
 It cannot satisfy the electromagnetic requirements, but this 
 failure, though immensely important in itself, is not the point 
 here. Even if it did satisfy them, it would probably be less 
 true than the electromagnetic theory, which, being abstract, 
 does not assert so much. There may be many "mechanical" 
 solutions of an abstract theory. Elastic solid theories are a 
 great deal too precise in saying what light consists of, and 
 mechanical speculations in general should be received with 
 much caution, and regarded rather as illustrations or analogies 
 than expressions of fact. We do not know enough yet about 
 the ether for dogmatising. 
 
 One can imagine that a clear-headed man might be able to 
 work his way through all the theories of light yet propounded, 
 assimilate what was useful and true, eliminate what was useless 
 
328 ELECTROMAGNETIC THEORY. CH. IV, 
 
 or false, and finally construct a purified theory, not professing 
 to explain what light is, but still connecting together in a per- 
 fectly unobjectionable manner, free from hypotheses, all the 
 principal facts, and of so compliant a nature as to readily adapt 
 itself to future discoveries, and finally settle down to a special 
 form expressing the theory of light. It is hardly likely that 
 the clear-headed man will be found for the purpose, and per- 
 haps his theory would not be very popular. It would be so 
 very abstract. What is far more likely is that the electro- 
 magnetic theory (itself abstract and possessing many of the 
 desired qualifications), which has already begun to find its 
 way into optical treatises at the end, will gradually work its 
 way right through them to the beginning, and in doing so, 
 oust out the most of the old-fashioned hypotheses. The 
 result will be to have optical theory expressed throughout in 
 electromagnetic language. To do it properly, it is hardly 
 necessary to say that the preposterous 47r, which the B.A. 
 Committee seem to want to perpetuate, should be ignored from 
 the beginning. 
 
 A Perfect Conductor is a Perfect Obstructor, but does not 
 absorb the Energy of Electromagnetic Waves. 
 
 184. So far, in varying the nature of the medium, we have 
 not introduced any property causing its local waste, such as 
 the existence of conductivity (finite) necessitates. In all the 
 varied journeys of electromagnetic waves in a (theoretical) non- 
 conducting dielectric with non-conducting obstacles there is no 
 local waste of energy, and the work done by impressed forces is 
 entirely accounted for by the electric and magnetic energy. 
 Assuming a certain amount of work done up to a certain 
 moment, and none later, that amount expresses the energy of 
 the electromagnetic field, and, however it may vary in distribu- 
 tion, the total remains the same, just as if it were a quantity 
 of matter moving about, having continuity of existence in time 
 and space. The useless complication introduced by the cir- 
 cuital indeterminateness of the energy flux may be ignored. 
 The only way, then, to get rid of the energy is to absorb it at 
 the sources by working against impressed forces. Putting 
 that on one side, there is the waste of energy by dissipation 
 in space, which cannot be stopped. The energy is still in exis- 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 329 
 
 tence, and is still in the electric and magnetic forms, and must 
 always be at a finite distance from its source and an infinite 
 distance from infinity ; but since it gets out of practical reach, 
 and is constantly going out further, it is virtually wasted. 
 Thus, ultimately, the energy of all disturbances generated in 
 a boundless non-conducting dielectric goes out of range and is 
 wasted. 
 
 To prevent this, we may interpose a reflecting barrier. 
 Imagine a screen to be introduced, at any finite distance, 
 enveloping the sources, of such a nature as to be incapable of 
 either transmitting or absorbing the energy of waves impinging 
 upon it. Then clearly the dissipation of the energy is stopped, 
 and it all keeps within the bounded region. The waves from 
 the source will be reflected from the boundary, and their sub- 
 sequent history will be an endless series of crossings and re- 
 crossings. The only way to destroy the induction and dis- 
 placement is to employ artful demons (or impressed forces), so 
 situated and so timing themselves as to absorb the energy of 
 waves passing them, instead of generating more disturbances. 
 In the absence of this demoniacal possession of the region, the 
 energy will remain within it in the electromagnetic form and 
 be in constant motion. 
 
 But any opening in the screen, establishing a connection 
 through ether between the inner and outer 'regions, will at 
 once put a stop to this local persistence of energy. For energy 
 will pass through the opening, and once through, cannot get 
 back again (though a part may), but will escape to infinity. A 
 mere pinhole will be sufficient, if time enough be given, to allow 
 all the energy to pass through it into the external region, and 
 there go out unimpeded to an infinite distance, in the absence 
 of a fresh complete screen to keep it within bounds. 
 
 A screen to perform the above-described functions may be 
 made of that very useful scientific substance, the perfect con- 
 ductor, which is possibly existent in fact at the limiting zero 
 of absolute temperature, the latest evidence being the recent 
 experiments of Profs. Dewar and Fleming, measuring the resist- 
 ances of metals at very low temperatures. If a perfect con- 
 ductor, it is also a perfect obstructor, or is perfectly opaque to 
 electromagnetic waves, without, however, absorbing their 
 energy superficially. 
 
330 ELECTROMAGNETIC THEORY. CH. IV. 
 
 Conductors at Low Temperatures. 
 
 185. For this reason, a perfectly conducting mass exposed 
 to electromagnetic radiation in ether cannot be heated thereby, 
 for heating would imply the absorption of energy. If, then, 
 the law (whose existence has long been suspected) that metals 
 tend to become perfect conductors when their temperature is 
 sufficiently lowered, were absolutely true, it would follow that 
 a metal, if once brought to zero temperature, would remain 
 there, provided its only source of heat-energy be electromag 
 netic vibrations. At the same time it is conceivable, and, in- 
 deed, inevitable, that it should receive energy from them to 
 some extent in another way, viz., by the electromagnetic stress 
 producing motion in bulk, even though the establishment of 
 irregular molecular motions be prevented. 
 
 Similarly, we should expect that a metal which obeys the law 
 approximately would show very small absorptive power for 
 radiation at very low temperatures. This refers to the recep- 
 tion of the energy of true radiation in ether that is, in vacuo. 
 A body may become heated in other ways, by the impacts of 
 air particles, for example, if air be present. Prof. De war's late 
 experiments are suggestive in this respect, but it is too soon 
 to draw conclusions from partially-published experiments. 
 
 But even as regards strongly absorptive bodies, the perfectly 
 black body of the thermal philosophers, for example, we may 
 expect a similar diminution of absorptive and emissive power 
 with fall of temperature. Stefan's law of radiation asserts that 
 the emissivity varies as the fourth power of the absolute tem- 
 perature. Since, however, in deriving this law from electro- 
 magnetic principles, as has been recently done by B. Galitzine,* 
 we seemingly require to invoke the aid of reversible cycles and 
 the second law of thermodynamics, whose range of application 
 is sometimes open to question, we may well be excused from 
 an overhasty acceptance of the law as the expression of abso- 
 lute fact. The second law of thermodynamics itself needs to 
 be established from electromagnetic principles, assisted by the 
 laws of averages, so that we may come to see more clearly the 
 validity of its application, and obtain more distinct notions of 
 the inner meaning of temperature. 
 
 * Phil. Mag., February, 1893. 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 331 
 
 Equilibrium of Radiation. The Mean Flux of Energy. 
 
 186. As before said, a perfectly conducting screen enclos- 
 ing a dielectric region supporting electromagnetic disturbances, 
 keeps in their energy, which remains in the electric and mag- 
 netic forms, and if there be no sources of energy present, the 
 total energy remains constant. Some interesting questions 
 arise regarding the subsequent history of the electromagnetic 
 disturbances, when left to themselves, subject to the obstruct- 
 ing and reflecting action of the screen. 
 
 In certain cases the initial state will continue absolutely 
 unchanged ; for example, when it is the steady state due to 
 electrification, or associated therewith. Similarly as regards 
 an initially steady state of magnetic force that, for instance, 
 associated with a linear current (without resistance) in the 
 enclosed region. Again, in other cases, although the subse- 
 quent history of an initial state may be one of constant 
 change, yet there will be regular recurrence of a series of 
 states ; as when a periodic state of vibration persists without 
 any tendency to degrade. It is sufficient to mention the very 
 rudimentary case of a plane wave running to and fro between 
 parallel plane reflecting boundaries, without the slightest ten- 
 dency to change the type of the vibrations. We see from 
 these examples, which may be multiplied, that there is no 
 necessary tendency for the initial state, even when vibratory, 
 to break up and fritter down into irregular vibrations. Never- 
 theless, there does appear to be a general tendency to this 
 effect, when the initial states are not so artfully selected as to 
 prevent it happening. Even when we start with some quite 
 simple type of electromagnetic disturbance, the general effect 
 of the repeated reflections from the boundary (especially when 
 of irregular form) and the crossing of waves is to convert the 
 initial simplicity into a highly complex and irregular state of 
 vibration throughout the whole region. This cannot happen 
 universally, as we have seen, and therefore a general proof of 
 conversion from any initial state to irregularity cannot be 
 given; but there can be little doubt as to the usual possi- 
 bility of the phenomenon. Especially will this be the case 
 if the initial state be itself of an irregular type, such as that 
 due to ordinary radiation from matter, when it is tolerably 
 
332 ELECTROMAGNETIC THEORY. OH. IV. 
 
 clear that the irregularity will persist, and become more 
 complete. 
 
 Let us, then, assume that we have got the medium within our 
 screen into this state in its extreme form, without enquiry into 
 the intermediate stages. Then the very irregularity gives rise 
 to a regularity of a new kind, the regularity of averages. The 
 total energy, which is a constant quantity, will be half electric 
 and half magnetic, and will be uniformly spread throughout 
 the enclosure, so that the energy density (or energy per unit 
 volume) is constant. As regards the displacement and the 
 induction, they take all directions in turn at any one spot, quite 
 irregularly, but so that their time-averages show no directional 
 preference. Similarly, the flux of energy, which is a definite 
 quantity at a given moment at a fixed spot, is constantly 
 changing in amount and direction. But in virtue of the con- 
 stancy of the mean density and the preservation of the normal 
 state by constant exchanges of energy, there is a definite mean 
 energy flux to be obtained by averaging results. This mean 
 flux expresses the flux of energy per second across a unit area 
 anywhere situated within the enclosure. 
 
 To estimate its amount, let the mean density of the energy 
 be U. This is to include both kinds. Now fix attention upon 
 a unit area, A, fixed in position anywhere within the enclosure, 
 and consider the flux of energy through it under different cir- 
 cumstances. First of all, if the energy all moved the same 
 way at the same speed v (that of propagation), as in simple 
 plane progressive waves, and the direction of its motion were 
 perpendicular to the fixed unit area A, then the energy passing 
 through it would belong to a ray (or bundle of rays) of unit 
 section, and the energy flux would be Uv simply. This is the 
 maximum. But this is impossible, because energy would accu- 
 mulate on one side of A at the expense of the other. The 
 next approximation, to prevent the accumulation, is to let half 
 the energy go one way and half the other ; still, however, in 
 the same line. This brings us down to JUv. To go further, 
 we must take all possible directions of motion into account. 
 The original ray conveying Uv must assume all directions in 
 turn, and the mean value of the flux through A must be 
 reckoned. Now, if the ray of unit section be turned round 
 so as to make an angle with the normal to A, the effective 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 333 
 
 section of the ray is reduced from 1 to cos ; that is, cos 6 is 
 the fractional part of the ray which sends its energy through 
 A, through which the flux per second is therefore Uv cos 0, due 
 to the ray at inclination 6. The true flux through the area A 
 is therefore the mean value of Uv cos for all directions in space 
 assumed by the ray. Now the mean value of cos for a com- 
 plete sphere is zero, and therefore the mean flux through A is 
 zero. This is right, as it asserts that as much goes through 
 one way as the other. To obtain the amount going either way 
 we must average over a hemisphere only. The mean value of 
 cos 6 is then J. But we are only concerned with half the total 
 energy, or JU, when we are confined to one hemisphere. Con- 
 sequently we have 
 
 W = JU*, ..... . (4) 
 
 to express the flux of energy W per second each way through 
 any unit area in the enclosure. 
 
 Another way of getting this result, which is, however, essen- 
 tially the same, is to divide the original ray of unit section 
 along which the flux is Uv into a very great number n of equal 
 rays of unit section, each conveying \/n part of the same, and 
 placed at such inclinations to the normal to A that no direction 
 in space is favoured. This amounts to dividing the surface of 
 a sphere whose centre is that of the area A into n equal parts, 
 the centre of every one of which defines the position of one of 
 the n rays. Any ray now sends (Uv/n) cos through A per 
 second. Now sum this up over the whole hemisphere and the 
 result is W. In the limit, when n is infinitely great, we have 
 
 e e=i 
 
 4?r 
 
 as before. The 4:r divisor in the integral is not the unspeak 
 able 4?r of the B.A. units. It is the area of the sphere of unit 
 radius, whilst the other factor sin ddd<j> is the area of an 
 element of the sphere. 
 
 As this is an important fundamental result in radiation, it 
 is desirable to establish it as generally and simply, and with as 
 much definiteness of meaning as possible. Bartoli made it 
 come to JU0 (apart from electromagnetic considerations, which 
 
334 ELECTROMAGNETIC THEORY. CH. IV, 
 
 are, however, only accidentally involved). On the other hand, 
 Galitzine makes it JUfl by considering a special case, viz., 
 that of a cylindrical enclosure with flat ends, one of which is a 
 radiant source (perfectly black) at constant temperature, whilst 
 the other end and the round surface are perfectly reflecting. 
 It would appear, however, from the above method, that the 
 result is general, and is independent of sources of heat, and of 
 the emissivity and temperature. Since we made no use of the 
 screen after introducing it to keep in the radiation, it may be 
 dispensed with, provided the stationary condition be still main- 
 tained. Thus, if a portion of the screen be made " perfectly 
 black," maintained at constant temperature, the quantity W, 
 which represents the amount of heat falling upon it per unit 
 area per second, is completely absorbed by it. But this is 
 perfectly compensated by the emission from the black surface 
 of an equal amount of heat. So %Uv measures the total emis- 
 sivity under the circumstances. 
 
 The Mean Pressure of Radiation. 
 
 187. Another important fundamental quantity is the mean 
 pressure of radiation. In a simple ray the electric and mag- 
 netic stresses unite to form a pressure U along the ray, with 
 no pressure or tension in lines perpendicular to the ray, that 
 is, in the plane of the wave, as described in 86. But when 
 the radiation is balanced as in the last paragraph, there is no 
 directional preference, and the pressure is all ways in turn, and 
 therefore, on the average, simulates a hydrostatic pressure. Its 
 value may be readily estimated in a manner similar to that 
 employed above concerning the mean flux of energy. When 
 we make the energy U go all one way in a ray of unit section 
 through the area A situated anywhere, the pressure in the ray 
 is U, and this is the pressure on A if the ray is perpendicular 
 to A. But when the ray is inclined to the normal to A at an 
 angle 6, only the fraction cos of the ray is concerned in the 
 action upon A. Furthermore, the line of pressure is inclined 
 to the normal to A at an angle 0, so that the effective normal 
 pressure is still further reduced by the factor cos 6 a second 
 time. The pressure on A is therefore only Ucos 2 0. This 
 must now be averaged for all directions in space that we may 
 give to the ray, without preference. Now the mean value of 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 335 
 
 cos 2 for the complete sphere is J. So the pressure, say p, is 
 given by 
 
 *> = JU ..... .... (6) 
 
 Notice particularly that we take the mean for the complete 
 sphere, not merely for the hemisphere as in the former calcula- 
 tion. The reason is that whether a ray goes through A from 
 right to left or from left to right, the pressure is the same : so 
 both ways have to be reckoned. 
 
 Or, we may divide the original ray in which the pressure is 
 U into a great number n of rays also of unit section, in each of 
 which the pressure is \J/n. Let the axes of these rays be denned 
 by the middle points of n small equal areas into which we may 
 divide the surface of a sphere, and sum up the normal pres- 
 sures on A. We obtain, in the limit, 
 
 This result was given by Boltzmann some years ago, and 
 Galitzine confirms it for the special cae of his straight cylinder 
 with a radiant surface at one end. By the above method we 
 see that the result is general, resulting from the uniformity 
 of radiation in all directions, as the previous formula for the 
 mean flux of energy did. 
 
 Emissivity and Temperature. 
 
 183. This pressure p is not only the mean pressure 
 throughout the enclosure, but also the pressure on its enve- 
 lope, which exerts an equal back pressure. If it move, then 
 work is done by or against the enclosed radiation through 
 the agency of its pressure, and the enclosure loses or gains 
 energy to an equal extent. Observe that the idea of tempera- 
 ture does not enter explicitly when the boundary of the enclo- 
 sure is of ideal perfectly reflecting material. But when it, 
 or a part of it, is made absorptive and emissive, the physics 
 of the matter becomes far more difficult and to some extent 
 dubious. The notion of temperature comes in, and with it the 
 isecond law of thermodynamics. Assuming its full applica- 
 bility, Stefan's law follows easily enough, as Galitzine has 
 shown. Let the enclosure bo a cylinder of unit section with 
 two pistons A and B. Let A be fixed, and be a perfectly black 
 
336 ELECTROMAGNETIC THEORY. OH. IV. 
 
 body, whilst B Is movable to and from A and is a perfect 
 reflector, as is also the round cylinder. 
 
 Now start with B close to A ; the volume of the enclosure 
 is then nil. Draw B very slowly away from A through the 
 distance h t keeping A at constant temperature t all the time, 
 and then stop it. During this operation the source A keeps 
 the enclosure filled with energy to density U, and pressure p, 
 corresponding to the temperature t at which A is maintained. 
 The pressure p does the external work ph. Besides that, there 
 is energy UA in the enclosure at the end of the operation. 
 Their sum is therefore the heat lost by A (excess of heat 
 emitted by A into the enclosure over that returned to A). 
 Say 
 
 H-(U+i>)A (8) 
 
 Now we know p in terms of U, so that we have, by (6), 
 H = |TO (9) 
 
 Now, B being fixed, let the cylinder cool down to zero tempera- 
 ture. The whole of the energy UA in the enclosure goes out 
 through A. Lastly, push B back to A without working, and 
 then raise A to temperature t. A cycle is then completed. 
 Applying the second law, we have 
 
 <> 
 
 where H is as before, and dR is an element of the heat lost in 
 the cooling process. On the left side put (U+p)h or Uh 
 for H, because external work was done in the first operation. 
 On the right side leave out the p term, because during the 
 cooling B was fixed. So we get 
 
 by omitting the factor h. Differentiate with respect to t, and 
 we get 
 
 5?-*-* (12) 
 
 dt t ' 
 
 from which we conclude that U, and, therefore, also p and W, 
 vary as the fourth power of the temperature, the result above 
 mentioned as applied to the emissivity W. 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 337 
 
 We tacitly assume that the ether is able to escape freely from 
 the cylinder through its envelope, or else that it is freely com- 
 pressible, without resistance. This difficulty in connexion with 
 the ether is a very old one. 
 
 Internal Obstruction and Superficial Conduction. 
 
 189. The properties of a perfect conductor ara derived from 
 those of common conductors by examining what would happen 
 if the resistivity were continuously reduced, and ultimately 
 became zero. In this way we find that a perfect conductor 
 is a perfect obstructor, for one thing, which idea is singu- 
 larly at variance with popular notions regarding conduc- 
 tors. But it is also a perfect conductor literally, though 
 in a different sense to that commonly understood. Ohm's- 
 law has played so important a part in the development of 
 electrical knowledge, especially on the practical side, that it 
 is really not at all a matter of wonder that some practicians- 
 should have been so reluctant to take in the idea of a con- 
 ductor as an obstructor. Scientific men who can follow the 
 reasoning by which the functions of conductors follow from 
 known facts have no difficulty in pursuing the consequences- 
 far beyond experimental observation. Again, younger men,, 
 with fewer prejudices to surmount, do not find much trouble- 
 with superficial conduction and internal obstruction. But the- 
 old established practitioner with prejudices, who could not 
 see the reason, was put into a position of some difficulty 
 resembling chancery. If you have got anything new, i 
 substance or in method, and want to propagate it rapidly,, 
 you need not expect anything but hindrance from the old 
 practitioner even though he sat at the feet of Faraday. 
 Beetles could do that. Besides, the old practitioner is apt 
 to measure the value of science by the number of .dollars- 
 he thinks it is likely to bring into his pocket, and if he 
 does not see the dollars, he is very disinclined to disturb his 
 ancient prejudices. But only give him plenty of rope, and 
 when the new views have become fashionably current, he may~ 
 find it worth his while to adopt them, though, perhaps, i 
 a somewhat sneaking manner, not unmixed with bluster, and 
 make believe he knew all about it when he was a little boy t 
 He sees a prospect of dollars in the distance, that is the- 
 
338 ELECTROMAGNETIC THEORY. CH. IV. 
 
 reason. The perfect obstruction having fciiled, try the per- 
 fect conduction. 
 
 You should make your converts out of the rising generation 
 and the coming men. Thus, passing to another matter, Prof. 
 Tait says he cannot understand my vectors, though he can 
 understand much harder things. But men who have no quater- 
 nionic prejudices can understand them, and do. Younger 
 men are born into the world with more advanced ideas, on 
 the average. There cannot be a doubt about it. If you had 
 taught the Calculus to the ancient Britons you would not 
 have found a man to take it in amongst the whole lot, Druids 
 and all. Consider too, what a trouble scientific men used 
 to have with the principle of the persistence of energy. They 
 could not see it. But everybody sees it now. The important 
 thing is to begin early, and train up the young stick as you 
 want it to grow. Now with Quaternions it is different. 
 You may put off till to-morrow what you cannot do to-day, 
 for fear you commence the study too soon. Of course, I" 
 refer to the Hamilton-Tait system, where you have to do 
 violence to reason by making believe that a vector is a 
 quaternion, and that its square is negative. 
 
 According to Ohm's law alone, a perfect conductor should be 
 one which carried an infinite current under a finite voltage, 
 and the current would flow all through it because it does so 
 ordinarily. But what is left out of consideration here is the 
 manner in which the assumed steady state is established. If 
 we take this into account, we find that there is no steady state 
 when the resistance is zero, for the variable period is infinitely 
 prolonged, and Ohm's law is therefore out of it, so far as the 
 usual application goes. In a circuit of no resistance containing 
 a finite steady impressed voltage E, the current would mount 
 up infinitely and never stop mounting up. On the other hand, 
 if we insert a resistance R in the former circuit of no resistance, 
 there will be a settling to a steady state, for the current in the 
 circuit will tend to the value E/R, in full obedience to Ohm's 
 law. The current is the same all round the circuit, although 
 a part thereof has no resistance. We conclude that that 
 portion has also no voltage. 
 
 But this is only a part of the story. Although we harmonise 
 with Ohm's law, we overlook the most interesting part. The 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 339 
 
 smaller the resistance the greater the time taken for the 
 current to get into the conductor from its boundary, where it 
 is initiated. In the limit, with no resistance, it never gets in 
 at all. Where, then, is the current? For, as we have said, 
 it mounts up to a finite value if there be a finite resistance 
 inserted along with the perfect conductor, and mounts up 
 infinitely if there be no resistance. 
 
 We recognise the existence of electric current in a wire by 
 the magnetic force round it, and in fact measure the current 
 by its magnetic force. Therefore, according to this, there is 
 the same total current in the wire, if the magnetic force out- 
 side it remains the same. If, then, the magnetic force stops 
 completely at the surface of the wire, whose interior is entirely 
 free from magnetic force, the measure of the current is just 
 the same. The uniformly distributed current of the steady 
 state appropriate to finite conductivity becomes a mere surface 
 current when the conductivity is infinite. In one case we 
 have a finite volume-density of current, and in the other a 
 finite surface-density. When the current inside the wire is 
 zero so is the electric force, in accordance with Ohm's law 
 again. The electric and magnetic phenomena are entirely in 
 the dielectric outside the wire, the entrance of any similar 
 manifestations into it being perfectly obstructed by the absence 
 of resistance. For this purpose the thinnest skin would serve 
 equally well. In the usual sense that an electric current is a 
 phenomenon of matter, it has become quite an abstraction, for 
 there is no matter concerned in it It is shut out completely. 
 In the circuit of finite resistance, a portion of which is a wire 
 of no resistance, supporting a steady current, there is no 
 difference whatever in the external, magnetic force outside the 
 resisting and non-resisting parts, though in one case there is 
 entrance of the magnetic force and waste of energy, whilst in 
 the other there is no entrance and no waste. These con- 
 clusions do not rest upon Maxwell's theory of dielectrics, but 
 upon the second circuital law of electromagnetism applied to 
 conductors. But it is only by means of Maxwell's theory that 
 we can come to a proper understanding and explanation of the 
 functions of conductors. 
 
 The sense in which a perfect conductor is a perfect con- 
 ductor in reality as well as in name is that it allows electro- 
 
 z2 
 
310 ELECTROMAGNETIC THEORY CIL IV. 
 
 magnetic waves to slip along its surface in a perfectly free- 
 manner, without waste of energy. Though perfectly obstruc- 
 tive internally, it is perfectly conductive superficially. It 
 merely guides the waves, and in this less technical sense of 
 conduction the idea of a perfect conductor acquires fresh life. 
 
 The Effect of a Perfect Conductor on External Disturb- 
 ances. Reflection and Conduction of Waves. 
 
 190. The conditions at the interface of a perfect conductor 
 and a dielectric are that the electric force in the dielectric has- 
 no tangential component and the magnetic induction no normal 
 component. Or 
 
 VNE = 0, NH = 0, 
 
 if N be the unit normal from the conductor. Thus, when 
 there is electric force at the boundary it is entirely normal, 
 with electrification to match ; and if there is magnetic force it 
 is entirely tangential, with electric current to match. Both 
 electrification and current are superficial. The displacement 
 measures the surface density a- of the one, and the magnetic* 
 force that of the other, say c, thus 
 
 <r = ND, c = VNH, 
 
 in rational units, without any useless and arbitrary 4?r constant,, 
 such as is required in the B.A. system of units, of amazing 
 irrationality. If, then, we have electromagnetic disturbances 
 given in a dielectric containing a perfect conductor, the latter 
 first of all is free from disturbance, and next causes such re- 
 flected waves as to annihilate the tangentiality of the electric 
 force and the normality of the magnetic force. 
 
 As regards steady states, the influence of a perfect conductor 
 on induction due to foreign sources is to exclude it in the same 
 manner as if the inductivity were made zero ; that is, the induc- 
 tion goes round it tangentially instead of entering it. This 
 is usually ascribed to an electric current-sheet induced upon it& 
 surface, whose internal magnetic force is the negative of that 
 due to the external field. This is right mathematically, but is 
 deceptive and delusive physically. There is no internal force,, 
 neither that of the external field nor that of the superficial 
 current. The current sheet itself merely means the abrupt 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 341 
 
 stoppage of the magnetic field, and cannot really be supposed 
 to be the source of magnetic force in a body which cannot 
 permit its entrance. The previously mentioned case of a per- 
 fectly conducting wire inserted in a circuit of finite resistance 
 supporting a steady current, will serve to bring out this point 
 strongly. The supposed induced superficial current is now 
 actually the main current in the circuit itself. 
 
 It is different with the steady state due to external electric 
 sources. The displacement is just as much shut out from the 
 perfect conductor (which may also be a dielectric) as was the 
 magnetic induction, but in a strikingly different manner, ter- 
 minating upon it perpendicularly, as if it entered it in the 
 manner that would happen were the conductor nonconducting, 
 but of exceedingly great permittivity, so that it drew in the 
 tubes of displacement. 
 
 Although a perfect magnetic conductor is, in the absence of 
 knowledge even of a finite degree of magnetic conductivity, a 
 very far-fetched idea, yet it is useful in electromagnetic theory 
 to contrast with the perfect electric conductor. A perfect 
 magnetic conductor behaves towards displacement just as a 
 perfect electric conductor does towards induction ; that is, the 
 displacement goes round it tangentially. It also behaves 
 towards induction as a perfect electric conductor does towards 
 displacement ; that is, the induction meets it perpendicularly, 
 as if it possessed exceedingly great inductivity, without 
 magnetic conductivity. This magnetic conductor is also per- 
 fectly obstructive internally, and is a perfect reflector, though 
 not quite in the same way as electric conductors. The tan- 
 gential magnetic force and the normal electric force are zero. 
 
 As regards waves, there are two extreme ways in which a 
 perfect conductor behaves that is, extreme forms of the gene- 
 ral behaviour. It may wholly conduct them, or it may wholly 
 reflect them. In the latter case we may illustrate by ima- 
 gining a thin plane electromagnetic sheet, consisting of crossed 
 electric and magnetic forces in the ratio given by E = /*vH, 
 moving at the speed of light, to strike a perfect conductor 
 flush that is, all over at the same time, by reason of parallel- 
 ism of the sheet and conducting surface. The incident sheet 
 is at once turned into another plane sheet, which runs away 
 from the conductor as fast as it came. If the conductivity be 
 
342 ELECTROMAGNETIC TEIEORY. OIL IV. 
 
 of the electric kind the reflected sheet differs from the incident 
 in having its displacement reversed, but in no other respect* 
 This is perfect reflection with reversal of E. During the act 
 of reflection, whilst the incident and reflected sheets partly coin- 
 cide, E is zero and H is doubled. Both are tangential ; but 
 there can be no tangential E, so the reflector destroys E and 
 initiates the reflected sheet, in which H is the same as in the 
 incident sheet, whilst E is reversed. 
 
 On the other hand, when the conductivity is of the magnetic 
 kind, the reflected wave sheet differs from the incident only in 
 having its induction reversed. The displacement persists, 
 being doubled during the act of reflection, whilst the induction 
 is then annulled. 
 
 The other extreme occurs when a plane electromagnetic sheet 
 hangs on to a conductor perpendicularly. It then slips along 
 the conductor at the speed of light, with perfect slip. This 
 may occur with a plane reflector, but, of course, the most 
 striking and useful and practical case is that of a straight 
 cylinder a wire, in fact, though it need not be round, but may 
 have any form of section. The wave then runs along the wire 
 at. constant speed v, and without change of type, at least so 
 long as the wire continues straight and of unchanged section. 
 If the section vary regularly, so that the wire is a cone, then it 
 is a spherical wave that is propagated along it without change 
 of type. This case includes an infinitely fine wire, when we 
 may have either spherical or plane waves. Other interesting 
 cases may be made up by varying the angle of the cone, or 
 using a double cone, or a cone and a plane, &c. 
 
 Now, in the first main case of perfect reflection (flush) the 
 incident and reflected sheets are wholly separated from one 
 another, except just at the reflecting surface, where there is 
 a momentary coincidence. On the other hand, when a plane 
 wave runs along a wire, or, say, more conveniently here, along 
 a plane, we only see one wave. It is the case of reflection at 
 grazing incidence, and may be considered the limiting case of 
 permanent union of the incident and reflected waves. Between 
 these two cases we have the general case of incidence and reflec- 
 tion at any angle. There are two plane waves (sheets, most 
 conveniently for reasoning and description) one going to and the- 
 other coming from the plane reflector, where they join together,. 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 343" 
 
 making equal angles with it. In the overlapping region, close 
 to the plane, the displacement due to the union of the two- 
 waves is normal to the surface (that is, with an electric con- 
 ductor) which is electrified, and the electrification runs along 
 the surface at a speed depending upon the angle of incidence,, 
 being v at grazing incidence (of rays) and v/cos 6 at incidence 
 angle 0, varying, therefore, between v and infinity. It may be, 
 perhaps, rather a novel idea to some readers that electrification 
 can run through space at any speed greater than that of light, 
 but the matter is made simple enough by considering the rate 
 of incidence upon the reflecting surface of different parts of the 
 plane sheet. In the case of nearly flash incidence of a sheet, 
 its different parts strike the surface nearly simultaneously, so- 
 that there is an immensely great speed of motion of the elec- 
 trification along the surface. The electrification is the same in 
 amount always, and is continuously existent, so we are some- 
 what justified in speaking of the electrification moving ; but 
 we may equally well regard it as a case of continuous genera- 
 tion of electrification at one end and of annihilation at the 
 other end of the part of the conducting surface which is 
 momentarily charged, the generation and annihilation being 
 performed by the different parts of the incident sheet and the 
 reflected sheet as they reach and leave tLe surface. Details of 
 these simple cases, leading to a plainer understanding, will 
 come later, when these general notions are got over. In the 
 limiting case of normal incidence of rays, when the incident 
 sheet strikes the surface flush, the electrification is non-existent, 
 It goes out of existence just as its speed becomes infinite. 
 
 The above describes one extreme kind of reflection of a plane 
 sheet at any angle, and is what occurs when H in the incident, 
 and, therefore, also in the reflected wave, is tangential to the 
 reflector, whilst E is in the plane of incidence. But when it 
 is H in the incident wave that is in the plane of incidence,, 
 and E is tangential, we have quite another kind of composition 
 in the overlapping part near the reflector. There is no E within 
 it at all, and also no electrification ; whilst the H within 
 it is parallel to the reflector, and simply joins together the H's 
 in the parts of the waves which do not overlap, the H in one 
 wave being directed towards the surface in the plane of inci- 
 dence, and in the other away from the surface. In both waves, of 
 
344 ELECTROMAGNETIC THEORY. CH. IV. 
 
 course, H is in the plane of the wave. In other respects we have 
 a similarity in propagation. It is now a surface current, instead 
 of electrification, that runs along the surface at speed v/cos 0. 
 
 If the reflector be a magnetic conductor, we have two very 
 similar main cases, in one of which magnetification (the ana- 
 logue of electrification), and in the other a magnetic current, 
 runs, or appears to run, along the surface. 
 
 The Effect of Conducting Matter in Diverting External 
 Induction. 
 
 191. The theory of the effect of a finite degree ot con- 
 ductivity in the medium on electromagnetic waves is far more 
 difficult and complicated than that of the effect of infinite con- 
 ductivity. Nevertheless, we may gain a general idea of the 
 nature of the effect by means of the substitution of simple 
 problems for the real ones that present themselves, and also in 
 this way obtain a knowledge of very important properties con- 
 cerned. There are two ways in which we may regard the ques- 
 tion of conductivity. First, starting from the theory of perfect 
 conductors surrounded by perfect dielectric non conductors, we 
 may examine the effect of introducing a slight amount of 
 resistivity, to be then increased more and more until at last 
 we come to conductors of high resistivity, or infinite, when 
 we have dielectrics merely. The other way is to start with 
 electromagnetic waves in a perfect dielectric, and examine the 
 effect produced upon them of introducing first a small amount 
 of conductivity, then more and more, until we come to perfect 
 conductors again. Both ways are instructive none the less 
 because they give very different views of the same matter. 
 
 In the first place, it may be readily conceived that if a con- 
 ductor have only a slight amount of resistivity it may behave 
 approximately the same as if it had none, and may obstruct 
 waves nearly perfectly internally, and likewise reflect and con- 
 duct them superficially nearly perfectly. This is true, but the 
 element of time has to be taken into account, as it becomes of 
 great importance when there is some resistivity, however little, 
 instead of quite none. Suppose, for example, we have an 
 initially steady magnetic field, and bring a conductor into it 
 very quickly from a distance, and keep it there. If this be 
 done quickly enough, the first effect is nearly the same as if 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 345 
 
 the conductor were perfect. That is, the induction of the 
 field will be driven out of the space it previously occupied, 
 which is now occupied by the conductor, and pass round 
 it tangentially, as if the conductor were of zero induc- 
 tivity. It will, therefore, be mechanically acted upon by 
 the stress in the field in a manner resembling the action 
 upon a perfect diamagnetic body that is, there will be a 
 force tending to drive it from stronger to weaker parts of the 
 field. In another form, we may say that external force must 
 be applied to the conductor to bring it into the field and 
 to hold it there. But this state of things will not continue. 
 The magnetic force, which is at first only skin deep, will pene- 
 trate into the interior in time, according to a law resembling 
 that of the diffusion of heat. Given time enough, it will 
 assume the same distribution as if there were no conductivity, 
 although our assumption is that there is nearly no resistivity 
 that is, in the ultimate state tended to, it is merely the induc- 
 tivity that settles how the magnetic induction will distribute 
 itself. Initially, there is a skin current, its total being mea- 
 sured by the difference in the magnetic force just outside and 
 a little way inside the conductor, in the bulk of which there 
 is practically no magnetic force or electric current. As in the 
 -case of a perfect conductor, Ohm's law is fully obeyed. There 
 is no internal current, because there is no internal electric or 
 magnetic force. They simply have not had time to get in, on 
 account of the obstructive action of the high conductivity. It 
 is as useless and misleading to say that one electric force is 
 -cancelled by another, as it is to ascribe the absence of magnetic 
 force to the counteracting magnetic force of the skin current. 
 As the magnetic force spreads into the conductor, so does the 
 electric current, which is the curl of the former. In the end, 
 when the magnetic force has got steady, the current ceases. 
 There may now still be moving force on the conductor, but not 
 of the same kind as before, it being simply the ordinary para- 
 anagnetic attraction or diamagnetic repulsion according as the 
 value of the inductivity of the body exceeds or is less than 
 that of the external medium. 
 
 It will be very much the same thing if we start with the 
 conductor at rest in a neutral state in a neutral field, and 
 then establish a steady magnetic field by some external 
 
346 ELECTROMAGNETIC THEORY. CH. IV. 
 
 cause. There will still be the same internal obstruction 
 offered by the conductor, and consequent delay in the assump- 
 tion of the steady state throughout it. In both cases, too, 
 there is a necessary waste of energy involved in the process, 
 according to Joule's law of the generation of heat by the exist- 
 ence of electric current in conducting matter. Thus two things 
 happen when the degree of conductivity is not infinite. First, 
 the reflection is imperfect at the boundary of the conductor, 
 a portion of an incident disturbance being transmitted into 
 it. Next, in the act of transmission and the attenuation 
 involved, there is a loss of energy from the electromagnetic 
 field. We shall see later more precisely the nature of the 
 attenuating process. At present we may note that if the 
 external field be not steady or do not tend to a steady state, 
 so that the conductor is exposed to fluctuating forces, then the 
 internal part of the conductor need never acquire any sensible 
 magnetic force. Thus, if the external field be the sum of a 
 balanced alternating, and of a field which would be steady in 
 the absence of the conductor, only the latter part will penetrate 
 fully into it. The former alternating part will penetrate 
 imperfectly, the more so the greater the conductivity, and, as 
 before said, not at all when the conductivity is perfect. The 
 other field is then also excluded. With rapid alternations the 
 region of sensible penetration is only skin-deep, consisting of 
 layers of opposite kinds (as regards direction of the magnetic 
 force, which is nearly tangential), with, however, so very rapid 
 an attenuation of intensity in going inward that practically 
 only one wave need be considered (except for short waves, like 
 light). Of course, the heat generation is now confined to the 
 skin. What goes in further does so by ordinary heat diffusion. 
 
 The time-constant of retardation of a conductor varies as the 
 conductivity, as the inductivity, and as the square of the linear 
 dimensions. This refers to the intervals of time required to 
 establish a definite proportion of the steady state under the 
 action of steady forces in bodies of different size, conductivity, 
 and inductivity, but geometrically similar. Here the two pro- 
 perties, conductivity and inductivity, act conjointly, so that, for 
 example, iron is far more obstructive than copper, although its 
 conductivity is much inferior. It is different with the heat- 
 generation. There the inductivity and conductivity act in 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 347 
 
 opposite senses, for, with the same electric force, the waste 
 varies as the conductivity, or, with the same current-density, 
 as the resistivity. It results that in cases of skin-conduction 
 of rapidly alternating currents, the resistance per unit area of 
 surface varies directly as the square root of the product of the 
 resistivity (not conductivity), inductivity, and frequency. 
 Thus, whilst we may increase the resistance by increasing the 
 frequency, with a given material, and also by increasing the 
 inductivity, we decrease it by increasing the conductivity, in 
 spite of the fact that the internal obstruction varies as the con- 
 ductivity and inductivity conjointly. The point to be attended 
 to here is that mere internal obstruction is no necessary bar to 
 effective skin conduction, although, of course, in a given case 
 the resistance is greater than if the conduction were more wide- 
 spread. It depends upon how it is brought about, whether by 
 conductivity or inductivity. This is how it comes about that 
 with the complete internal obstruction of a perfect conductor, 
 with the effective skin reduced to nothing, there is still no 
 resistance, and the slip of electromagnetic waves along them is 
 perfect. But it is different when we obtain the internal obstruc- 
 tion by increasing the inductivity, preserving the conductivity 
 constant. Perfect internal obstruction then means infinite 
 resistance, and no proper slipping of waves at all. If the 
 obstruction be not complete, it will be accompanied by very 
 rapid attenuation of waves running along the surface when the 
 obstruction arises from high inductivity, and by relatively very 
 slight attenuation when it arises from high conductivity. 
 
 The repulsive force which was referred to in the case of a 
 perfect conductor brought into a magnetic field, or when a 
 magnetic field is created outside the perfect conductor, arises 
 from its obstructive action, combined with the fact that it is 
 only the lateral pressure of the magnetic stress that acts on the 
 conductor, owing to the tangentiality of the magnetic force. 
 This repulsive force is naturally also operative, though in a less 
 marked form, when the conductivity is not perfect. In fact, it 
 is operative to some extent whenever there is a sufficiently rapid 
 alternation of the field for the conductance to cause a sensible 
 departure from the undisturbed state of the magnetic force, 
 and is, therefore, strongly operative with ordinary metallic 
 conductors with quite moderate frequency of vibration. The 
 
CIS ELECTROMAGNETIC THEORY. CH. TV. 
 
 remarkable experiments of Prof. Elihu Thomson on this " elec- 
 tromagnetic repulsion " will be remembered. The phenomenon 
 has nothing specially to do with electromagnetic waves. It is 
 magnetic repulsion, rather than electromagnetic, using the 
 word " magnetic " in its general sense, apart from the special 
 fact of magnetisation when the iiid activity of the conductor 
 is not the same as that of the ether.* 
 
 When a conductor is brought into an electric instead f a 
 magnetic field, the case is somewhat different. There is in 
 any case merely skin conduction, for there is an actual destruc- 
 tion of the flux displacement by conductivity. The final result 
 therefore, is that the ultimate permanent state in the con- 
 ductor is a state of perfect neutrality, just as if the conduc- 
 tivity were perfect. Electric conductivity destroys displace- 
 ment, but it cannot destroy induction. Similarly, magnetic 
 conductivity would destroy induction, but would be unable to 
 destroy displacement. Thus, a magnetic conductor brought 
 into an electric field would, in time, permit its full penetra- 
 tion, but if brought into a magnetic field the final result would 
 be a state of internal neutrality, however low the conductivity. 
 If, however, it be very low, then, whether it be of the electric 
 or the magnetic kind, there will be an initial nearly complete 
 penetration (owing to the removal of the obstruction), followed 
 by subsidence to zero of the flux appropriate to the conduc- 
 tivity, electric or magnetic respectively. The persistence of 
 magnetic induction, in spite of the presence of electric con- 
 
 * This reference to Elihu Thomson's experiments must not be under- 
 stood as a full explanation, which is sometimes complex. The idea in the 
 text has been of a lump of metal. When made a disc or a linear circuit 
 we have special peculiarities, and the theory may be perhaps best done 
 in terms of inductances and resistances. The principle concerned of 
 the temporary diversion of magnetic induction by conducting matter 
 remains in force, however, whether the matter be in a lump or in a closed 
 line. In the latter case the tendency of the conductance is to keep the 
 total induction through it constant. Consider first an infinitely con- 
 ducting disc which completely diverts induction ; and next, a ring made 
 by removing nearly all the disc except the outer part. Induction now 
 goes through the circuit, of course, when brought into a magnetic field, 
 but its total is zero, by reason of the infinite conductance and the current 
 " induced " in the ring. As is well known, Maxwell considered perfectly 
 conducting linear molecular circuits in applying his views to Weber 'A 
 theory of diamagnetism. 
 
THEORY OF PLANE ELECTROMAGNETIC WAVE3. 349 
 
 ductivity, is a very important and significant fact, of which I 
 shall give a simple proof in a later Section, in amplification of my 
 proof of 1887-8. The present method of passing from perfect 
 to finite conductivity is unsuitable for the purpose, because 
 in good conductors the dielectric permittivity is altogether 
 swamped, and is therefore ignored ; whilst, on the other hand, 
 in very bad conductors the permittivity is a factor of the 
 greatest importance. Now we can pass continuously from a 
 non-conducting dielectric to a conducting one, up to perfect 
 conduction, but we cannot pass the other way without having 
 the permittivity in view all the time, which makes the matter 
 difficult. 
 
 Parenthetical Remarks on Induction, Magnetisation, Indue- 
 tivity and Susceptibility. 
 
 192. As people's memories are very short, and there is some 
 discussion on the subject, I may repeat here that the so- 
 frequently-used word inductivity is not intended for use as a 
 mere synonym for permeability. The latter is the ratio of the 
 inductivity of a medium to that of ether, and is therefore a 
 mere numeric. On the other hand, inductivity has a wider 
 meaning, namely, such that j/*H 2 is the density of the magnetic 
 energy, irrespective of dimensions. We can only make it a 
 numeric by assumption. Even then, it has only a fictitious 
 identity with permeability a forced numerical identity. 
 Similar remarks apply to some other quantities, but they 
 are particularly necessary in the case of inductivity, on 
 account of the obscure and misleading manner in which the 
 connections between induction, magnetic force, and magnetisa- 
 tion were formerly commonly presented (and still are some- 
 times), together with the misleading connection between the 
 susceptibility and the permeability. We should write 
 
 if p is the permeability and K the susceptibility, instead of 
 
 where the suffix letter refers to the common irrational reckon- 
 ing. The 4?r is, as usual, simply nonsense, unworthy of scientific 
 men near the end of the nineteenth century. Now introduce 
 
350 ELECTROMAGNETIC THEORY. CH. IV. 
 
 Maxwell's ether theory, and make /? = /V/V where // is the 
 inductivity of ether, and p that of some other substance (with 
 the usual reservations), then 
 
 takes the place of the common 
 
 -which is misleading in two respects, first as regards the obstre- 
 perous 47T, and next in making p and K be quantities of the 
 same kind. But K is always and essentially a numeric, whilst //, 
 is not. We see that the use of inductivity rather than permea- 
 bility is necessary in electromagnetic theory, as a matter of 
 logical common sense as well as for the purpose of scientific 
 clearness. But this need not interfere with the use of permea- 
 bility in its above-described sense of a ratio. If, on the other 
 hand, one of the two words should be abolished, there can, I 
 think, be little doubt as to which should go. 
 
 In the case of purely elastic magnetisation (without intrinsic) 
 we have 
 
 where B is the induction and F the magnetic force. Here 
 is what the induction would be in ether, so that the additional 
 part /* KF expresses the effect of the matter present, which 
 becomes magnetised. The ratio K-, therefore, naturally expresses 
 the susceptibility for magnetisation of the matter. Perhaps, in 
 passing, it might be thought that /A O K should express the sus- 
 ceptibility. But this will not do, because magnetisation and 
 induction are similar. The induced magnetisation is />t KF. In 
 strictness it should not be called the intensity of magnetisation, 
 but rather the density, if we properly carry out Maxwell's prin- 
 ciples about forces and fluxes, or intensities and densities. B is 
 a flux, therefore so is /* KF, to be measured per unit area. 
 Now, the common form is 
 
 or i = i + 7r i , 
 
 if 1. = K^i. Here we have, apart from the 4?r absurdity, an 
 irrationality of a different kind, viz., that induction and mag- 
 netisation are made identical in kind with magnetic force, since 
 we have the difference of two flux densities expressed by an 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 351 
 
 "intensity," which is referred to unit length. Now, this may 
 matter very little in practical calculations, but it is more than 
 mischievous in theory. Suppose, for example, we are working 
 with a kind of mathematics that takes explicitly into account 
 the two ways of measuring vector magnitudes, with reference 
 to length and to area respectively, according as they are re- 
 garded as " flux " densities or " force " intensities. Then, if we 
 do not recognise and take account of the radical difference 
 between B; and F f in the last equation, we may expect to be led 
 to singular and unaccountable anomalies. This is, I think, 
 what has happened in Mr. MacAulay's recent paper on the 
 theory of electromagnetism. The remedy is easy. There 
 should be no special limitations imposed upon the quantities 
 concerned such as occur when permeability and inductivity are 
 made the same. 
 
 It is also highly desirable, for the same purpose of obtaining 
 scientific clearness and freedom from distressing anomalies, 
 that the distinction between " induced " and intrinsic magneti- 
 sation should be clearly recognised and admitted in the formula. 
 In the above there has been no intrinsic magnetisation. Let 
 this now be I . An equivalent form is /xh , where h is the cor- 
 responding intrinsic magnetic force. This I is of the same 
 nature as B. The complete induction becomes 
 
 where H is the complete force of the flux B. This is the best 
 way of exhibiting the induction. If H be split at all, let it be 
 into the part involving the intrinsic force h and the rest. Or, 
 
 The other separation, namely, of //F into the ether induction 
 /x F and the additional part due to matter, is less useful. If 
 done, then 
 
 If we now amalgamate I and I, to make, say, I 1} the total 
 magnetisation, intrinsic and induced, we have 
 
 which, translated into irrational units, makes 
 
352 ELECTROMAGNETIC THEORY. CH. IV. 
 
 and lastly, omitting the /t* by assuming it to be unity, we 
 obtain the common 
 
 B i -F < +47rI li , 
 
 containing three faults, the arbitrary 47r, the equalising of an 
 intensity and a flux, and the unnatural union of physically dis- 
 tinct magnetisations. 
 
 " Different men have different opinions some like apples, 
 some like inions." But can anyone possibly really like the 
 roundabout and misleading way of presenting the magnetic 
 flux relations which I have above criticised? There is no 
 excuse for it, except that it was employed by great men when 
 they were engaged in making magnetic theory, before they 
 had assimilated its consequences thoroughly. When the rough 
 work of construction is over, then it is desirable to go over 
 it again, and put it in a better and more practical form. We 
 should copy the virtues of great men, if we can, but not their 
 faults. 
 
 Men who are engaged in practical work can hardly be expected 
 to fully appreciate the importance of these things, because their 
 applications are of so highly specialised a nature, in the details 
 of which they may become wholly absorbed. They may even 
 go so far as to say that the paper theory of magnetic induction is 
 not of the least moment, because they are concerned with iron, 
 and although there may be a certain small range of application 
 of the theory even in iron, yet they are scarcely concerned 
 with it, and, therefore, it is of no consequence. There could 
 not be a greater mistake. A complete theory of magnetic 
 Induction, including hysteresis, must necessarily be so con- 
 structed as to harmonise with the limited theory that has 
 already been elaborated, which is understandable when exhibited 
 in a purified form, freed from 47r's and other anomalies. First 
 of all, we have the ether, in which B = /* F or B = /x H, because 
 of the absence of intrinsic magnetisation. Next we come to 
 elastically magnetised bodies in which the relation between flux 
 and force is linear. Then B = /xH, where ^ differs from /x , being 
 either greater or smaller, and is either a constant scalar, or else 
 (with eolotropy) a linear operator. If there is no intrinsic mag- 
 netisation, F and H are still the same, and the curl of either is 
 the current density. But should there be intrinsic magnetisa- 
 tion, then H = h ( , + F, whilst B = /xH still, and it is the curl 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 353 
 
 of P that is the current density. Or, B = I + /*F. The next 
 .step is to make fiF be not a linear, but some other function 
 of F, to be experimentally determined, if it be possible to 
 express B - 1 , which is the free induction, as a function of F. 
 Of course, it can be done approximately. Then comes the 
 difficult question of hysteresis. This involves the variation of 
 I with F, with consequent waste of energy. If this little 
 matter be satisfactorily determined, we may expect to have a 
 sound mathematical theory of magnetic induction in an 
 extended form which shall properly harmonise with the 
 rational form of the elementary theory. The divergence of 
 B is zero all through. The experimental justification of this 
 generalisation is the fact that no unipolar magnets have yet 
 been discovered. 
 
 Effect of a Thin Plane Conducting Sheet on a Wave. 
 Persistence of Induction and Loss of Displacement. 
 
 193. Coming now to the effects produced on electromag- 
 netic waves by a small amount of conductivity, to be after- 
 wards increased, we shall adopt a particular device for simplify- 
 ing the treatment. Imagine, first, the dielectric medium to 
 possess a uniformly-distributed small conductivity. Evidently, 
 che action of the conductivity on a wave is a continuous and 
 cumulative one. Next, localise the conductance in parallel 
 sheets that is, substitute for the uniform conductivity a great 
 number of parallel plane conducting sheets, between which the 
 medium is non-conducting. If we increase their number suffi- 
 ciently, their action on a wave whose plane is parallel to that 
 of the sheets will approximate, in the gross, to the effect of the 
 uniform conductivity which the conductance of the sheets 
 replaces. We have, therefore, to examine the influence of a 
 single very thin conducting sheet upon a wave. This is not 
 difficult. 
 
 Imagine, then, a simple plane electromagnetic sheet to be 
 running through the ether at the speed of light. This is the 
 natural state of things ; and, in the absence of conductivity or 
 other disturbing causes, there will be no change. Now insert 
 a plane conducting sheet in the path of the wave. It should 
 be so thin that the retarding effect of diffusion within it is 
 quite insensible. Let the wave strike it flush. The theory 
 
 AA 
 
354 BLECTROMAQNBTIC THEORY. OH. IV. 
 
 shows that it is immediately split into two similar waves, one 
 of which is transmitted beyond the plate, whilst the other is 
 reflected. The transmitted wave differs from the incident in 
 no respect except strength. It is attenuated in a certain ratio 
 depending upon the conductance of the sheet, being greatlv 
 attenuated when the conductance is large, and slightly atten- 
 uated when it is small. The formulee are reserved. This 
 transmitted wave moves on just as the incident wave did, and 
 nothing further happens to it. The reflected wave, on the 
 other hand, having its direction of motion opposite to that of 
 the incident and transmitted waves, travels back the way it 
 came, and nothing further happens to it. 
 
 The direction of the magnetic force in the three waves is the 
 same. This is one general property, irrespective of the amount 
 of conductance. But a much more striking one connects the 
 intensity of the magnetic force in the three waves. The sum 
 of the intensities in the reflected and transmitted waves equals 
 the intensity in the original incident wave. That is, the con- 
 ductance, with dissipation of energy, has had no effect whatever 
 on the total induction. It has merely redistributed it, by 
 splitting it into two parts, which then separate from one 
 another. The " number of tubes " in the reflected wave may 
 be made to bear any ratio we please to the number in the 
 transmitted wave by altering the conductance of the plate : 
 but their sum is always exactly the number of tubes in the 
 incident wave. This property exemplifies, in a manner which 
 may be readily understood, the persistence of induction, in 
 spite of conduction and waste of energy. 
 
 But as regards the displacement, the case is quite different. 
 From the fact that the reflected wave runs back, whilst its 
 magnetic force preserves its original direction, we see that the 
 electric force must be reversed. On the other hand, it is 
 unchanged in the transmitted wave. If, then, their sum 
 equalled the electric force in the incident wave, it would imply 
 that the transmitted wave was of greater amplitude than the 
 incident. But it is smaller, invariably. So there is a loss of 
 displacement. Thus, if H in the incident becomes (1 - 7i)H in 
 the transmitted wave, where n is some proper fraction, it 
 becomes nH in the reflected wave. At the same time E in the 
 incident becomes (1 -?t)E in the transmitted, and - wE in the 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 355 
 
 reflected wave. The loss of E is, therefore, 2nE, or the loss of 
 displacement is 2nD. By the loss we mean the excess of the 
 displacement in the incident over the sum of the displacements 
 in the two resulting waves, transmitted and reflected. This 
 loss occurs at the very moment the incident wave coincides 
 with the plate. The plate itself may be regarded as a 
 dielectric homogeneous with the ether outside, or perhaps of 
 different permittivity, but with the conducting and dissipating 
 property superposed. When thin enough, the permittance of 
 the plate is of insensible influence, and may be disregarded. 
 But strictly, a conducting dielectric is a dielectric which cannot 
 support displacement without wasting it, so that a continuous 
 supply of fresh displacement is needed to keep it up. The rate 
 of waste of energy is proportional to the electric stress. 
 
 But it should be carefully noted that the loss of energy and 
 the loss of displacement are entirely distinct things, which are 
 not proportional ; and that the loss of displacement itself may 
 sometimes require to be understood in a somewhat artificial 
 sense. For the loss may be greater than what there is to lose. 
 It must then be understood vectorially. This occurs when n 
 is greater than \. When n = J, the reflected and transmitted 
 waves are equally strong, and only differ in the direction 
 of the displacement. The loss of displacement is, there- 
 fore, complete. The loss of energy in the plate is simul- 
 taneously at its maximum, being equal to one-half of the 
 energy of the original wave. If we reduce the conductance of 
 the plate, we increase the transmitted wave, and reduce the 
 waste of energy in it and the loss of displacement. The 
 amount of the latter still remains positive, therefore, assuming 
 it to be positive in the incident wave. The extreme is reached 
 when the plate has no conductance. Then the incident wave 
 goes right through without any splitting and reflection, and 
 therefore without attenuation, and there is no waste of energy. 
 On the other hand, if we increase the conductance above the 
 critical value making n = J, we reduce the transmitted wave 
 and increase the reflected, whilst we simultaneously reduce the 
 waste of energy in the plate and increase the loss of displace- 
 ment. The extreme is reached when the plate is a perfect 
 conductor. There is then no transmitted wave and no loss of 
 energy, whilst the reflected wave is of full size, but with the 
 
 AA 2 
 
356 ELECTROMAGNETIC THEORY. OH. IV. 
 
 displacement reversed as compared with the incident, so that 
 the loss of displacement, in the sense described, is the greatest 
 possible, viz., 2D. 
 
 It will be seen from these details that whilst the absorption 
 or dissolving of induction by conductance is a myth, the idea 
 of an absorption of displacement is not without its incon- 
 veniences when the conductance is great, and that this becomes 
 extreme when there is really no loss of energy in the plate, 
 when, in fact, the incident wave does nothing in it, but is wholly 
 rejected with its displacement reversed. It seems, then, more 
 natural to consider the waste of energy from the field caused 
 by the plate as loss. This takes place equally from the electric 
 and magnetic energies, since they are necessarily equal in every 
 one of the three waves. But in the application made later, 
 the plate is to have very slight conductance (in the limit an 
 infinitely small amount), so that the total displacement cannot 
 change sign, but merely suffers a slight loss. Then the idea 
 of loss of displacement by conductance becomes useful again. 
 
 The loss of energy takes place as the incident wave is travers- 
 ing the plate. Its successive layers each cause a minute attenua- 
 tion of the wave passing, and this applies equally to the induc- 
 tion and displacement, so that the transmitted wave emerges 
 from the plate a pure electromagnetic wave, a reduced copy of 
 the incident. The successive layers, too, each cast back a 
 minute portion of the wave traversing them, with unchanged 
 sign of the induction, but with displacement reversed; and 
 these rejected fluxes make up the reflected wave. There are 
 evidently residual effects due to the internal reflections of 
 minute portions of the main reflected wave, but these residuals 
 tend to vanish when the plate is thin enough. 
 
 If the plate be a magnetic instead of an electric conductor, 
 the theory is quite similar. The transmitted wave is an 
 attenuated continuation of the incident. The reflected wave 
 is also a copy of the incident, also reduced. But it is now 
 the induction that suffers loss, because its direction in the 
 reflected wave is opposite to that in the incident and trans- 
 mitted. On the other band, the displacement now fully per- 
 sists, being merely split into two parts by the plate. 
 
 Notice that if the plate be both an electric and a magnetic 
 conductor, its attenuating effect from these two causes on the 
 
OP PLANE ELECTROMAGNETIC WAVES. 357 
 
 transmitted wave will be additive, so that it will emerge a 
 pure wave with extra attenuation. But as regards the reflected 
 wave, we have a peculiar result. The action of the magnetic 
 conductance is to reverse the induction whilst keeping the 
 displacement straight ; whilst that of the electric conductance 
 is to reverse the displacement and keep the induction straight. 
 The result is that the reflected wave is reduced in magnitude 
 by the addition of magnetic conductance to previously existent 
 electric conductance. With a proper proportioning of the two 
 conductances, the reflected wave may be brought nearly to 
 evanescence from a plate of finite conductance. In the limit 
 the compensation is perfect, and the incident wave goes right 
 through without reflection, though it suffers extra attenua- 
 tion. This is the explanation of the distortionless propagation 
 of waves in a dielectric medium possessing duplex conducti- 
 vity, electric and magnetic. Whilst there is no reflection in 
 transit, there is a continuous loss both of displacement and of 
 induction. 
 
 The Persistence of Induction in Plane Strata, and in general. 
 Also in Cores and in Linear Circuits. 
 
 194. Now return to the case of electric conductivity alone, 
 and, as described, let it be locally condensed into the conduc- 
 tance of any number of parallel plates. We know that the 
 effect of any one of them on a thin electromagnetic sheet is to 
 split it, as previously described. If we like, therefore, we can 
 follow each of the resulting waves, and observe how they are, 
 in their turn, split by the first plates they meet, giving rise to 
 four waves, to be a little later split into eight, and so on. This 
 process may seem cumbrous, but it is also an instructive one. 
 
 Thus, consider what happens to the total induction. We 
 know that it persists in amount and direction when a single 
 split occurs. Now the same property applies to every succes- 
 sive split a wave suffers in our dielectric medium containing 
 parallel conducting plates. So the total induction remains 
 constant. It is redistributed and spreads out both ways, but 
 without the least loss. There is a small loss of energy at every 
 split, but this does not affect the total induction. This applies 
 when we start from a single pure electromagnetic sheet moving 
 either way. It therefore applies when the initial state consists 
 
358 ELECTROMAGNETIC THEORY. CH. IV. 
 
 of any number of such sheets, of any strengths, forming a per- 
 fectly arbitrary initial distribution of induction and displace- 
 ment in parallel plane layers. There is still persistence of 
 the induction. Finally, the same applies when we split up 
 the conducting plates themselves into plates of smaller con- 
 ductance, and spread them out at uniform distances. The 
 ultimate limit of this process is reached when the conduct- 
 ance is quite uniformly spread, so that we have a perfectly 
 homogeneous medium under consideration. It is, fundament- 
 ally, a dielectric propagating disturbances at speed v ; but it 
 is, in addition, a conductor as well, and distorts the waves and 
 dissipates their energy. The speed v is (MC)"*, with the proper 
 values of //. and c. The conductivity does not interfere with 
 this property of propagation at finite speed. But observe that 
 if we choose to ignore the displacement, then the corresponding 
 speed is infinitely great. We conclude from the above that 
 plane sheets of induction in electric conductors always preserve 
 the total induction constant in amount, irrespective of the 
 amount of elastic displacement, or whether there is any at all. 
 That is, induction cannot be destroyed by conductance. 
 
 If, then, it suffers destruction, this must be due to some 
 other cause. It may be merely a cancellation by the union of 
 oppositely-directed inductions. This may be termed a vectorial 
 cancellation. It may occur, of course, with plane strata of 
 induction. Thus if, in an infinitely large conductor, the total 
 induction be initially zero, which does not require the induction 
 density to be zero, the final effect will be a complete annihila- 
 tion of the induction by mutual cancellations. Should, how- 
 ever, the total induction be not zero, it will persist. The 
 induction density will tend to zero, but that will be merely on 
 account of its attenuation by spreading, not because there is 
 any destruction by the conductance or resistance when either 
 of them is finite. To prevent the attenuation to zero we may 
 interpose infinitely conducting barriers, one on each side, in 
 planes parallel to the sheets of induction. Then the final 
 result will be that the induction will spread itself out uniformly 
 between the barriers and maintain a finite density, 
 
 To illustrate this property in a somewhat less abstract man- 
 ner, consider a large ring, say of copper, though iron will do 
 equally well except as regards some complications connected 
 
THEORY OP PLANE! ELECTROMAGNETIC WAVES. 359 
 
 with its magnetisation. Let it be inductised by an enveloping 
 coil-current so that the induction goes along the core in a 
 complete circuit. When it is steadily set up, if we remove 
 the coil- current (and the coil too, preferably for our present 
 purpose) the induction in the core will, in time, all come out of 
 it. But if we clap an infinitely-conducting skin upon the core, 
 it will not come out. Then we have a certain flux of induction 
 locked up, as it were, in a conducting material, which has no 
 effect upon it. It can neither be destroyed by the conductance 
 of the core nor can it get through the perfectly-obstructive 
 skin. If the skin is clapped on after the induction has 
 partially escaped (which escape begins on the outside, before 
 the interior is sensibly affected), there is a redistribution of 
 induction, which continues until a new state of equilibrium is 
 reached. During this process there is electric current in the 
 core and some waste of energy. But there is no waste of the 
 induction. The final induction is the mean value of .the 
 original induction across the section of the core. 
 
 In further illustration, let the core be hollow and be induc- 
 tised circularly instead of along its length by means of two 
 currents on its boundaries, inner and outer, oppositely directed, 
 following the length of the core. When this is done, remove 
 the currents and clap on perfectly conducting skins internally 
 and externally. There will be a similar persistence of the 
 induction, although its tubes now go round the inner boundary 
 circularly. There may be an initial settling down, but the 
 outer skin will not let the induction expand outwardly, and the 
 inner skin will not let it contract inwardly. If the latter could 
 happen we might have cancellation. To get this effect remove 
 the inner skin. Then, whether we fill up the hollow with 
 finitely conducting matter or leave it nonconducting, we allow 
 the induction to spread internally and permit cancellation. The 
 induction will now wholly disappear, in spite of the external skin. 
 That is to say, there will be a continuous passage of the induc- 
 tion out of the initially inductised region, accompanied by elec- 
 tric current therein, which will continue until the whole of the 
 magnetic energy is wasted as heat in the core. 
 
 The same property is exemplified, though in a less easily under- 
 standable manner, with a single closed line or circuit of infinite 
 conductance. If it embrace a certain amount of Induction it 
 
360 ELECTROMAGNETIC THEORY. OH. IV. 
 
 will always do so, in the absence of impressed force to alter 
 the amount. The induction is locked in and cannot pass 
 through the infinitely- conducting circuit to dissipate itself. 
 If the conductance of the circuit be finite, then it can get 
 through. The time-constant varies as the conductance. The 
 disappearance of the induction is manifested by the waste of 
 energy in the circuit, the electric current in which is sup- 
 ported by the voltage of the decreasing induction through it. 
 But the current is there all the same (measured magnetically) 
 when the conductance is infinite. The induction is steady, 
 and there is no voltage in the circuit. But none is needed. 
 
 On the other hand, if there is initially no induction through 
 the circuit, there will continue to be none when a magnetic field 
 is created in its neighbourhood. But although the tubes of induc- 
 tion cannot cut through the infinitely conducting circuit so as to 
 make the induction through it be a finite quantity, yet they do 
 pass through a surface bounded by the circuit, as much positively 
 as negatively. The resulting induction distribution is to be got 
 by superimposing the external induction and that due to a cur- 
 rent in the circuit of such strength as to make the total induction 
 through it be zero. The property is a general one, for if the cir- 
 cuit be moved about in a magnetic field, there is always, in virtue 
 of its impermeability to the magnetic flux, zero total induction 
 through it if its conductance be infinite ; whilst if it be finite 
 but great, there is an approximation to this result so long as the 
 motion is kept up, or the external field be kept varying. At the 
 same time, the least amount of resistance in the circuit will be 
 sufficient, if time enough be given, to allow the external induc- 
 tion, when due to a steady cause, to get past it to the full 
 extent, when of course the current in the circuit will cease. 
 
 In a similar manner, displacement can be locked up by a 
 circuit of perfect magnetic conductance. There is also per- 
 sistence of displacement in spite of a finite degree of magnetic 
 conductivity in a continuous medium, unless it" be electrically 
 conducting as well. 
 
 The Laws of Attenuation of Total Displacement and Total 
 Induction by Electric and Magnetic Conductance. 
 
 195. Next consider the effect of a conducting medium upon 
 the total displacement. We know that the latter decreases 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 361 
 
 with the time, and the law of decrease may be readily found 
 from the theory of a single conducting plate. We found that 
 when its conductance exceeded a certain value, the loss of dis- 
 placement exceeded the original. But, in regarding the action 
 of a homogeneous conductor upon a wave as the limit (in the 
 gross) of that of an assemblage of parallel plates in which the 
 conductance is localised (which process may not seem unassail- 
 ably accurate beforehand, but which is justified by the results), 
 it is easy to see that we have merely to deal with plates of 
 such very low conductance that the loss at each is extremely 
 small, so that the above-mentioned difficulty does not enter. 
 Thus, let the loss at one plate be such as to reduce the initial 
 displacement D in a wave to mD, where m is a fraction nearly 
 equal to unity. Here mD is the sum of the displacements in 
 the transmitted and reflected waves, the latter being very 
 small and of the opposite sign to the initial D. As these waves 
 separate, they reach other plates and are split anew. If these 
 plates have each the same conductance as the first, the total 
 mD is further attenuated by them to w 2 D when the two waves 
 become four. Next, when these four waves are split into eight 
 by the next plates that are reached, the total displacement 
 becomes m 3 D ; and so on. These successive displacement totals 
 decrease according to the law of a geometrical series. It fol- 
 lows that, in the limit, we shall have the total displacement 
 represented by an exponential function of the time, say by 
 
 D = D oe -<, (1) 
 
 where D is the initial value, and D what it becomes at time t. 
 To find the value of n, we have merely to examine the form of 
 the fraction m, observe how it depends on the conductance of 
 one plate, and proceed to the limit by making the number 
 of plates infinite, whilst their conductances are infinitely small. 
 The result is that the constant n has the value &/c, where k is 
 the conductivity and c the permittivity of the homogeneous 
 conducting medium. 
 
 In the irrational units of the B.A. Committee this quantity 
 is represented by 47r&/c, which is, of course, nonsense, like the 
 quaternionic doctrine about the square of a vector. They are 
 both going to go. The above reasoning applies to any initial 
 distribution of displacement in plane layers, instead of merely 
 
362 ELECTROMAGNETIC THEORY. CH. IV. 
 
 one elementary sheet. Therefore, equation (1) shows that the 
 total displacement subsides according to the time-factor 
 e ~ ' c ' Now, this represents Maxwell's law of subsidence of 
 displacement in a conducting condenser (apart from " absorp- 
 tion" and hysteresis), or of the static distribution of displace- 
 ment associated with electrification in a conducting medium. 
 We see that the law has a far more general meaning. The 
 initial displacement need not be static, but may be accom- 
 panied by magnetic induction, and may consequently move 
 about in the most varied manner, whilst its total amount 
 decreases according to the static law. A homogeneous medium 
 is presupposed, and modifications may be introduced by the 
 action of boundaries. 
 
 Passing next to the analogous case of a magnetic conductor, 
 in which the total displacement remains constant whilst 
 the total induction subsides, it is unnecessary to repeat the 
 argument, but is sufficient to point out the law according to 
 which the subsidence occurs. If B be the initial total 
 induction, and B what it becomes at time t, we shall have 
 
 (2) 
 
 where g is the magnetic conductivity and p. the indue tivity. 
 The time-constant cjk of the former case has become p/g. 
 
 Returning to the former case, it should be noted that when 
 the initial distribution is of the static nature, unaccompanied 
 by magnetic force, it retains this property during the sub- 
 sidence. For, since the displacement subsides everywhere 
 according to the same time-factor, its distribution does not 
 alter relatively, or it remains similar to itself. Since, then, 
 there is no magnetic force, there is also no true electric current. 
 There is also no flux of energy. That is, the electric energy 
 is converted into heat on the spot. 
 
 A considerable extension may be given to this property. If 
 there be a conducting dielectric in which the permittivity 
 varies from place to place, containing a static distribution of 
 displacement, then, if the conductivity vary similarly from 
 place to place, so that the time-constant cjk is the same every- 
 where, the displacement will subside everywhere alike, without 
 magnetic force or flux of energy, and with purely local dis- 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 363 
 
 sipatipn of the electric energy. For the solution is repre- 
 sented by 
 
 E = E , -**, H = 0, 
 
 where E is the initial electric force of the static kind, having 
 no curl, and E that at time t. Both the fundamental circuital 
 laws are satisfied, the first because the true current is the 
 sum of the conduction and displacement currents, and the 
 second because E has no curl and k/c is constant. If it were 
 not constant then, obviously, the property considered would not 
 be true ; there would be different rates of subsidence at 
 different places, and the distribution of displacement would 
 change, along with magnetic force, electric current and 
 transfer of energy. 
 
 The corresponding property in a magnetic conductor 
 requires the constancy of the time-constant fi/g. Then, 
 whether p and g are themselves constant or variable from 
 place to place, a static distribution of induction subsides 
 everywhere alike, and without the generation of electric force. 
 
 Returning again to plane strata of displacement in an 
 electrically-conducting homogeneous dielectric, it may be 
 inquired how the property (1) of the subsidence of the total 
 displacement will be affected by the simultaneous existence of 
 magnetic conductivity. This will undoubtedly affect the 
 phenomena in detail, but will have no effect on the property in 
 question. Similarly, the law (2) of the subsidence of total 
 induction will not be affected by the presence of electric con- 
 ductivity. That is, in general, when there are both conduc- 
 tivities present, and both the fluxes displacement and induction 
 present, the total displacement subsides according to one law 
 and the total induction according to the other, without inter- 
 ference. These properties have their parallels in the theory of 
 telegraph circuits, as we shall see later. 
 
 It should be remembered that we are dealing always with 
 matter in the gross, and not with molecules at all ; or, equiva- 
 lently, we assume a homogeneous constitution of the elements 
 of volume. Thus, when displacement subsides in an electric 
 conductor without generating magnetic force, the possibility 
 and necessity of which are clearly indicated by the two cir- 
 cuital laws, it may be that if we go in between the molecules 
 
364 ELECTROMAGNETIC THEORY. CH. IV. 
 
 there is magnetic force. It is, in fact, difficult to conceive how 
 displacement in a heterogeneous medium of molecular consti- 
 tution could be done away with without the generation of 
 magnetic force, considering that the energy of the displace- 
 ment is converted into heat energy. 
 
 This matter, however, does not belong to the skeleton theory 
 of electromagnetism, but is rather to be considered as a side- 
 matter involving physical hypotheses to account for the influ- 
 ence of matter upon the electromagnetic laws. 
 
 The Laws of Attenuation at the Front of a Wave, due to 
 Electric and Magnetic Conductance. 
 
 196. Besides the above simple laws relating to the subsi 
 dence of the total fluxes (sometimes true for the elementary 
 parts) there are equally simple laws relating to the subsidence 
 of the fluxes at the front of a wave advancing into previously 
 undisturbed parts of the medium, which sometimes admit of 
 extension to the body of the wave. To understand this it may 
 be mentioned first, that the front of a wave in a non-conduct- 
 ing dielectric is always pure ; that is, the electric and magnetic 
 fluxes are in the wave-plane, and are perpendicular and in 
 constant ratio. The body of the wave need not be of this 
 pure type, owing to the change of form of the wave-front and 
 other causes, but the property of purity always characterises 
 the wave-front. This may be disguised in the case of a thin 
 electromagnetic shell, when it is regarded as the front, for 
 the shell itself may be complex. Then the mere front of the 
 shell may be the only quite pure part. But taking cases 
 free from this complication, we should next note that the 
 introduction of conductivity into the medium makes no 
 difference in the form of the wave-front or its position at a 
 given stage of its progress, provided, of course, that the two 
 quantities upon which the speed of propagation depends the 
 inductivity and permittivity are not altered. Now, as has 
 been already explained in connection with the theory of a thin 
 conducting plate, as the wave advances through a continuously 
 conducting medium its successive layers are being continuously 
 subjected to a reflecting process, a minute portion of every 
 layer being thrown back, whilst the bulk is transmitted. In 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 365 
 
 the body of a wave, therefore, there is a mixed-up state of 
 things. At the very front, on the other hand, there is no such 
 mixture, for the disturbance consists wholly of what has been 
 transmitted of the front layer. We may, therefore, fully expect 
 that the law of its attenuation in transit is of a simple nature. 
 To find it, locally condense the conductance into that of 
 any number of equal conducting plates. Let any one of these 
 plates attenuate a wave traversing it from E to wE. If 
 initially pure it emerges a pure wave, and passes on to the 
 next plate, where it suffers a second attenuation viz., to 
 w 2 E, and again emerges pure. At the third plate it becomes 
 w 3 E, and so on. The reflected portions we wholly ignore at 
 present. The limit of this process, when the plates are in- 
 finitely closely packed and of infinitely small conductance, so 
 as to become a homogeneous dielectric possessing finite con- 
 ductivity, is that the time-factor of attenuation takes the 
 exponential form. The result is 
 
 E = E e-*</ 2 <, (3) 
 
 E being the initial, and E the value at time t. The time- 
 constant 2c/k is just double that of the subsidence of total 
 displacement. Whilst, for example, the total displacement in 
 a plane wave attenuates to, say, y^- of its initial value, the 
 disturbance at the wave front has only attenuated to j 1 ^ of its 
 original value. 
 
 The property (3) applies to the magnetic as well as to the 
 electric force and flux. It does not apply merely to plane 
 waves, but to any waves, because the superficial layer only is 
 involved, and any elementary portion thereof may be regarded 
 as plane. So it comes about that the exponential factor given 
 in (3) makes its appearance in all investigations of waves in 
 electrical conductors when the permittivity is not ignored. It 
 is a more fundamental formula than the previous one with the 
 time-constant c/&, which is the final result of the complex pro- 
 cess of mixture of reflected waves, or is equivalent thereto. 
 
 The corresponding property in a magnetic conductor is that 
 the disturbance at the front of a wave is attenuated in time t 
 according to the time-constant 2p/g. Thus, 
 
 (4) 
 
366 ELECTROMAGNETIC THEORY. CH. IV. 
 
 Here the time-constant is twice that of the subsidence of the 
 total induction. Like the former formulae, these modified 
 ones, (3) and (4), have their representatives in the theory of a 
 telegraph circuit, in spite of the absence of magnetic con- 
 ductance. It is replaced by something that produces approxi- 
 mately the same result. 
 
 In a conductor possessing duplex conductivity, electric and 
 magnetic, their attenuative actions at the wave-front are 
 independently cumulative, or additive. The attenuation is 
 expressed by 
 
 : ... (5) 
 
 It is really the attenuative actions of a single conducting plate 
 that are additive. This applies separately to every successive 
 thin conducting layer through which the front of the wave 
 runs, with the result (5), where the time-factor is the product 
 of the two former time-factors of (3) and (4). 
 
 In the theory of coils and condensers, not only do we meet 
 with the time-constants L/R and S/K, the ratios of inductance 
 to resistance and of permittance to conductance, but also with 
 the double values. Their ultimate origin may be traced in the 
 theory of the effect of a thin conducting plate upon a wave. 
 
 The exponential time-factors concerned in (3) and (4), and 
 the more complex one in (5), also make their appearance in 
 connection with the disturbance in the body of a wave, though 
 in a less simple manner. This will be returned to. 
 
 The Simple Propagation of Waves in a Distortionless 
 Conducting Medium. 
 
 197. Coming now to the influence of conductivity on a 
 wave elsewhere than at its extreme front, where we have 
 recognised that the influence is simply attenuative, the easiest 
 way of treating the matter is not to pass from the known to 
 the unknown, but to reverse the process and pick out the 
 cases which theory indicates are most readily understandable. 
 This is to be done by a process of generalisation. The theory 
 of a conductor with duplex conductivity is, in a certain case, 
 far simpler than that of a real electric conductor. We have 
 already mentioned that the reflective actions of two plates, 
 one an electric, the other a magnetic conductor, are of oppo- 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 367 
 
 site natures. The first reverses the displacement, and the 
 second reverses the induction when throwing back a portion 
 of the wave. The joint action of the two plates when 
 coexistent and coincident, or the action of a single plate with 
 duplex conductance, results in a complete disappearance of the 
 reflected wave when the conductances are in proper ratio and 
 the plate is infinitely thin. We then have transmission with 
 attenuation but without reflection. This occurs, in a homo- 
 geneous medium, when k/c=g/p. Reflex action being abolished, 
 we are reduced to a kind of propagation of unique simplicity. 
 
 To see the full meaning of this, start from any initial distribu- 
 tions of induction and displacement in a non-conducting dielec- 
 tric. Imagine that we have obtained the full solution showing 
 the subsequent history of the disturbances. Now, if we introduce 
 only one kind of conductivity, say electric, we shall, with the 
 same initial state, have a profoundly different subsequent his- 
 tory. Again, with magnetic conductivity alone, we shall have 
 a course of events different from both the previous. But if we 
 add on magnetic conductivity to previously existent electric 
 conductivity, we shall partly counteract the distorting influ- 
 ence of the latter. This counteraction becomes complete when 
 the value of the magnetic conductivity is raised so high as to 
 produce equality of the time-constants of attenuation due to 
 the two conductivities separately. Further increase of the 
 magnetic conductivity will overdo the correction and bring on 
 distortion again, though of a different kind. 
 
 Similarly, the distortion due to magnetic conductivity alone 
 is diminished by introducing electric conductivity, and becomes 
 completely abolished when there is enough of the latter to 
 equalise the time-constants. Further increase brings on the 
 distortion again, which is now of the electric kind. 
 
 When the state of balance occurs, and the distortion is 
 wholly removed, the course of events following any initial state 
 is precisely the same as in a non-conducting medium, but with 
 a continuous attenuation expressed by equation (5) above spe- 
 cialised to suit the equality of the time-constants. That is, the 
 time-factor of attenuation is now ~*^ c . This removal of dis- 
 tortion applies to every kind of wave. 
 
 This distortionless state in conducting media furnishes a 
 sort of central basis for investigating the more recondite effects 
 
368 ELECTROMAGNETIC THEORY. CH. IV. 
 
 accompanying distortion. Nevertheless, its consideration would 
 possess only a theoretical value, on account of the non-existence 
 of the second kind of conductivity involved, were it not for the 
 remarkable practical imitation of the distortionless state of 
 things which is presented in the theory of telephone and other 
 circuits under certain circumstances. If we abolish the ficti- 
 tious magnetic conductivity throughout the medium traversed 
 by the waves, we should, to have distortionless transmission, 
 also abolish the electric conductivity. This is only to be 
 attained by using wires of no resistance to guide the waves 
 through a non-conducting medium. But they have resistance, 
 of greater or lesser importance according to circumstances. Of 
 what nature, then, is the distortion of waves produced by the 
 resistance of a wire along which they run? The answer is, 
 that it is approximately of the kind due to magnetic con- 
 ductivity in the medium generally. On the other hand, 
 the different kind of distortion due to electric ;onduc- 
 tivity in the medium generally remains in action, being 
 the effect of the leakage-conductance of the insulating 
 medium surrounding the wire, or the average effect of other 
 kinds of leakage at distinct and separate spots along the 
 circuit. Thus we obtain an approximate reproduction of the 
 theory of magnetic conductivity acting to neutralise the 
 distorting effect of electric conductivity. The time-constants 
 fj./g and c/k become L/R and S/K in a telegraph circuit, 
 L being inductance, R resistance, S permittance, and K 
 leakage-conductance. Their equalisation produces the distor- 
 tionless circuit, which may turn up again later on. In the 
 meantime I may remark that if the reader wishes to under- 
 stand these things, he must give up any ancient prejudices he 
 may be enamoured of about a " KR law " and the consequent 
 impossibility of telephoning when "KR" is over 10,000. 
 When pointing out, in 1887, the true nature of the telephonic 
 problem and the absurdity of the " KR law " applied thereto 
 generally, I predicted the possibility of telephoning with 
 " KR " several times as great. It has since been done. In 
 America, of course. A short time since, in noticing the KR = 
 32,000 reached by the New York-Chicago circuit, I further pre- 
 dicted that it would go up a lot more. It did very shortly after. 
 The record is now about 50,000 (Boston-Chicago) for practical 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 3G9 
 
 work, I believe. It means a good deal more for possible work. 
 But there is no need to stop at 50,000. That can be largely 
 exceeded in an enterprising country. 
 
 The Transformation by Conductance of an Elastic Wave to 
 a Wave of Diffusion. Generation of Tails. Distinct 
 Effects of Electric and Magnetic Conductance. 
 198. We are now prepared to somewhat understand the 
 nature of the changes suffered by electromagnetic waves in 
 transit through a conducting medium. It being the distortion 
 due to the conductance alone that is in question, we eliminate 
 that due to other causes by choosing plane waves for examina- 
 tion, since these do not suffer any distortion in a homogeneous 
 dielectric when it is non-conducting. Imagine, then, a simple 
 electromagnetic plane wave-sheet of small depth to be running 
 through a dielectric at the natural speed conditioned by its 
 inductivity and permittivity. At any stage of its progress, let 
 the medium become slightly electrically conducting all over, 
 not merely in advance of the wave but behind it as well, for a 
 reason that will presently appear. What happens to the wave 
 now that the fresh influence is in operation ? 
 
 A part of the answer we can give at once, by the pre- 
 vious. The wave-sheet will move on just as before, but will 
 attenuate as it goes, according to the time- factor e~ kt i 2c . Since 
 we suppose the conductivity to be slight, it follows that a great 
 distance may be traversed before there is notable attenuation. 
 We also know that the total induction remains constant. 
 The rest of it that is, what is not in the sheet at any moment 
 is therefore left behind. The rejecting process commences the 
 moment the conductivity is introduced, and continues to act 
 until the plane wave is attenuated to nothing. The rejected 
 portions travel backwards. But they are themselves subject 
 to the same laws as the main plane wave, and so get mixed up. 
 The result is that at time t after the introduction of the con- 
 ductivity, the whole region of disturbance extends over the 
 distance 2vt, half to the right and half to the left of the 
 initial position. At the advancing right end we have a strong 
 condensed disturbance, viz., the original wave attenuated, and 
 behind it a weak diffused one. We can therefore, without 
 
 BB 
 
370 ELECTROMAGNETIC THEORY. CH. IV. 
 
 misunderstanding refer to them as the head and the tail, with- 
 out any body to complicate matters. Now the nature of the 
 tail is quite different as regards the displacement and the 
 induction. It is therefore convenient to regard one of them 
 alone in the first place, and, of course, we select the induction, 
 on account of the simple property of persistence that it 
 possesses. We can distinguish three or four different stages in 
 its development. 
 
 The first stage is when the attenuation of the head is not 
 great, say, whilst the head decreases from 1 to - 75. Whilst 
 this occurs, the total induction in the tail rises from to 0'25. 
 The tail is long and thin, and tapers to a point at its extreme 
 end, or tip, at distance 2vt behind the head, and is thickest 
 where it joins on to the head. 
 
 The second stage roughly belongs to the period during which 
 the head further attenuates from 0'75 to 0'5 or 0'4. The total 
 induction in the tail then increases from 0"25 to 0-5 or 0'6. 
 During this stage we find that the tail, which has, of course, 
 greatly increased in length, does not go on increasing in thick- 
 ness at the place where it is developed, but stops increasing 
 and shows a maximum at or near that place. 
 
 The third stage occurs during the further attenuation of the 
 head to, say, 0*1, whilst the total induction in the tail increases to 
 0*9. The maximum thickness of the tail is now a long way from 
 the head, and at the end of the stage is nearer to the middle than 
 to the head. Of course, since the head itself is now so small, 
 the additions made to the tail must also become smaller. 
 
 The fourth stage is when the head practically disappears 
 and all the induction is in the tail. The maximum thickness 
 is now nearly in the middle on the right side, however and 
 the tail is nearly symmetrical with respect to its middle, where 
 there is a swelling, beyond which the tail tapers off both ways 
 to its two tips.* 
 
 The final state is the consummation of the previous, and is 
 one of perfect symmetry with respect to the middle of the tail, 
 which is situated exactly where the plane wave was when the 
 
 * As the division into distinct stages is somewhat arbitrary, this descrip- 
 tion of the transition from an elastic to a diffusion wave should be under- 
 stood to be only roughly approximate. It is made up, not from the 
 formula, but by a numerical process of mixture. 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 371 
 
 spreading began. The spreading now takes place according to 
 the pure diffusion law, as of heat by conduction. 
 
 Now as regards the displacement in the head and tail in the 
 different stages. In the first and second stages the displace- 
 ment is wholly negative in the tail, assuming it to be positive 
 in the head, where, it should be remembered, it attenuates in 
 the same manner as the induction which accompanies it. Thus, 
 when the head has fallen to 0'9, the total displacement in the 
 head and tail has fallen to (0'9) 2 or 0-81, so that the total nega- 
 tive displacement in the tail is of amount 0'09, which is not 
 much less than the coincident induction. And when the head 
 has attenuated to 0*8, and the total displacement to (0*8) 2 or 
 0-64, the negative displacement in the tail amounts to 0'16. 
 But, unlike the induction, the displacement increases in the 
 tail from the head up to not far from the tip, where, of course, 
 it falls to zero. There is no tip at the forward end. But as 
 the tail stretches out further to the left, and has fresh addi- 
 tions made to it on the right side, the decrease of the density 
 of displacement in passing towards the head continues, until 
 somewhere about the end of the second stage, it becomes zero 
 next the head. This node is approximately at the place where 
 the induction has its maximum. When the head has fallen to 
 0*4, we have the total displacement attenuated to 0*16, so that 
 the negative displacement in the tail amounts to 0'24. 
 
 In the third stage the displacement is negative from the tip 
 up to somewhere near and beyond the maximum of induction, 
 and increasingly positive in the remainder, up to the head. 
 That is, the region of positive displacement now extends itself 
 from the head a good way into the tail. At the same time the 
 place of maximum negative displacement moves forward. 
 
 During the fourth stage, the place of maximum negative 
 displacement shifts itself to nearly the middle of the region 
 between the tip and the node, beyond which the positive dis- 
 placement has a nearly similar distribution, with a maximum. 
 But this positive distribution is only three-parts formed, as the 
 bead is still of some importance. The fifth stage completes 
 the formation of the tail, with the displacement negative in 
 one half and positive in the other, and nearly symmetrical 
 with respect to the middle. Finally, we come to a state of 
 perfect symmetry, with one maximum of induction and two 
 
 BB2 
 
372 ELECTROMAGNETIC THEORY. CH. IV. 
 
 (or a maximum and a minimum) of displacement. This may 
 be readily understood by considering that in the expression for 
 the true electric current, &E + cE, the second term finally be- 
 comes a small fraction of the first, or the current is practically 
 the conduction-current only. Then B, and therefore the dis- 
 placement, is proportional to the space-variation of the induc- 
 tion. Owing to the tail being made up of a complicated mix- 
 ture of infinitesimal electromagnetic waves going both ways, 
 we lose sight of the fundamental property of elastic wave-pro- 
 pagation in a dielectric, the resultant effect being propagation 
 by diffusion now, or very nearly so. The approximation to this 
 result is closest in the middle of the tail. At the tips, on the 
 other hand, we still have elastic waves, but, of course, of 
 insensible strength. 
 
 Now, suppose that the initial state is one of induction only, 
 though still in a plane sheet. It may be regarded as the coin- 
 cidence of two plane electromagnetic sheets of half the strength, 
 with similar inductions and opposite displacements. The sub 
 sequent history may, therefore, be deduced from the preceding. 
 The initial induction splits into two halves, of which one moves 
 to the right at speed v, and the other to the left. The history 
 of the first wave as distorted by the conductance has been 
 given. That of the second wave is the same, if we allow for 
 the changed direction of motion and sense of the displacement. 
 So we have only to superpose the two systems to show how an 
 initial distribution of induction in a plane sheet splits and 
 spreads, and the accompanying electric displacement. We 
 have two equal heads, separating from one another at speed 2v, 
 whilst the two tails unite to make a stouter kind of tail 
 (referring to the induction) joining the two heads. This tail 
 is always thickest in the middle. In fact, the distributions of 
 induction and displacement are symmetrical with respect to 
 the initial position from the first moment. In the final state 
 that is tended to, when the tails have vanished, the induction 
 is distributed in the same way as in the former case, in spite 
 of the remarkable difference in the initial phenomena. 
 
 If, again, the initial state is one of displacement alone in a 
 plane sheet, this generates two oppositely-travelling electro- 
 magnetic waves in which the displacements are similar and the 
 inductions are opposite. The result is therefore to be got by 
 
THEORY OP PLANE ELECT ROMAGNET1C WAVES. 53 
 
 combining the previous solution for a wave of positive induc- 
 tion and displacement moving to the right, with a similar wave 
 of negative induction and positive displacement moving to the 
 left. 
 
 Whatever be the magnitude of the conductivity, if finite, the 
 same phenomena occur during the conversion of elastic waves 
 to diffusion waves, and they may be represented by the same dia- 
 grams, if suitably altered in the relative scale of ordinates and 
 abscissae. But by increasing the conductivity from the small 
 value which makes the above-described process take place over 
 a considerable interval of time, we make it occur in a small 
 interval only; and when we come to what are usually con- 
 sidered good conductors, viz., metals, then the interval of time 
 is so small that we may say practically that the heads vanish 
 almost at 6nce, and before the propagation has proceeded any 
 notable distance. The rest of the story is the spreading by 
 diffusion or mixture. Thus we have an important practical 
 distinction between the very good and the very bad conductor, 
 as regards the manner of propagation of induction. In the 
 former the electric displacement is of no account hardly, 
 except perhaps for very short waves, and the practical theory 
 is the theory of diffusion. At the other extreme are perfect 
 non-conductors, in which the propagation is entirely by elastic 
 waves. Between the two we have, in bad conductors, a mix- 
 ture of the two kinds, though with a continuous transition 
 from one kind to the other. The mathematical treatment of 
 elastic waves is the easiest. The next in order of difficulty are 
 the waves of pure diffusion, by the ordinary Fourier mathe- 
 matics. The most troublesome are the intermediate forms of 
 changing type, and the full analytical results are best got by 
 a generalised calculus. But the general nature of the results 
 may be obtained approximately by easy numerical calculations, 
 with diagrammatical assistance. This is explainable without 
 difficulty, and may be entered upon later on, when we come to 
 detailed problems. 
 
 la the above the conductivity has been electric only, and 
 we have seen that there is a great difference between its effects 
 on the electric and on the magnetic flux, which arises from the 
 positive reflection and persistence of the latter, and the nega- 
 tive reflection and subsidence of the former flux. The corres- 
 
374 ELECTROMAGNETIC THEORY. CH. IV. 
 
 ponding effects in a magnetic conductor may be readily 
 deduced by the proper transformations, and exhibit analogous 
 differences. Thus, starting with a pure plane wave sheet 
 moving to the right, it is now the displacement that is con- 
 served and divided between the head and tail, being initially all 
 in the head, and finally all in the tail. On the other hand, the 
 induction is negative in the tail at first and until the attenua- 
 tion of the head is considerable. After this, the region of posi- 
 tive induction extends from the head into the tail. As time 
 goes on, and the head disappears, we have negative induction 
 in one half and positive in the other (forward) half, whilst the 
 displacement is positive all along it, with its maximum nearly 
 in the middle. In the final state of diffusion we have symmetry 
 with respect to the initial position of the wave, and the induc- 
 tion is proportional to the space-variation of the displacement. 
 The second term of the magnetic current ^H + juH is then a 
 small fraction of the first term. 
 
 Now, in the medium with duplex conductivity we must take 
 the distortionless condition for the standard state. When this 
 obtains there is no tailing, although the head (displacement and 
 induction together) attenuates according to the time-factor c~^/ c . 
 But if tho electric conductivity be in excess (that is, greater than 
 is required to make k/c=g/fjL) there is tailing of the kind de- 
 scribed as due to electric conductivity, with the induction posi- 
 tive and the displacement negative in the tail at first. And if 
 the magnetic conductivity is in excess, the tailing is of the 
 other kind, with positive displacement and negative induction 
 at first. But of course the details are not the same, because 
 both the fluxes now attenuate to zero in time, in the tail as well 
 as in the head. 
 
 Application to Waves along Straight Wires. 
 
 199. We must now endeavour to give a general idea of 
 the tailing of waves when they run along conducting wires, in 
 co-ordination with the previous. In the first place, observe 
 that although the lines of electric and magnetic force in a pure 
 electromagnetic sheet in ether must be always perpendicular to 
 one another, yet they need not be straight lines, except in their 
 elementary parts. On the contrary, we may have an infinite 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 375 
 
 variety of distributions of the electric and magnetic fluxes in 
 curved lines in sheets which shall behave as electromagnetic 
 waves. Keeping to plane sheets, we may take any distribution 
 of displacement in a sheet that we please, and make it an 
 electromagnetic wave by introducing the appropriate distribu- 
 tion of induction, also in the sheet. If left to itself it will 
 move through the ether one way or the other at speed v, 
 according to the directions of the fluxes. It should be noted, 
 however, that the fluxes should have circuital distributions. 
 For if not, and there is electrification or its magnetic analogue, 
 they, too, should be moved through the ether so as to exactly 
 keep up with the wave. If the electrification does not move 
 thus, we have a changed state of things. But there is a way 
 out of this difficulty. Let the displacement, when it is discon- 
 tinuous, terminate upon infinitely-conducting lines, or surfaces 
 of cylinders, according to circumstances. The interference 
 produced by holding back the electrification will then be done 
 away with. It is sufficient for explicitness to take a single 
 definite case. 
 
 Let the displacement in a plane electromagnetic sheet ter- 
 minate perpendicularly upon a pair of perfectly-conducting 
 cylindrical tubes placed parallel to one another. One is for the 
 positive and the other for the negative electrification. The 
 induction will then go between and round the tubes. One 
 tube may enclose the other, but, preferably, we shall suppose 
 this is not the case, so that our arrangement resembles a pair 
 of parallel wires. Now the wave will run through the ether 
 in the normal manner, without distortion or change of type, 
 and will carry the electrification (on the conductors) along 
 with it. 
 
 Similarly if the tubes be magnetic conductors, if the induc- 
 tion terminates perpendicularly upon them, whilst the displace- 
 ment goes between and round them. So far, then, the theory 
 of the propagation of the wave is unaltered. 
 
 But this property may be greatly extended. The medium 
 outside the tubes through which the wave is moving may be 
 made electrically conducting. The theory is then identical 
 with that of 198, as regards the distortion produced and the 
 gradual destruction of the wave, ending in the process of diffu- 
 sion. Thid will be true whether the tubes are perfect electric 
 
376 ELECTROMAGNETIC THEORY. CH. IV. 
 
 or magnetic conductors, provided we have the displacement 
 terminating upon them in the first case and the induction in 
 the second. 
 
 Again, if the medium is magnetically conductive, the theory 
 still holds good, with the same reservations as regards termi- 
 nating the fluxes. But now the distortion is of the other kind, 
 with persistence of the displacement and subsidence of the 
 induction. 
 
 Finally, the theory is still true when the medium is a duplex 
 conductor, whilst the tubes are either perfect electric or mag- 
 netic conductors, according to choice. The distortion is now 
 of the electric or of the magnetic kind, according as kfc is 
 greater or less than g/fj.. 
 
 But not one of these cases is quite what we want to repre- 
 sent propagation along real wires. The nearest approach is 
 the case of finite electric conductivity in the medium combined 
 with perfect electric conductance of the tubes. If we make 
 their conductance be imperfect, we then come close to the real 
 problem. Now when we do this, the theory, when done pre- 
 cisely, becomes excessively difficult, for two reasons. First, 
 the waves in the ether outside the tubes are no longer plane ; 
 and next, the penetration of the disturbances into the sub- 
 stance of the tubes in time by means of cylindrical waves has 
 to be allowed for. The latter is, of course, more necessary 
 when solid wires are in question. A practical working theory 
 is seemingly impossible of attainment by strictly adhering to the 
 actual conditions. But one is possible by taking advantage of 
 the fact that the waves in the ether are very nearly plane under 
 ordinary circumstances. Of this we may assure ourselves 
 by considering that the tangential component of the electric 
 force at the surface of a wire (upon which the penetration 
 depends) is usually a very small fraction of the normal com- 
 ponent of the same outside it. The practical course, then, is 
 to treat the waves in the dielectric as if they were quite plane. 
 This does not prevent our allowing for the distorting effect of 
 the resistance of the wires. It has the effect of making our 
 solutions approximate instead of complete. But the important 
 thing is to have a theory that, whilst sufficiently accurate, is 
 practically workable, and harmonises with more rudimentary 
 theories. This is precisely v;hat we do get, as done by me in 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 377 
 
 1886. The theory is brought to such a form that we may 
 employ it in several ways, according as circumstances allow us 
 to ignore this or that influence. We may, for example, treat 
 the wires as mere resistances (constant), and this is quite suffi- 
 cient in a variety of applications. Or, using the same equations 
 with more general meanings attached to the symbols, we may 
 find the effects due to the imperfect penetration of the mag- 
 netic induction and electric current into the wires when sub- 
 jected to varying forces at their boundaries, either simple 
 harmonically or otherwise. Furthermore, the theory is in such 
 a form that it admits readily and without change of the intro- 
 duction of terminal or intermediate conditions of the kind that 
 occur in practice, whose effects are brought in according to the 
 usual equations of voltage and current in arrangements of 
 apparatus. 
 
 What we are immediately concerned with here, however, is 
 the connection between this theory and the general theory of 
 waves in a medium of duplex conductivity. When the tubes 
 are so thin that penetration is practically instantaneous (for 
 the wave-length concerned), a constant ratio is fixed (assisted 
 by Ohm's law) between the intensities of magnetic and of tan- 
 gential electric force at the boundary of the tube, where it 
 meets the dielectric. And when we incorporate this result in 
 the second circuital law applied to any section of the circuit 
 formed by the parallel tubes, we find that the result is to turn 
 it to the form expressing the existence of magnetic conductivity 
 in the outer medium, without resistance in the wires. That is, 
 the resistance of the wires has the same effect in distorting 
 and dissipating the waves outside as the fictitious magnetic 
 conductivity in the medium generally. This is true in the 
 first and most important approximation to the complete theory. 
 It is a point that is not altogether easy to understand, because 
 the magnetic conductivity is fictitious, whereas the resistance 
 is real. This, however, may be noted, that the theory of 
 propagation of plane waves in a medium of duplex conduc- 
 tivity bounded by perfect conductors for slipping purposes, 
 professes to be a precise theory, whereas the other, although 
 concerning the same problem in most respects, professes to be 
 only approximate so far as the influence of the wires is con- 
 cerned. 
 
378 ELECTROMAGNETIC THEORY. CH. IV. 
 
 Transformation of Variables from Electric and Magnetic 
 Force to Voltage and Gaussage. 
 
 200. In the consideration of the transmission of waves 
 along wires, especially, of course, in practical calculations, it is 
 more convenient to employ the line-integrals of the electric and 
 magnetic forces as variables rather than these forces them- 
 selves. This transformation of variables involves other neces- 
 sary transformations, and the connection between the old 
 quantities, suitable for waves in general, and the new, should 
 be thoroughly understood, if something more than a super- 
 ficial knowledge of the subject be desired. 
 
 Thus, commencing with a circuit consisting of a pair of 
 parallel straight conducting tubes, of no resistance in the first 
 place, take for reference any plane which crosses the tubes per- 
 pendicularly. It is the plane of the wave at the place, and the 
 lines of electric and magnetic force lie in it, the former starting 
 from one (the positive) and ending upon the other (the negative) 
 tube, the latter passing between and round them. These lines 
 always cross one another perpendicularly, but their distribution 
 in the plane may be varied by changing the size and form of 
 section of the tubes by the plane and the distance of separation. 
 Now there is no axial magnetic induction that is, induction 
 parallel to the tubes. Nor is there any tangential component 
 of electric force on the surface of the tubes. It follows, by the 
 second circuital law, that the line-integral of the electric force 
 from one tube to the other is the same by any path in the 
 reference plane. It is not the same by any path with the 
 same terminations if we depart from the reference plane, but 
 that is not yet in question. Call this constant line-integral V. 
 It is the transverse voltage. In another form, we may say 
 that the circuitation of the electric force in the reference plane 
 is zero. 
 
 Next consider the magnetic force. Its circuitation is also 
 zero in the reference plane, provided the one tube or the other 
 is not embraced, because there is no axial electric current in 
 the dielectric. But when a tube is embraced the circuitation 
 is finite. Call this quantity C. It is " the current " in the 
 tubes, or, at any rate, is the measure thereof, and is positive 
 for one tube and negative for the other. These two quantities, 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 379 
 
 V and C, the transverse voltage and the current, are the new 
 variables. They are as definite as E and H themselves in a 
 given part of the reference plane under the circumstances. 
 
 Next, as regards the fluxes, displacement and induction. Intro- 
 duce a second reference plane parallel to the first and at unit 
 distance in advance of it along the tubes. Anywhere between 
 the planes we have D = cE, where E and D vary according to 
 position. Now, as we substitute the complete voltage for E, 
 so we should substitute the complete displacement for D. The 
 complete displacement is the whole amount leaving the posi- 
 tive and ending upon the negative tube, between the reference 
 planes. That is, it is the " charge " per unit length of the 
 tubes, say Q. As D is a constant multiple of E, so is Q a con- 
 stant multiple of V. Say Q = SV. This S is the permittance 
 of the dielectric, per unit length axially. It is proportional to 
 c, the permittivity, but of course involves the geometrical data 
 (brought in by the tubes) as well. 
 
 Similarly, we have B = /xH everywhere between the reference 
 planes, H and B varying from place to place. But if we sub- 
 stitute the complete circuitation of H, we should also substi- 
 tute the complete integral of B. This means the total flux of 
 induction passing between and round the tubes, between the 
 reference planes. Call it P. It is the magnetic momentum 
 per unit length axially. As B is a constant multiple of H, so is 
 P a constant multiple of C. Thus P = LC, where L is the 
 inductance per unit length axially. This L varies as the induc- 
 tivity /*, and involves the geometrical data. 
 
 In virtue of the relation /*cv 2 = 1, the inductance and permit- 
 tance are reciprocally related. Thus, LSv 2 = 1. Since v is con- 
 stant, it might appear that the inductance was merely the 
 reciprocal of the permittance, or the elastance, with a constant 
 multiplier, only to be changed when the dielectric is changed. 
 But there is much more in it than this. Different physical 
 ideas and effects are conditioned by inductance and permit- 
 tance. Inductance and inductivity involve inertia, whilst 
 permittance and permittivity involve compliancy or elastic 
 yielding. 
 
 Observe that P and Q are analogous, being the total mag- 
 netic and electric fluxes, whilst V and C are also analogous, the 
 voltage and gaussage respectively. So the ratios L and S axe 
 
380 ELECTROMAGNETIC THEORY. CH. IV. 
 
 also strictly analogous, in spite of their quantitative recipro- 
 cality. They are both made up similarly to conductance, only 
 in the case of permittance it is reckoned across the dielectric 
 from tube to tube, and in the case of inductance round and be- 
 tween the tubes that is, the magnetic circuit is closed, and 
 the electric circuit unclosed. There are other ways of setting 
 out the connections, but the above way brings out the analogies 
 between the electric and magnetic sides in a complete manner. 
 The quantity C may be quite differently regarded when the 
 magnetic field penetrates into the substance of a tube (then to 
 be of finite conductance), viz., as the total flux of conduction 
 current, or sometimes of displacement current as well, in a 
 tube. But the above method is independent of the penetration, 
 holding good whether there is penetration or not, saving small 
 corrections due to the waves not being quite plane. The mag- 
 netic view of C, as the gaussage, is also the nearest to expe- 
 rimental electrical knowledge. The other view is darker, 
 because we cannot really go inside metals and observe what 
 is going on there, or the forces in action. They must be 
 inferential in a greater degree than the external actions. 
 
 The electric energy per unit volume being JED or JrE 2 , 
 when this is summed up throughout the whole slice of the 
 medium contained between the reference planes at unit dis- 
 tance apart, we obtain the amount JVQ or JSV 2 , which is 
 therefore the electric energy per unit of length axially. 
 
 Similarly, the magnetic energy density JHB or J/*H 2 , when 
 summed up throughout the slice, amounts to JCP or JLC 2 . 
 The reader should, in all these transformations, compare the 
 transformed expressions with the original, and note the proper 
 correspondences. It is all in rational units, of course, to avoid 
 that unmitigated nuisance, the 4?r factor of the present B.A. 
 units. 
 
 The density of the flux of energy, which is the vector pro- 
 duct of E and H in general, is of amount simply EH when 
 the forces are perpendicular, as at present. When this is 
 summed up over the reference plane, the result is VC, the pro- 
 duct of the voltage and current. The flux of energy is parallel 
 to the guiding tubes, and EH is the amount for a tube in 
 the dielectric of unit section, whilst VC is the total flux of 
 energy. 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 381 
 
 The relation E = p>H, or, which means the same, H = cvE, 
 which obtains in a pure electromagnetic wave, becomes converted 
 to V = LvC, or C = SvV. We see that Lv is of the dimensions of 
 electric resistance, and Sv of electric conductance. Similarly, 
 because V and C are the line-integrals of E and H, we see that 
 fjiv is of the dimensions of resistance, and cv of conductance. 
 The activity product VC is what engineers have a good deal to 
 do with, now that " electrical energy," or energy which has been 
 conveyed by electromagnetic means, is a marketable commodity. 
 However mysterious energy (and its flux) may be in some of its 
 theoretical aspects, there must be something in it, because it is 
 convertible into dollars, the ultimate official measure of value.* 
 
 Transformation of the Circuital Equations to the Forms 
 involving Voltage and Gaussage. 
 
 201. Now, still under the limitation that the guiding tubes 
 have no resistance, and that the dielectric has no conductance, 
 consider the special forms assumed by the circuital laws. These 
 are, in general, 
 
 curlH=cE, .,,,.. (1) 
 
 (2) 
 
 Now, in applying these to our plane waves, we may observe 
 at the beginning that E and H and their time- variations are in 
 the reference plane, and have no axial components. So the 
 only variations concerned in the operator " curl " are such as 
 occur axially, or parallel to the guides, f Let x be distance 
 along them, then the circuital laws become 
 
 -^? = cE, (3) 
 
 dx v ' 
 
 ~ = ^ (4) 
 
 dx x ' 
 
 * See the Presidential Address to the Institution of Electrical Engineers, 
 January 26, 1893. 
 
 t That is, curl reduces from Vv, where y is i\?i +JV^ + kVs in general, to 
 % r iVi simply. It is now more convenient to use the tensors E and H, as 
 in (3) and (4), ignoring their vectoiial relations. 
 
382 ELECTRO MAGNETIC THEORY. CH. IV. 
 
 which apply anywhere in the reference plane. Transforming 
 to V and C, we obtain 
 
 (5) 
 
 By (3) and (4) we see that the space-variation of H is the 
 electric-current density, and the space- variation of E the mag- 
 netic-current density in the dielectric. By (5) and (6) we 
 express the same truths for the totals concerned. 
 
 We may see the meaning of (5) thus. Draw a closed line in 
 the reference plane on and embracing the positive guiding tube. 
 The circuitation of H there is the total surface current C. Let 
 the closed line be shifted to the next reference plane at unit 
 distance forward. Then dC/dx is the amount by which C 
 decreases during the shift, and (5) tells us that it is equiva- 
 lently represented by the time-rate of increase of the charge on 
 the tube between the reference planes, or the rate of increase 
 of the total displacement outward. This process may be applied 
 to any closed line in the reference plane, provided it embraces 
 the tube. When it is shifted bodily forward through unit 
 distance to the next reference plane, the amount by which the 
 circuitation of H, that is, C, decreases from the first to the 
 second position measures the displacement current through 
 the strip of the cylinder swept out by the closed line. This 
 is the time-rate of increase of the charge per unit length of 
 the tubes, and is the total transverse current from one tube 
 to the other. 
 
 The meaning of (6) requires a somewhat different, though 
 analogous, elucidation. Join the tubes by any line in the first 
 reference plane. The line-integral of E along it is V, the 
 transverse voltage. Shift the line bodily along the tubes to 
 the next reference plane. The transverse voltage becomes 
 V + dV/dx, still reckoned from the positive tube to the 
 negative. But if we join the two starting points on the 
 positive tube together, and likewise the ending points on the 
 negative tube, we make a complete circuit, having two trans- 
 verse sides and two axial sides. Now reckon up the voltage in 
 this circuit. The axial portions contribute nothing, because 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 383 
 
 the electric force is perpendicular to them. The voltage 
 required is, therefore, simply the difference in the values 
 of the transverse voltage in the other two sides, or dNjdx. 
 By the second circuital law, it is also measured by the rate 
 of decrease of induction through the circuit, that is, by -LC. 
 Whence follows equation (6). 
 
 The reader who is acquainted with the (at present) more 
 "classical" method of treating the electromagnetic field in 
 terms of the vector and scalar potentials cannot fail to be 
 impressed by the difference of procedure and of ideas involved. 
 In the present method we are, from first to last, in contact with 
 those quantities which are believed to have physical signi- 
 ficance (instead of with mathematical functions of an essentially 
 indeterminate nature), and also with the laws connecting them 
 in their simplest form. Notice that V is not the difference of 
 potential in general. It sometimes degenerates to difference of 
 potential, viz., in a perfectly steady state. But when the state 
 changes, we cannot express matters in terms of an electric 
 potential. 
 
 Now, still keeping the guiding tubes perfectly conducting, 
 let the medium in which they are immersed be slightly con- 
 ducting. The electric-current density, which was cE before, now 
 becomes &E + cE, where the additional &E is the conduction- 
 current density. Along with this there is waste of energy at 
 the rate E 2 per unit volume. This waste of energy may also 
 be regarded as a storage of energy, viz., as heat in the medium. 
 But as it is not recoverable by the same means (reversed) which 
 stored it, it is virtually wasted, and we have no further concern 
 with it. 
 
 We have next to consider the total waste in the slice of the 
 medium between the two reference planes, in terms of the trans- 
 verse voltage. It sums up to KV 2 , where K is the transverse 
 conductance of the medium per unit length axially. This is 
 proportional to the conductivity k, and involves geometrical 
 data in the same way as the transverse permittance, as may be 
 readily seen without symbolical proof, on considering that the 
 conduction-currrent lines and the displacement lines are simi- 
 larly distributed, whilst both are controlled by the transverse 
 voltage. In fact, the conduction-current density &E sums up 
 
384 ELECTROMAGNETIC THEORY. CH. IV. 
 
 to KV, the complete transverse conduction-current, in the same 
 way as the displacement density cE sums up to the total trans- 
 verse displacement SV, in terms of the permittance and 
 voltage. 
 
 The first circuital law (1) becomes changed to 
 
 (7) 
 
 by the addition of the conduction current. And its modified 
 form (3) for our plane waves becomes 
 
 (8) 
 
 whilst in terms of the new variables, voltage and gaussage, we 
 have this extended form of the equation (5), 
 
 (9) 
 dx 
 
 Next, let the medium become magnetically conductive as 
 well, no other change being made. The conduction-current 
 density is ^H, and its distribution resembles that of the induc- 
 tion itself, whilst its amount is controlled by the quantity G, 
 the gaussage. The total magnetic-conduction current may 
 therefore, be represented by KG, where E, is the magnetic 
 conductance, which varies as the magnetic conductivity g, and 
 involves the geometrical data in the same manner as the 
 inductance does. 
 
 Similarly, the rate of waste of energy due to the magnetic 
 conductivity is 0rH 2 per unit volume, and the total rate of 
 waste in the slice of the medium between the two reference 
 planes amounts to RC 2 correspondingly. 
 
 Finally, the effect of the magnetic conductivity is to turn 
 the second circuital law from the elementary form (2) to 
 
 -curlE = #H + /iH, .... (10) 
 
 and, correspondingly, the special form (4) for our plane waves 
 becomes turned to 
 
 ..... (11) 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 385 
 
 This, in terms of V and C, is equivalent to 
 
 -4^ = RC + LC, ...... (12) 
 
 dx 
 
 which is the proper companion to the equation (9) expressing 
 the first circuital law. These equations (9) and (12) are the 
 practical working equations. 
 
 We have already remarked that the structure of the electric 
 permittance and conductance are similar. From this it follows 
 that the time-constant c/k is identically represented by S/K. 
 Similarly, the time- constant njg is identical with L/R. 
 
 The density of the flux of energy is, with the two conduc- 
 tivities, still represented by the vector product of E and H, and 
 therefore the total flux is still the activity product VC. To 
 corroborate this statement, and at the same time show the 
 dynamical consistency of the system, consider that if the quan- 
 tity VC is really the flux of energy across a reference plane, the 
 excess of its value at one plane over that at a second plane 
 further on at the same moment represents the rate of storage 
 of energy between the planes. Therefore, the rate of decrease 
 of VC with x is the rate of storage between two reference planes 
 at unit distance apart. Now, 
 
 _i(VC)=-vf-cf ..... (13) 
 dx dx dx 
 
 On the right side use the circuital equations (9) and (12), and 
 it becomes . . 
 
 V(KV + SV) + C(RC + LC), . . . (14) 
 
 or, which is the same, 
 
 . . (15) 
 
 By the previous, the first term is the total electric waste, the 
 second is the total magnetic waste, the third is the increase of 
 total electric energy, and the fourth is the increase of total 
 magnetic energy, per unit of time, between the two reference 
 planes. This proves our proposition, with a reservation to be 
 understood concerning the circuital indeterminateness of the 
 flux of energy. There is, therefore, no indistinctness anywhere, 
 nor inconsistency. We have now to show in what manner 
 the above is affected by the resistance, &c., of the guides. 
 
 oo 
 
386 ELECTROMAGNETIC THEORY. CH. IV. 
 
 The Second Circuital Equation for Wires in Terms of V and 
 C when Penetration is Instantaneous. 
 
 202. Let the parallel conducting tubes be of finite resist- 
 ance. As a result, the external disturbance penetrates into 
 them, and a waste of energy follows. As a further result, the 
 external disturbance becomes modified. This occurs in a two- 
 fold manner. By cumulative action, the nature of a wave sent 
 along the leads may be profoundly modified as it progresses. 
 Besides this, there is what we may term a local modification, 
 whereby the wave at any place is no longer strictly a plane 
 wave. It is merely with this effect that we are now concerned. 
 The departure from planarity requires a considerably modified 
 and much more difficult theory, no longer expressible in terms 
 of V and C simply, in order to take it into account. But in 
 the construction of a practical theory, we take advantage of the 
 fact that the departure from planarity is slight, to which we 
 have already referred towards the end of 199. A line of elec- 
 tric force does not now start quite perpendicularly from the 
 positive lead and end similarly on the negative lead. It has a 
 slight inclination to the perpendicular, and therefore curves out 
 of the reference plane to a small extent. In the dielectric 
 itself, this peculiarity is of little importance. But the slight 
 amount of tangentiality of the electric force at the surface of 
 the leads, which is conditioned thereby, is of controlling import- 
 ance as regards the leads themselves, and eventually through 
 them, to the waves in the dielectric, by attenuating and altering 
 the shape of waves (considered axially), as just referred to. We 
 should now consider what form is assumed by the second cir- 
 cuital law in terms of V and C, when we admit that there is 
 tangential electric force on the surface of the leads, but on the 
 assumption that the minor effects in the dielectric itself, which 
 are associated with the presence of the tangential force, are of 
 insensible influence. 
 
 We may still regard V as the transverse voltage in the 
 reference plane. Whether we go straight across from the 
 positive to the negative lead, keeping in the reference plane, 
 or leave it slightly in order to precisely follow a line of force in 
 estimating the voltage, is of no moment, because the results 
 differ so slightly. The quantity C is subject to a similar slight 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 387 
 
 difference, according to the way it is reckoned. We may, per- 
 haps, most conveniently reckon it as the circuitation of the 
 magnetic force round either lead upon its surface and in the 
 reference plane, because this way gives the exact value of the 
 conduction current in the lead. But the circuitation in other 
 paths in the reference plane will not give precisely the same 
 value now that the waves are not quite plane. This difference 
 we also ignore in the practical theory. In truly plane waves 
 the electric current in the dielectric is wholly transverse. There 
 is now really an axial component as well, but being a minute 
 fraction of the transverse current, it is ignored. In short, we 
 have to make believe that the waves are planar when consider- 
 ing their propagation through the dielectric, whilst at the same 
 time we take into account the departure from planarity in 
 considering the influence of the leads and what occurs in them. 
 It is unfortunate to have to refer to small corrections, as it 
 confuses the statement of the vitally important matters. Let 
 us, then, set them aside now. 
 
 Construct the second circuital equation in the manner fol- 
 lowed in 201, in elucidating the meaning of equation (6). 
 Consider a rectangle consisting of two transverse sides in 
 reference planes at unit distance apart, beginning upon the 
 positive and ending upon the negative lead, and of two axial 
 sides of unit length upon the leads themselves. Reckon up the 
 voltage in this rectangle. The transverse sides give V and 
 V + dV/dx. The axial sides give E x and E 2 say, if Ej is the 
 tangential component in the direction of increasing x of the 
 electric force at the boundary of the positive tube, and E 2 the 
 same on the negative tube, but reckoned the other way. The 
 complete voltage is then 
 
 or + 1 + 2 . 
 
 dx 
 
 It is also equal to - LC, as in 101. (There is no magnetic 
 
 conductance of the dielectric now.) So 
 
 .... (16) 
 dx 
 
 is the form assumed by the second circuital law. 
 
 cc2 
 
388 ELECTROMAGNETIC THEORY. CR. IV. 
 
 Observe that we have made no specification of the nature of 
 the leads, so that the equation possesses a high degree of 
 generality. As regards the first circuital law, that is unchanged, 
 being expressed by (5) when the dielectric is non-conductive, 
 and by (9) when conductive, since the minor changes alluded 
 to as regards V and C are not significant. 
 
 Being general, the equation (16) needs to be specialised 
 before it can be worked. If our electromagnetic variables are 
 to continue to be V and C, we require to express E : and E 2 as 
 functions of C. Fortunately this can be done, sometimes very 
 simply, and at other times in a more complicated way. To 
 take the simplest case, let the leads be tubes (or sheets) of so 
 small depth that penetration is practically instantaneous as 
 waves pass along them. Then E x is not only the boundary 
 tangential electric force, but is also the axial electric force 
 throughout the substance of the positive tube. Similarly as 
 regards E 2 for the negative tube. It is true that, in virtue of 
 the transverse electric current from tube to tube, there is also 
 transverse current in the tubes, and, therefore, transverse 
 electric force, but this is to be ignored, because it is a small 
 fraction of the axial. The current density in the positive tube 
 is therefore axial, of strength ^E x , if ^ is the conductivity 
 of the material; and the total current in the tube is KjEj, 
 where Kj is the axial conductance per unit length, or, which is 
 the same, Ej/R^ if R x is the resistance per unit length. But 
 this quantity is also the previously investigated quantity C, 
 the circuitation of the magnetic force on the boundary of the 
 tube. So we have the elementary relations 
 
 E^R^, E 2 = R 2 C 2 , . . . (17) 
 
 which are, be it observed, essentially the connections of the 
 electric and magnetic forces at the boundaries, though brought 
 to a particularly simple form by the instantaneous penetration. 
 The second circuital equation (16) therefore takes the form 
 
 (18) 
 ax 
 
 Or, finally, if R is the resistance per unit length of the two 
 leads, 
 
 -^ = RC + LC, ...... (19) 
 
 dx 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 389 
 
 which is the practical equation for most purposes. Observe, 
 comparing it with (12), that there is an identity of form, but 
 with a changed meaning of the symbol R. In the exactly stated 
 problem of plane waves running through a medium of duplex 
 conductivity bounded by perfect conductors, the quantity R 
 is the external magnetic conductance per unit length axially. 
 In the present approximately stated problem of very nearly 
 plane waves running through an electrically conducting medium 
 bounded by resisting wires, the quantity R is the resistance of 
 the wires, per unit length axially. What we have to do, there- 
 fore, in order to turn the real problem into one relating to 
 strictly plane waves admitting of rigorous treatment, is to 
 abolish the resistance of the leads and substitute equivalent 
 magnetic conductance in the dielectric medium outside. The 
 two materially different properties are nearly equivalent in 
 their effects. Equations (13), (14), (15), still hold good, only 
 RC 2 is now to be the rate of waste in the leads, instead of in 
 the medium generally due to the (now suppressed) magnetic 
 conductance. The flux of energy is now not quite parallel to the 
 leads everywhere, but has a slight slant towards them. But 
 the usual formula gives the waste correctly. The product 
 EjHj of the tangential electric force E x and the magnetic force, 
 say H 1 , at the boundary of the positive lead, is the rate of 
 supply of energy to the lead per unit surface. Therefore, by 
 circuitation, the product EjC is the rate of supply per unit 
 length axially. This is the same as the previous RjC 2 . 
 
 The equivalence of effect of magnetic conductance externally 
 and of electric resistance in the leads, is undoubtedly a some- 
 what mysterious matter, principally because it is hard to see 
 (apart from the mathematics) why it should be so. As regards 
 the above reasoning, however, it is essentially simple, and is a 
 direct application of fundamental electrical principles. It may, 
 therefore, cause no misgivings, except the doubt that may 
 present itself whether the ignored small effects in the dielectric 
 due to departure from planarity of the waves are really ignor- 
 able. As a matter of fact, they are not always of insensible 
 effect, and plenty of problems can be made up and worked out 
 concerning tubes and wires which do not admit of the compa- 
 ratively simple treatment permissible when V and C are the 
 variables, and the results are exceedingly curious and interest- 
 
390 ELECTROMAGNETIC THEORY. CH. IV. 
 
 ing, though quite unlike those at present in question. But 
 these are not practical problems, and have little bearing upon 
 the question of the free propagation of waves along long parallel 
 straight wires. They do not start as full grown plane waves 
 from the source of disturbance, but they very soon fit them- 
 selves on to the wires properly, and then follow the laws of 
 plane waves pretty closely. 
 
 Although thin tubes for leads were specially mentioned in 
 connection with equation (19), yet any sort of leads will do 
 provided the penetration be sufficiently rapid to be instan- 
 taneous " within the meaning of the Act." It is obviously true 
 for steady currents, when the inertial term disappears and B, is 
 the steady resistance. And if the variations of current be not 
 sufficiently rapid to cause a sensible departure from uniformity 
 of distribution of current in the conducting wires, then, of 
 course, (19) may still be safely used. At the same time, it 
 should be mentioned that the inductance L should, under the 
 circumstances, be increased by a (usually) small amount due 
 to induction in the conductors themselves, as will be presently 
 noted more closely. Furthermore, remember that no allow- 
 ance has been made for the influence of parallel conductors, 
 should there be any. The earth does not count if the leads be 
 alike and equidistant from the ground, as its influence can be 
 embodied in the values of L and S, the inductance and per- 
 mittance. 
 
 The Second Circuital Equation when Penetration is Not 
 Instantaneous. Resistance Operators, and their Definite 
 Meaning. 
 
 203. The next question is what to do when penetration is 
 not instantaneous within the meaning of the Act. We should 
 first go back to (16), which remains valid, and inquire whether 
 the tangential electric forces cannot be expressed in terms of 
 the current (or conversely) in some other way than by a linear 
 relation. Suppose we say 
 
 E^B^C, E 2 = R" 2 C. . . . (20) 
 
 Is it possible to give a definite meaning to the symbol R," ? Is 
 there a definite connection between C, regarded as a function 
 of the time, and E x or E 2 also regarded in this way ? 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 391 
 
 Imagine a wire to be free from current, and, therefore, elec- 
 trically neutral. Now expose its boundary to tangential 
 electric force, beginning at a certain moment, varying in some 
 particular way with the time later, and then ceasing. The 
 result is that C, the total current in the wire, will run through 
 a particular sequence of values, and then finally cease. If we 
 begin again with the applied force, and make it run through 
 the same values in the same manner, we shall again obtain the 
 same values of C at corresponding moments. So far, then, the 
 connection is a definite one. Moreover, if we make the applied 
 force run through the same values increased in a certain ratio, 
 the same for all, then the current will have its previous values 
 increased in the same ratio. But if we change the nature of 
 the applied force as a function of the time (irrespective of size) 
 we shall find that C is not merely changed as a function of the 
 time, but also as a function of the applied force. That is, the 
 mere value of one does not necessitate any particular simulta- 
 neous value of the other. So, if we keep to the usual sense 
 meant when algebraists say u =/(#), or u is a function of #, we 
 cannot say that C is a function of E. It is, nevertheless, true 
 that the march of C is strictly connected with that of E, so 
 that when the latter is given, the former is obligatory. To 
 deny this would be equivalent to the denial of there being 
 definite controlling laws in operation. The full connection 
 between E and C, however, involves not merely their values, 
 but also the values of their first, second, third, <fec., differential 
 efficients up to any order. That is, the symbol R", taken by 
 -,self, is a function of the differentiator d/dt. To illustrate by 
 simple example, suppose 
 
 R" = R + L P + ($p)-\ .... (21) 
 
 ere R, L, S, are constants, and p stands for the differentiator. 
 's means that 
 
 }0, .... (22) 
 
 or E = RC + LC + S/Ccft, .... (23) 
 
 in the common notation of integrals. Now imagine the march 
 of C to be given. This implies that the march of C is also 
 known, and likewise that of /Cdt. Consequently the march of 
 
392 ELECTROMAGNETIC THEORY. CH. IV. 
 
 E is explicitly known. It is not obvious that when the march 
 of E is given, that of C is known, by the same operator impli- 
 citly. That is, by (22) we have 
 
 (24) 
 
 and given E as a function of the time, C is known as a function 
 of the time, or if not known, can be found without ambiguity. 
 It is not obvious, because we do not immediately see how the 
 operation indicated in (24) is to be carried out, whereas, in the 
 case of (22), it is visible by inspection. Nevertheless, the fact 
 that the march of E, physically considered, conditions that of 
 C, makes the above equation (24) not only definite, but com- 
 plete. In the usual treatment of the theory of differential 
 equations, there is no such definiteness. Arbitrary constants 
 are brought in to any extent, to be afterwards got rid of. Now 
 this is all very well in the general theory of differential 
 equations, where arbitrary constants form a part of the theory 
 itself, but for the practical purpose of representing and obtaining 
 solutions of physical problems, the use of such arbitrary and 
 roundabout methods (which are too often followed, especially 
 by elementary writers) leads to a large amount of unnecessary 
 work, tending to obscure the subject, without helping one on. 
 It would not, perhaps, be going too far to say that such a 
 misuse or inefficient application of analysis often makes rig- 
 marole. 
 
 When we say that E = R"C, where IT is the resistance 
 operator (so called because it reduces to the resistance in 
 steady states), we assert a definite connection between E and 
 C, so that when G is fully given as a function of the time, and 
 the operations contained in R" are performed upon it, the func- 
 tion E results, and similarly, when it is E that is given, then 
 the inverse operator (R")" 1 (or the conductance operator) act- 
 ing upon it will produce C. There may be an infinite number 
 of differentiators in R", as p, p 2 , p*, and so on, where p n mean? 
 d^/df 1 . But there is not a single arbitrary constant involved 
 in E R"C, nor, indeed, anything arbitrary. 
 
 Returning to the wires, the form of the resistance operator. 
 as a function of p, depends upon the electrical and geometrical 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 393 
 
 data. It has been determined for round wires and round tubes, 
 and plane sheets. We may take 
 
 . . . (25) 
 ax 
 
 as the form of the second circuital law, and make the determi- 
 nation of the operators a matter of separate calculation. 
 
 As already mentioned, when C is steady, R"j degenerates to 
 R x and R" 2 to R 2 , the steady resistances (per unit length) of 
 the wires, tubes, rods, or cylinders of any shape that may be 
 employed. At the same time the inertial term disappears. 
 Also, when C varies, it is sometimes sufficient to take into 
 account only the first approximation to the form of the opera- 
 tors. This is, for a solid round wire, 
 
 ...... (26) 
 
 and similarly for R" 2 . This J/^ is the value of the steady 
 inductance of the wire, ^ being its inductivity. When not 
 solid, or not round, some other expression is required. Notice 
 that this brings (25) to the elementary form (19), because 
 the inductance of the wires may be included in L itself, which 
 then becomes the complete steady inductance (per unit length 
 axially), including that due to the dielectric and that due to 
 the two wires. This usually means only a small increase in 
 the value of L, unless the wires be of iron. 
 
 Simply Periodic Waves Easily Treated in Case of Imperfect 
 Penetration. 
 
 204. But besides the above simplification, there is an ex- 
 ceedingly important general case in which a similar reduction 
 takes place. This occurs when the sources of disturbance vary 
 simple periodically with the time. Then the electric and mag- 
 netic fluxes everywhere vary ultimately according to the same 
 law with the same period, provided the relations of the fluxes to 
 the forces are linear ; that is, when the conductivity, permit- 
 tivity, and inductivity, are constants at any one place. Now, 
 when this comes to pass, both E and C in the equation E = R"C 
 vary simple periodically with the time. But when the sine or 
 cosine is differentiated twice, the result is the same function 
 negatived, and with a factor introduced. Thus, if the frequency 
 
394 ELECTROMAGNETIC THEORY. CH. IV. 
 
 is ?&/27r, so that sin (nt + 0) may be considered to be the time- 
 factor in the expression for either E or C, we have the property 
 d 2 /dt 2 = - n 2 . Put, therefore p 2 = - n 2 in the expressions for the 
 resistance operators, and we reduce them to 
 
 B'j-B'j + L'ip, ll" 2 = R' 2 + L>, . . (27) 
 
 where R' and L' are functions of n\ They are, therefore, con- 
 stants at a given frequency, and this is a very valuable property, 
 as it is clear at once that we reduce the equation (25) to the 
 simple form 
 
 . (28) 
 ax 
 
 or, more briefly and clearly, 
 
 _^I = R'C + L'C, .... (29) 
 dx 
 
 which is the same as (19), valid when the penetration is in- 
 stantaneous, but with different values of the constants involved. 
 The steady resistance R is replaced by R', the effective resist- 
 ance (of both tubes per unit length) at the given frequency, 
 and L the steady inductance (inclusive of the parts due to the 
 wires) by L', the effective inductance. 
 
 Owing to the reduction of the second circuital equation to 
 the primitive form, we are enabled to express the propagation 
 of simply periodic waves along wires by the same formulae, 
 whether there be or be not imperfect penetration. We simply 
 employ changed values of the constants, resistance and induct- 
 ance, which may be independently calculated, or left to the 
 imagination, should the calculation be impracticable. This is, 
 when it can be effected, the best way of making extensions of 
 theory. Do the work in such a way that harmony is produced 
 with the more rudimentary results, and so that they will work 
 together well, and the new appear as natural extensions of th* 
 old. An appearance of far greater originality may, indeed, be 
 produced by ignoring the form of the elementary results, but 
 the results would be cumbrous, hard to understand, and un- 
 practical. 
 
 The effect of increasing the frequency from zero to a high 
 degree is to first lessen the penetration, and end in mere skin 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 395 
 
 penetration. Consequently, the quantity L', the effective in- 
 ductance, goes from the full steady value L + L^ + L'g, and 
 finishes at L simply, the inductance of the dielectric. But the 
 difference need not be great. In the case of suspended copper 
 telephone wires the inductance of the dielectric is far larger 
 than the rest, so that there is no important variation in the 
 value of the inductance possible, nor would there be were the 
 frequency increased up to that of Hertzian vibrations. But as 
 regards the effective resistance, the case is different. As the 
 distribution of current in a wire changes from that appropriate 
 to the steady state, the resistance increases, and the increase is 
 not always a negligible matter. If long-distance telephony 
 were carried on along iron wires it would be a very important 
 effect. But the Americans, who were the introducers of long- 
 distance telephony, soon found that iron would not do, and that 
 copper would. The reason of the failure is mainly the largely 
 increased resistance of iron. In copper, on the other hand, it 
 is an insensible effect at the lower limit of telephonic frequency 
 of current waves, with the size of wire employed, and is not 
 very important at a frequency three or four times as great. 
 On the other hand, in the numerous experiments with very 
 rapid vibrations of recent years, due to Hertz, Lodge, Tesla^ 
 and many others, the increased resistance due to imperfect 
 penetration becomes a very important matter, and is one of the 
 controlling factors that should be constantly borne in mind. 
 
 Long Waves and Short Waves. Identity of Speed of Free 
 and Guided Waves. 
 
 205. By replacing fictitious magnetic conductance of the 
 medium outside the pair of leads by real electric resistance of 
 the leads themselves, we have obtained the same results as 
 regards the propagation of waves, subject to certain reserva- 
 tions referring to the practical applicability of the theory. It 
 is worth while noticing, in passing, a certain peculiarity show- 
 ing roughly when we may expect the theory to be admissible, 
 and when it should fail. We know that in the transmission of 
 waves along perfectly conductive leads the waves are con- 
 tinuously distorted if the medium be electrically conductive. 
 Also, that by introducing magnetic conductivity into the 
 
396 ELECTROMAGNETIC THEORY. CH. IV 
 
 medium we may reduce this distortion, and ultimately abolish 
 it when the magnetic conductivity reaches a certain value. 
 Now observe here that the correcting influence of the magnetic 
 conductivity is exerted precisely where it is required, namely, 
 in the body of the wave itself. The wave is acted upon in 
 every part by two counteracting distorting influences, so that 
 the correction is performed exactly ; in other words, we obtain 
 an exact theory of distortionless transmission. 
 
 But, on the other hand, when we employ the resistance of 
 the leads to perform the same functions, the correcting in- 
 fluence is not exerted uniformly throughout the body of the 
 wave, but outside the wave altogether ; at its lateral boun- 
 daries, in fact. Nevertheless, for reasons before stated, we still 
 treat the waves as if they preserved their planarity under the 
 influence of the resisting leads. Whilst, therefore, we fully 
 recognise and employ the finite speed of propagation of disturb- 
 ances axially, or parallel to the leads, we virtually assume that 
 the correcting influence of the leads id transmitted laterally 
 outwards instantaneously. We may, therefore, perceive that 
 the wave-length is a matter of importance in determining the 
 applicability of the practical theory. It is of no moment what- 
 ever in the exact theory employing magnetic conductance ; but, 
 when we remove this property from the medium generally, and 
 (virtually) concentrate it at the leads, we should at the same 
 time keep the wave-length a considerable multiple of the dis- 
 tance between the leads if the practical theory is to be applic- 
 able. To see this, it is sufficient to imagine the case of waves 
 whose length is only a small fraction of the distance between 
 the leads, when it is clear that their correcting influence could 
 no longer be assumed to be exerted laterally and instantaneously, 
 as if the small portion of the leads between two close reference 
 planes belonged to and was associated solely with the slice of 
 the medium between the same planes. 
 
 We have referred in the above to the action of the resistance 
 of the leads as a correcting one, neutralising the distortion due 
 to another cause. But the same reasoning is applicable when the 
 action is not of this nature. Thus, when the external medium 
 is non-conductive, and the leads are non-resistive, we have 
 perfect transmission. Making the leads resistive therefore now 
 brings on distortion. We may now say that in order that this 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 397 
 
 distortion should occur in the same way as under the influence 
 of magnetic conductivity in the external medium, the wave- 
 length should not be too small, as specified above. The prac- 
 tical theory is therefore the theory of long waves, and in the 
 interpretation of the word "long," some judgment may be 
 exercised as regards the leads and other matters, because there 
 is no hard and fast distinction, and what may be a long wave 
 under some circumstances may be a short one under others. 
 
 To illustrate this point, imagine that we have got an inge- 
 nious instrument (not yet made), for continuously recording 
 the electromagnetic state of a non-conducting medium, say the 
 air at a certain place (just as we have thermometric and 
 barometric recorders), and that this instrument is so immensely 
 quick in its action as to take cognisance of changes happen- 
 ing in very short intervals of time, say one thousand-millionth 
 of a second. Now in applying this instrument to register the 
 state of air traversed by electromagnetic waves, it is clear that 
 the size of the waves must be considered in relation to the size of 
 the instrument. If the instrument were one decimetre across, 
 then waves of one kilometre in length could as well be regarded 
 as of infinite length. But if only one metre in length, though 
 it could still be used, there would be no longer the same accu- 
 racy of application. And if of only a centimetre in length, 
 then it is plain that several waves would be acting at once on 
 the apparatus in different parts, and the resultant effect 
 recorded would not represent the history of the waves by any 
 means. 
 
 Now as regards waves sent along parallel leads, it is obvious 
 that light waves are totally out of the question, being im- 
 mensely too short. On the other hand, telephonic waves are to 
 be treated as long waves very long, in fact though they are 
 short compared with telegraphic waves. But it is somewhat 
 curious that the electromagnetic waves investigated by Hertz are 
 sometimes so short as to come within the scope of the above 
 reservatioDal remarks, or of others of a similar nature. Whether 
 the generation of waves by an oscillator be considered, or their 
 effect on a resonator, or their transmission along leads, their 
 shortness in relation to the apparatus employed may some- 
 times vitiate the results of approximate theories, and render 
 caution nec r ssary. For example, the plane-wave theory indi- 
 
398 ELECTROMAGNETIC THEORY. CH. IV. 
 
 cates that the attenuation factor for waves running along 
 parallel leads is e- Ra; / 2Lt> in the distance x, or c- 1 "/ 21 ' in the 
 equivalent time of transit t. This is when R and L are con- 
 stants. If not, and the waves be simply periodic, then we may 
 use the same formula with the effective values of R and L at 
 the frequency concerned. But if this be true for long waves, 
 we cannot expect it to continue true on shortening the waves 
 to the transverse distance of the leads, more especially if we 
 are ignorant of the precise type of the waves. At the best, we 
 should not expect more than results of a similar kind. But 
 not much has been done yet in the quantitative examination of 
 Hertzian waves, for sufficiently obvious reasons. 
 
 In one respect, however, a formerly very strange anomaly 
 has been cleared up satisfactorily. When Hertz opened people's 
 eyes and made them see the reality of Maxwell's ether as a 
 medium propagating electromagnetic disturbances at the speed 
 of light, by showing their transmission across a room and 
 reflection by a metallic screen, the full acceptation of Maxwell's 
 theory was considerably hindered for a time by his finding 
 that the speed of waves sent along wires was much less than 
 that of free waves. The discrepancy was a large one, 
 and gave support apparently to the old view regarding 
 the function of wires, which made the wires the primary 
 seat of transmission, and effects outside secondary, due 
 to the wires. And it came to pass that people, whilst 
 admitting the truth of Maxwell's theory, yet made a distinc- 
 tion between waves in free space and " in wires." This was 
 thoroughly out of harmony with Maxwell's theory, which 
 makes out that the wires, though of great importance as 
 guides, are nevertheless only secondary. On the other hand, 
 it should be mentioned that Lodge found no such large de- 
 parture from the speed of light in his experiments. But the 
 matter has been explained by the discovery that an erroneous 
 estimate was made of the permittance of the oscillator in the 
 experiments which apparently showed that the speed was largely 
 reduced. When corrected, there is not left any notable differ- 
 ence between the speed of a free wave and of one guided by 
 wires. Of course, there is no reason why reference to waves 
 " in wires " should not be dropped, unless the laterally-propa- 
 gated cylindrical waves are meant. These are secondary, and 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 399 
 
 have no essential connection with the propagation of the 
 primary waves through the external dielectric, although modi- 
 fying their nature. 
 
 The Guidance of Waves. Usually Two Guides. One suf- 
 ficient, though with Loss. Possibility of Guidance 
 within a Single Tube. 
 
 206. When waves are left to themselves in ether without 
 the presence of conductors, they expand and dissipate them- 
 selves. Even if they are initially so constituted as to converge 
 to centres or axes, they will subsequently expand and dissipate. 
 To prevent this we require conducting guides or leads. Now 
 this usually involves dissipation in the leads ; but the point at 
 present under notice is the property of guidance only. We 
 can stop the expansion in a great measure, and cause a wave 
 to travel along wherever we wish it to go. Practically there are 
 two leads, as a pair of parallel wires ; or if but one wire be 
 used, there is the earth, or something equivalent, to make 
 another. But it is still much the same, as regards guidance, 
 when there is but one wire, if we choose to imagine the case of 
 a single infinitely-long straight wire alone by itself in ether. 
 If we make it the core of a plane electromagnetic sheet, this 
 sheet will run along the wire just as well and in the same way 
 as if there were a second guide. But the energy of such a 
 sheet, even though of finite depth, and containing electric and 
 magnetic forces of finite intensity, would be infinite. The 
 quantity L, the inductance per unit length of guide, is infinite 
 under the circumstances. We could not, therefore, set up such 
 a wave from a finite local source. If we cause an impressed 
 voltage to act axially for a very short interval of time across 
 any section of the guide, say in a reference plane, the result is 
 an approximately spherical wave. (To be perfectly spherical, 
 the wire should be infinitely fine. The case is then that of a 
 spherical wave-sheet with conical boundaries, already referred 
 to, with the angle of the cone made infinitely small.) Its 
 centre is at the origin of the wave, and as it expands, the por- 
 tions of the wave-sheet nearest the wire become approximately 
 parallel plane waves, one going to the right, the other to the 
 left along the wire. But not being pure plane waves they are 
 weakened as they progress, by the continuous expansion of the 
 
400 ELECTROMAGNETIC THEORY. CH. IV. 
 
 spherical wave of which they form a part. Nevertheless, we 
 have the propagation of nearly plane waves of finite energy, or 
 of a perfectly plane wave-sheet of infinite energy, along a single 
 guide. 
 
 Now, this takes place outside the conducting guide, and the 
 question presents itself whether we cannot transmit an electro- 
 magnetic wave along the interior of a tube, in a manner resem- 
 bling a beam of light ? We can certainly do so if we have a 
 second guide within the first tubular one, for this does not 
 differ substantially from the case of two parallel wires, each out- 
 side the other. But it does not seem possible to do without 
 the inner conductor, for when it is taken away we have nothing 
 left upon which the tubes of displacement can terminate inter- 
 nally, and along which they can run. A theoretical expedient 
 is to carry the electrification forward at the proper speed. But 
 we want the process to be automatic, so to speak, hence convec- 
 tion will not do. Again, if we make the displacement start from 
 one portion gf an electrically conducting tube and terminate 
 upon the rest, we must insulate the two portions from one 
 another, and then there will be a division of the charges 
 between the interior and exterior, so that the result will be an 
 external as well as an internal wave, or rather, one wave occu- 
 pying both regions. 
 
 It would appear that the only way of completely solving the 
 problem of the automatic transmission of plane waves within a 
 single tube is a theoretical one, employing magnetic as well as 
 electric conductance. To see this, imagine any kind of purely 
 plane wave being transmitted in the normal manner through 
 the ether, and fix attention upon a tube of the flux of energy, 
 or a beam, using optical language. This beam cuts perpen- 
 dicularly through the reference planes, in which the lines of 
 electric and magnetic force lie, which again cross one another 
 perpendicularly. The shape of the section of the beam by a 
 reference plane may be arbitrary. But let it be quadrilateral, and 
 so that it is bounded by magnetic lines on two opposite sides, 
 and by electric lines on the other two. Now let the two sides 
 of the tube upon which the displacement terminates perpen- 
 dicularly, be electrically conductive thin sheets, and the other 
 two sides, upon which the induction terminates perpendicularly, 
 be magnetically conductive sheets. If the conduction be perfect, 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 401 
 
 we . shall not interfere with the transmission of the beam 
 within the tube. That is, we have solved the problem stated. 
 We may also notice that the external portion of the wave is 
 not intrvfered with by the tube. But the interposition of the 
 tube in the manner described renders the external and internal 
 waves quite independent of one another, and either of them 
 may be suppressed. 
 
 Similarly, if the interior region be made finitely conductive, 
 electrically or magnetically, or both together, we can still in- 
 vestigate the transmission of waves along it in the same way 
 as previously described for complete plane waves in a homo- 
 geneous medium made conductive, and the same applies to the 
 external region, independently of the internal. Going further, 
 we may do away with the diffused conductances, and concen- 
 trate equivalent resistances in the plates bounding the tubes, 
 in the same way as we replace magnetic conductance of the 
 external medium by equivalent electric resistance of the wires 
 in passing from the exact plane-wave theory to the practical 
 theory of wires in terms of V and C. That is, the two plates 
 on which the displacement ends may be made electrically resis- 
 tive, the resistance taking the place of the magnetic conduct- 
 ance in the interior ; whilst the other two plates should be 
 made magnetically resistive, if the interior electric conductance 
 is also to be abolished. We may then express the propagation 
 of waves in the tube in a manner resembling the practical 
 theory of wires, though it will no longer be an exact theory. 
 Nor will the interior and exterior regions be quite independent 
 of one another now that the tube is only finitely conductive. 
 
 Interpretation of Intermediate or Terminal Conditions in 
 the Exact Theory. 
 
 207. Leaving these somewhat abstruse considerations, re- 
 turn to the practical theory concerning wires and its connection 
 with the exact theory involving magnetic conductance. In 
 the working out of the practical theory (which is not yet, how- 
 ever, the theory of official representatives of practice, though 
 they are decidedly getting on), we have often to consider the 
 effects due to intermediate insertions of resistance in the circuit 
 of the leads, or of shunts across them, and other modifications. 
 
 DD 
 
402 ELECTROMAGNETIC THEORY. OH. IV. 
 
 The question now is, what do these insertions represent in the 
 exact theory ? 
 
 Consider first a pair of parallel leads of no resistance. This 
 will admit of the application of the exact theory, provided there 
 be either no leakage or the external medium be uniformly con- 
 ductive electrically. Now we know that when the leads have 
 resistance, we make the transformation to the exact theory by 
 substituting uniform magnetic conductivity of the external 
 medium of the proper amount. Plainly, then, if we insert an 
 electrical resistance in a lump in the circuit of the leads at any 
 place, we should, in the exact theory, transfer it equivalently 
 (changed to magnetic conductance) to the whole of the corre- 
 sponding reference plane. That is to say, we must make the 
 reference plane (outside the leads) uniformly conductive mag- 
 netically, so that the total magnetic conductance of the equi- 
 valent plate inserted equals the electric resistance which it 
 replaces. We may then investigate the action of the plate on 
 the waves traversing it in the manner described in a previous 
 paragraph ( 193 and after). 
 
 Similarly, the effect of an electrically conducting bridge 
 across the circuit is equivalent to that of an electrically con- 
 ducting plate at the reference plane. Their conductances are 
 here not only equal, but of the same kind. We merely re- 
 arrange the leakage conductance so that it shall act uniformly, 
 to suit plane waves. 
 
 Thus, a terminal short-circuit should be replaced by a 
 terminal perfectly conductive plate, reflecting H positively and 
 E negatively, or with reversal of sign ; or, which is the same, 
 C positively and V negatively. This maintains E and V per- 
 manently zero at the short-circuit, unless there be impressed 
 force there. And a terminal disconnection may be represented, 
 in the exact theory, by a terminal perfectly magnetically con- 
 ductive plate, reflecting E positively and H negatively, or V 
 positively and C negatively, thus maintaining C permanently 
 zero at the disconnection. 
 
 A somewhat different kind of transformation is required in 
 connection with impressed forces. Suppose, for instance, we 
 insert an impressed voltage in the circuit of the leads at a given 
 reference plane. How is it to be equivalently transferred to 
 the whole plane, or to that part of it outside the leads, so that 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 403 
 
 it may generate plane waves of the precise type belonging to 
 the given leads ? This is by no means so easy to follow up as 
 the previous plate substitutions ; but the following may suffi- 
 ciently describe the essence of the transformation. First, when 
 the generation of disturbances is concerned, it is not impressed 
 electric force e, but its curl, that is effective. We must there- 
 fore find the curl of the given impressed force. If the latter 
 is half in each wire, acting opposite ways, and uniform across 
 their sections by the reference plane, then the curl of e, 
 say f, is situated on the boundaries of the wires, and encloses 
 them. Having ascertained the integral amount of f, it must 
 be transferred to the whole reference plane outside the wires. 
 Then the question arises how to distribute it properly. To 
 answer this, it may be mentioned that when a distribution of 
 f starts into action, the first effect is to generate induction 
 following the lines of f. This will be made clear later. If, 
 therefore, we distribute the lines of f over the reference plane 
 in such a way as to exactly copy the natural distribution of 
 the magnetic lines in plane waves having the given leads for 
 cores, we shall obtain what we want, viz., the sources so 
 distributed as to generate plane waves of the required type on 
 the spot. 
 
 By means of the above and similar devices, most telephonic 
 and telegraphic problems may be converted to problems in an 
 exact plane-wave theory. But we must always be careful to 
 distinguish between a theory and the application thereof. The 
 advantage of a precise theory is its definiteness. If it be dyna- 
 mically sound, we may elaborate it as far as we please, and be 
 always in contact with a possible state of things. But in 
 making applications it is another matter. It requires the 
 exercise of judgment and knowledge of things as they are, to 
 be able to decide whether this or that influence is negligible or 
 paramount. 
 
 The Spreading of Charge and Current in a long Circuit, and 
 their Attenuation. 
 
 208. The detailed application of the principles already 
 discussed to the data which occur in practice, or which may 
 occur in later practice, would lead us into technical complica- 
 tions which would be out of place in the present stage of 
 
 DD 2 
 
404 ELECTROMAGNETIC THEORY. CH. IV. 
 
 development of this work. Only a few general considerations 
 can be taken up here in connection with the theory of plane 
 waves. We have seen that there are four distinct quantities 
 which fundamentally control the propagation of " signals " or 
 disturbances along a circuit, symbolised by R, K, L, and S, 
 the resistance, external conductance, inductance, and per- 
 mittance ; whilst there are two distinct variables, the transverse 
 voltage V and the circuital gaussage C, or the difference of 
 potential (in an extended sense) of the leads in a reference 
 plane, and the current in the leads. The words voltage and 
 gaussage were proposed in 27 to represent electromotive force 
 and magnetomotive force, and have been employed experi- 
 mentally to see how they would work. They work very well 
 for theoretical purposes, but I observe that they are not 
 generally or universally approved of, perhaps on account of 
 the termination " age," or because of the commencements volt 
 and gauss being unitary names. It is open, of course, to any- 
 one to find names which shall be more suitable or meet with 
 general approval. 
 
 But I stick to inductance. Of all the words which I have 
 proposed, that one seems to me to be, more than any other 
 except impedance and conductance, the right word in the right 
 place, and I am bound to think it possible that some of those 
 who prefer the old " coefficient of self-induction," do not fully 
 appreciate the vital significance of inductance in a theory of 
 intermediate action of a medium. The idea of the direct 
 mutual action of current-elements upon one another is played 
 out. And as regards the name of the practical unit of induct- 
 ance, I think the best name is mac, in honour of the man who 
 knew something about self-induction, of course. Many other 
 names have been proposed, but none so good as mac. Some 
 critic has made etymological objections. But what has 
 etymology got to do with it 1 The proper place for etymology 
 is the grammar book. I always hated grammar. The teach- 
 ing of grammar to children is a barbarous practice, and should 
 be abolished. They should be taught to speak correctly by 
 example, not by unutterably dull and stupid and inefficient 
 rules. The science of grammar should come last, as a study 
 for learned men who are inclined to verbal finnicking. Our 
 savage forefathers knew no grammar. But they made far 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 405 
 
 better words than the learned grammarians. Nothing is more 
 admirable than the simplicity of the old style of short words, 
 as in the A sad lad, A bad dog, of the spelling book. If you 
 transform these to A lugubrious juvenile, A vicious canine, 
 where is the improvement ? 
 
 Now, as we have already explained the characteristics of dis- 
 tortion of plane waves by conductance in the medium generally, 
 of two kinds, acting differently on the electric and magnetic 
 fluxes, the corresponding relations in a long circuit of leads 
 need be only briefly mentioned. Consider an infinitely long 
 circuit of parallel and equal resisting leads under uniform 
 external conditions, and governable by the relations laid down. 
 Let it be, at a given moment, wholly free from charge or cur- 
 rent except at one place, where there is merely a charge, say 
 Q . Or, say Q = SV on unit length only at the place. Left 
 to itself, this will spread. But it will do so in widely diverse 
 manners under different circumstances, although there are some 
 common characteristics. The speed of propagation of disturb- 
 ances is v = (LS)~*, independent of the values of R and K, the 
 dissipation constants. Thus, at time t after the spreading 
 begins, we are bound to find the charge, or what is left of it, 
 within the region of length 2vt, being vt on each side of the 
 origin. Beyond this region there is no disturbance. 
 
 Next as regards the amount of the charge. It cannot in- 
 crease, but it may diminish. If the insulation be perfect, which 
 means that K = 0, the charge remains constant. It is unaffected 
 by the resistance of the leads, although energy is wasted in 
 them. It only decreases by the external conductance, or by 
 equivalent leakage. Then we have 
 
 ...... (1) 
 
 to express the charge Q at time t, decreasing according to the 
 well-known exponential law. 
 
 As regards the manner of spreading, either with or without 
 the subsidence, this is represented in the exact theory by the 
 spreading and subsidence of displacement in a medium which 
 is magnetically as well as electrically conductive, although the 
 former property does not affect the total amount of displace- 
 ment. We see, therefore, that the distortionless case, which 
 occurs when the time constants L/R and S/K are equal, forms 
 
406 ELECTROMAGNETIC THEORY. CH. IV. 
 
 a natural division between two distinct kinds of resultant 
 distortion. In this neutral case, the charge Q splits into two 
 halves which at once separate from one another at relative 
 speed 2v. At time t later they are each attenuated to 
 
 KHiQo^" 8 (2) 
 
 The energy was wholly electric to begin with, but in each of 
 the resulting waves it is half electric and half magnetic, be- 
 cause they are pure waves. As they attenuate, their energy is 
 wasted, and this occurs so that half is wasted by the resistance 
 of the leads, and half by reason of the external conductance. 
 This is necessary in order that a pure wave shall remain pure. 
 The final result is attenuation of the two charges to zero, when 
 they are infinitely widely separated, and with this a complete 
 simultaneous disappearance of the other characteristics, as the 
 magnetic force. In the exceptional case of no resistance and 
 no leakage, there is, of course, an everlasting persistence of the 
 two charges and of the waves of which they form a feature. 
 They go out of range, but not out of existence. In all other 
 cases but the distortionless one, the charge (or the remains of 
 the original charge) is only partly in the plane waves at the 
 two ends of the disturbed region, the rest being diffused 
 between them. But this may happen in two ways. The total 
 charge at time t is expressed by (1) above. But the charge in 
 the terminal waves, half in each, is given by 
 
 Q . -(B/2L+K/28X f ..... (3) 
 
 reducing to (2) doubled in the distortionless case. Therefore 
 the excess of (1) over (3) is the amount of the diffused charge. 
 
 It is 
 
 Q .e- K '/ s (l-e-(R/2i--K/2S)*) ) ... (4) 
 
 and is positive or negative according as R/L is greater or less 
 than K/S. 
 
 That is, when the resistance of the leads is in excess, the 
 charge is positive between the waves as well as in them. But 
 when the leakage is in excess the intermediate charge is nega- 
 tive. This happens equivalently when the inductance is in 
 excess, or the permittance in deficit. How this reversal comes 
 about has been already explained in the exact theory. In the 
 present circumstances we may say that an intermediate resist- 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 407 
 
 ance in the circuit of the leads reflects V positively and C 
 negatively, whilst an intermediate leak reflects C positively and 
 V negatively. The same applies to the elements of the circuit, 
 and from this follows the reversal of V when leakage is in 
 excess. Whether the tailing be of the usual positive or of 
 the negative kind (referred to the charge), the maximum or 
 minimum density is in the middle ; that is, at the origin. The 
 density decreases symmetrically both ways to the two ends, 
 where the plane waves are to be found. When they have at- 
 tenuated to practically nothing, there is left merely the widely 
 diffused charge, unless it has attenuated similarly. With the 
 resistance largely in excess, there may be nearly all the charge 
 left when the terminal waves have become of insensible sig- 
 nificance. But if the leakage be in excess, the disappearance 
 of the terminal waves is accompanied by a similar disappear- 
 ance of the intermediate negative charge, because the total 
 charge is always positive. 
 
 So far regarding the spreading of a charge, next consider the 
 spreading of induction. The magnetic analogue of Q is P = LC, 
 the magnetic momentum per unit length of circuit. So, as L 
 is constant, the spreading of P is represented by that of C. 
 Suppose initially there be current C in unit length at the 
 origin, with momentum P = LC , and no charge there or else- 
 where. This also splits immediately into two plane waves, 
 which only differ from those arising from a charge in having 
 similar C's and opposite V's instead of similar V's and opposite 
 C's. 
 
 The total induction remains constant only if the leads 
 have no resistance. In general it decreases, so that at time t 
 we have 
 
 P = P .-**0- (5) 
 
 At the same time the amount in the terminal waves, half in 
 each, is 
 
 P . c -(*/2L+K/2S)e ...... (6) 
 
 The amount between the terminal waves is, therefore, the 
 excess of (5) over (6), which is 
 
 p o -R/L(l_ e -(K/2S-R/2LX) ... (7) 
 
 These equations (5), (6), (7), may be instructively compared 
 with the set (1), (3), (4), their analogues. Notice that in 
 
408 ELECTROMAGNETIC THEORY. CII. IV. 
 
 transforming from one set to the other, we exchange V and C, 
 Q and P, R and K, L and S. In the exact theory the R in 
 these formulae would also be a conductance, as before described. 
 
 By (7) we see that in the distortionless case all the induction 
 left is in the terminal waves, which is to be expected. But we 
 require the leakage to be in excess for the intermediate induc- 
 tion to be positive, and conversely the resistance must be in 
 excess for it to be negative. Of course, the initial induction is 
 assumed to be positive. Also observe that leakage by itself 
 has no effect on the total induction except to redistribute it, 
 just as the resistance of the leads has no attenuating effect 
 on the total charge. As before, the maximum or minimum 
 density of induction is at the origin, though, of course, the 
 terminal inductions may have greater or smaller density than 
 this, according to the extent of attenuation at the moment 
 concerned. Remember that the destruction of induction by 
 the resistance of the leads is represented, in the exact theory, 
 by its destruction by the equivalent magnetic conductance 
 externally. 
 
 Putting together the preceding results, we may readily see 
 the effect of having both V and C initially at any spot. One 
 case is specially noteworthy. Take V = L^C^ initially. This 
 means a pure plane wave sheet, travelling in the positive 
 direction. There is, therefore, no initial splitting, and the 
 wave just goes on. To what extent this will continue depends 
 upon the relation the constants of the circuit bear to the 
 distortionless state. When the latter obtains, the wave is 
 transmitted without spreading out behind, so that at time t 
 the initial state, if existent over unit length, will be found 
 over unit length at distance vt to the right, attenuated to 
 
 V -K</L, ..... (8) 
 
 where for t may be substituted its equivalent x/v. In all other 
 cases there is reflection in transit, and the reflected portions 
 travel back, besides getting mixed together, thus making a tail 
 of length Zvt, half on each side of the origin, but now, of 
 course, without a head at the negative end to balance the one 
 at the positive end. 
 
 If the resistance be in excess, the charge in the tail is positive 
 and the current is negative, at least initially ; whilst if the 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 409 
 
 leakage be in excess, the current is positive and the charge 
 initially negative. See 195 to 199 for more details, which may 
 be readily translated to suit the present case. The total charge 
 subsides according to equation (1) above, and the total induc- 
 tion according to (5). Similarly (3) represents the charge in 
 the one terminal wave (instead of both, as there), and (6) 
 shows the induction in the terminal wave. 
 
 If initially V = - L^CQ, this means a negative wave (going 
 to the left) to start with. This needs no separate notice in 
 detail. 
 
 The Distortionless Circuit. No limiting Distance set by it 
 when the Attenuation is ignored. 
 
 209. The distortionless state forms a simple and natural 
 boundary between two diverse kinds of propagation of a com- 
 plicated nature, in each of which there is continuous distortion 
 which is ultimately unlimited. Given time enough, and a 
 circuit of infinite length to work with, the least departure 
 from the distortionless condition would be sufficient to allow 
 all the natural changes to be gone through, with ulti- 
 mate unlimited distortion. The attenuation is a separate 
 matter in this connection, though itself of great importance. 
 If, however, we are only concerned with the distortion produced 
 in a finite interval of time, which may be quite small, then the 
 matter may be differently regarded. There is much or little 
 distortion, generally speaking, according as the ratios R/L and 
 K./S are nearly equal or widely different. But this needs to be 
 understood with caution, for obviously there may be next to no 
 distortion of signals even when the distortionless state is widely 
 departed from, provided the changes of charge and current are 
 made slowly enough. But when this is not the case, and there 
 is marked distortion of waves in transit, then we shall increase 
 it by making the time-constants more unequal than they are, 
 and decrease it by a tendency to equalisation. Furthermore, 
 this process is a continuous one, so that, starting from the dis- 
 tortionless state connected with equal time-constants, we may, 
 by continuously increasing one time-constant, bring on con- 
 tinuously increasing distortion of one kind, or else, by increasing 
 the other, whilst the first is kept constant, bring on con- 
 tinuously increasing distortion of the other kind. And this is 
 
410 ELECTROMAGNETIC THEORY. CH. IV. 
 
 true by variation of any one of the four quantities concerned, 
 the resistance, leakage, inductance, and permittance, to prove 
 which we need only remark that with any values given to these 
 constants we can equalise the time-constants and abolish the 
 distortion altogether by altering any one only of the four. 
 
 When a circuit has been brought to this state, then arbitrary 
 signals of any size and manner of variation, originated at the 
 beginning of the circuit, will run along it at the speed of light, 
 and in doing so suffer no alteration save a weakening according 
 to the exponential law given above. The voltage and gaussage 
 at any spot will be always in the same phase, and the suc- 
 cession of values will faithfully repeat those at the origin. If 
 the circuit be infinitely long, the train of disturbances will run 
 out to infinity, of course, though it may be still a long way 
 from infinity when the attenuation is so great as to make the 
 disturbances insensible. Therefore, disregarding the attenua- 
 tion, there is no limiting distance of possible signalling. More 
 than that, it is not only possible, but perfect signalling, accord- 
 ing to this theory. 
 
 But there is no perfection in this world. Even if a circuit 
 were constructed with constant R, K, L, and S, and with R/L = 
 K/S exactly, so that it should be truly distortionless in this 
 practical theory, it would not be so in reality. There are 
 several disturbing causes, which, though they might not be of 
 much importance in general, would -serve to prevent the attain- 
 ment of the clean-cut perfection of the theory which ignores 
 them. Instead of zero distortion, then, it is practically only a 
 state of minimum distortion that is attainable. But it may be 
 a very good imitation of the theoretical ideal. It is unnecessary 
 to enlarge upon the necessary failure of a professedly approxi- 
 mate theory when pushed to extremes. It is sufficient to say 
 that the ignored disturbing influences alone would serve to set 
 a practical limit, even if the condition R/L = K/S were truly 
 attained, and attenuation were of no importance. 
 
 But there is an influence in full action in the practical theory 
 itself which must not be overlooked, namely, the attenuation 
 due to resistance and leakage. Of what use would it be to 
 have a distortionless circuit if it took nearly all the life out of 
 the current in the first 100 miles, when you want it to go 
 1,000? Now, on paper nothing is easier than to increase the 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 411 
 
 battery power to any extent you want, till, in fact, the minute 
 fraction which got to the desired distance became magnified 
 up to a recognisable size. But practically there would usually 
 be strong reasons against this process. At any rate, it is im- 
 possible to overlook the attenuation. It is not merely enough 
 that signals should arrive without being distorted too much ; 
 but they must also be big enough to be useful. If an Atlantic 
 cable of the present type were made distortionless by the 
 addition of leakage, there would be no sign of any signals at 
 the distant end with any reasonable battery power. Nor can 
 we say that telephony is possible on a circuit of given type to 
 this or that distance merely because a certain calculable and 
 small amount of distortion occurs at that distance. Nor can 
 we fix any limiting distance by consideration of distortion 
 alone. And even if we could magnify very weak currents, say 
 a thousandfold, at the receiving end, we should simultaneously 
 magnify the foreign interferences. In a normal state of things 
 interferences should be only a small fraction of the principal or 
 working current. But if the latter be too much attenuated, 
 the interferences become relatively important, and a source of 
 very serious distortion. We are, therefore, led to examine the 
 influence of the different circuit constants on the attenuation, 
 as compared with their influence on the distortion. 
 
 The two Extreme Kinds of Diffusion in one Theory. 
 
 210. Although the equations connecting the voltage V and 
 gaussage 0, through the line constants R, S, K, and L, are of 
 a symmetrical character, so that, abstractedly considered, a 
 study of the nature of their propagation on either side of the 
 distortionless state is as desirable as on the other side ; yet 
 when we examine the conditions prevailing in practice, we see 
 at once that one side only presents itself actively. The reason 
 is easily to be seen. It is in all cases desired to get plenty of 
 received current, because that is the working agent, and leak- 
 age is detrimental to this consummation. There are, besides, 
 practical inconveniences connected with leakage. Thus it comes 
 about that we are mainly concerned with the state of things 
 existing when K/L is greater than, or at the least equal to, K/S, 
 and scarcely at all with the exceptional case of an excess of 
 
412 ELECTROMAGNETIC THEORY. CH. IV. 
 
 leakage, unless it be as a curiosity, and for the sake of its 
 theoretical connection with other electromagnetic problems. 
 
 Now this concerns the propagation of V and C through the 
 external dielectric. The influence of R prevails over that of 
 K, and controls matters. It is, however, somewhat remarkable 
 that the other side of the propagation problem does actually 
 present itself actively, when we regard the important secondary 
 effect of the propagation into the guides of the cylindrical 
 waves, which result from the passage over them of the plane 
 waves in the external dielectric. Suppose, for example, there 
 is no external conductance. This gives one extreme form of 
 propagation of V and C, the quantity K/S (or equivalently k/c) 
 being zero, and R/L finite. Now, remember that in the exact 
 plane-wave theory R stands for the magnetic conductance of 
 the external medium, so that R/L is the same as g//j,, and there- 
 fore we are, when considering V and C, also discussing the 
 influence upon the plane waves of the fictitious property of 
 magnetic conductance. 
 
 On the other hand, in the guides themselves, which are elec- 
 trical conductors, and perhaps, and most probably, dielectrics 
 as well, it is k/c that is greater than g/fj., since the latter is 
 zero. Here, then, we have the other extreme form of the 
 theory. In the external dielectric k/c is zero and g/jj. (equiva- 
 lent to R/L) finite, whereas in the guides g/p is zero and k/c 
 finite. The distortion of the external plane waves by the 
 resistance of the guides is, therefore, of the opposite kind to 
 that of the cylindrical waves in the guides, for E and H change 
 places when the form of the distortion is discussed. 
 
 Now, in the guides, on account of their being very good con- 
 ductors, the permittivity is swamped, and may be ignored 
 in calculating resultant effects. Then we have the theory of 
 pure diffusion (as of heat according to Fourier), controlled by 
 the two constants k and //,, since both g and c are now zero. 
 Similarly, in the external dielectric, it may happen that the in- 
 fluence of fj. (or equivalently of L) is insensible, as in the slow 
 working of long cables ; then the propagation of V and C is 
 controlled by the two constants g and c (or actually and equiva- 
 lently by R and S), whilst the other two, k and \^ are zero. 
 Here, then, we have a curious exchange of active properties 
 between the dielectric and the conductor. In both, the resul- 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 413 
 
 tant propagation follows the diffusion law, but is controlled by 
 different properties. The propagation of V or E outside is like 
 that of H inside, and the propagation of C or H outside is like 
 that of E inside. 
 
 We see by the above illustrations that in spite of the absence 
 of magnetic conductivity in reality, we are, nevertheless, obliged 
 to consider both sides of the wave-theory it involves, even in 
 one and the same actual problem, provided it be treated com- 
 prehensively. 
 
 The Effect of varying the Four Line-Constants as regards 
 Distortion and Attenuation. 
 
 211. As, however, we are not immediately concerned with 
 the internal state of the guides, and, so to speak, eliminate it 
 by treating R and L as constants, we may now dismiss its 
 consideration, and return to V and C only, on our understand- 
 ing that the quantity K/S is always less than R/L, or does not 
 exceed it in the limit. 
 
 Now we have shown that the distortion in transit depends 
 upon the difference 
 
 R K. /n \ 
 
 2L-2S (9) 
 
 whilst the attenuation of the front of an advancing wave 
 depends upon the sum 
 
 o = A + JL. (10) 
 
 2L 2S 
 
 Both a- and p should be as small as possible, of course. The 
 four line-constants act upon p and o- in different ways, and it is 
 rather important to understand them. So, remembering that 
 the first term of the right members of (9) and (10) is bigger 
 than the second, observe the effect of varying the line-con- 
 stants, one at a time. 
 
 [R]. By increasing R we increase both p and <r. Thus the 
 resistance of the guides is wholly prejudicial, since it not only 
 weakens signals in transit, but distorts them. It is, therefore, 
 a fundamental notion that the resistance of the guides should 
 be reduced if possible, but never increased, if the object be to 
 facilitate signalling. 
 
414 ELECTROMAGNETIC THEORY. CH. IV. 
 
 [L]. By increasing L we decrease both p and or. Thus the 
 inductance of the circuit (other things being equal) is wholly 
 beneficial, since it diminishes both the attenuation and the dis- 
 tortion in transit, or makes the received signals both larger and 
 plainer. These are not necessarily insignificant effects, but 
 may be very large when applied to very rapid vibrations on a 
 long circuit. 
 
 [K]. Increasing K increases p and reduces or. Thus leakage 
 is both beneficial and prejudicial, according to circumstances, 
 since although it reduces the distortion, it simultaneously in- 
 creases the attenuation in transit. This action is most pro- 
 nounced when the signalling is slow. 
 
 [S]. Increasing S increases cr and reduces p. It may be, 
 therefore, both prejudicial and beneficial. But practically the 
 important effect of S, notably on cables, is the prejudicial one 
 of distorting the signals, which is a large effect.* 
 
 Now the above statements, though obviously true enough as 
 deductions from (9) and (10), are merely qualitative. The 
 practical import of varying one quantity when three others are 
 constant, depends upon the actual immediate values of all four. 
 The length of the line also comes in as an important factor, 
 and likewise the frequency of the waves, to settle under what 
 circumstances this or that variation of the line constants is 
 important in a special case. Nevertheless, the above properties 
 are important, and should be most usefully borne in mind in 
 general reasoning, as well as in the detailed examination of 
 formulae, for instance, the important solutions for simply 
 periodic waves. 
 
 Some of these effects have been known and understood from 
 the earliest times, or, more correctly, since the time when 
 William Thomson taught telegraph-cable engineers the prin- 
 ciples of their business (in its electrical aspects), and, to a 
 great extent, the practice too. They were previously in quite a 
 benighted state, generally speaking, though there were a few, 
 whose names it is needless to mention, who combined enough 
 
 * Every rule is said to have an exception. It is, at any rate, possible 
 for the above rules to fail, or appear to fail, when the making and reception 
 of the signals is not of a simple kind. We may then attribute the failure 
 to instrumental peculiarities. An example will occur a little later in which 
 K reduces working speed largely. 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 415 
 
 electrical science with their practical knowledge to have fairly 
 correct ideas about retardation in submarine cables. I do 
 not mention Faraday in this connection, for that great genius 
 had all sorts of original notions, wrong as well as right, and 
 not being a mathematician, could not effectually discriminate, 
 especially as he had so little practical experience with cables. 
 
 A complete history of the subject of the rise of Atlantic 
 telegraphy, including the scientific side, written by a man fully 
 and accurately acquainted with the facts, preferably with per- 
 sonal knowledge, and devoid of personal bias, would be of great 
 value and of permanent utility. It will not be very long before 
 no contemporary will be left to do it. Perhaps, on the other 
 hand, it might be more fairly done at second hand. In the 
 meantime, there is a somewhat bulky volume, the Report of the 
 Submarine Cable Committee of 1859, which serves to show the 
 state of knowledge at that date. But it is the history of the 
 march of knowledge in the several preceding years that is 
 important. 
 
 The two effects referred to are that resistance and permit- 
 tance (R and S) act conjointly in producing "retardation," so 
 that both R and S should be reduced to improve the electrical 
 efficiency of a cable. This is embodied in Lord Kelvin's theory 
 of 1855, and gave rise to a rather neat law of the squares, 
 something that practicians could grasp. This law soon became 
 matter of common knowledge amongst cable engineers and 
 electricians generally. But when once grasped they could not 
 let go, and in later times the law has been most grossly abused 
 and misapplied. The fact still remains, however, that R and 
 S are prejudicial. But allowance has to be made for other 
 influences, and it may be very large allowance. 
 
 But, very curiously, there has been an exception to the rule. 
 There have not been wanting electricians who, when telephony 
 was extended to circuits containing underground work, sup- 
 posed that high, not low resistance, was good for telephony. 
 This was giving up the law of the squares with a vengeance. 
 But there was really no warrant for such a conclusion, either 
 theoretical or practical. As this view turned out manifestly 
 wrong by practical failure, it was followed by a precipitate re- 
 vulsion, the law of the squares being again seized, and grasped 
 more firmly than ever, and against all reason. The fact that R 
 
416 ELECTROMAGNETIC THEORY. CII. IV. 
 
 and S are prejudicial does not make the law of the squares. To 
 have that, R and S must be the only controlling electrical factors. 
 
 All the other effects are of later recognition, and were mostly 
 discovered, and their laws investigated, and consequences 
 worked out by myself. They completely remodel the subject. 
 The old views remain right, in their proper application, bub 
 have a very limited application in general, as in modern tele- 
 phony particularly. My extended theory opened out a wide 
 field for possible improvements in signalling over long circuits 
 in advance of what could be possible were R and S the only 
 controlling factors, and so far as it goes, modern telephony 
 already goes far beyond the supposed restrictions of the law of 
 the squares. 
 
 We should not only consider the effects of varying the four 
 line-constants one at a time, but also of combinations. Thus, 
 K may be counteracted by L. The former is beneficial, inas- 
 much as it lessens the distortion, but it is prejudicial in the 
 attenuation it causes in transit. But if we simultaneously in- 
 crease L, we counteract the attenuation caused by K ; more- 
 over, we still further lessen the distortion. This is corrobora- 
 tive of the preceding remark as to the unreserved benefit due 
 to L. We see, therefore, that [K] and [L] above point the way 
 to salvation, if that be by means of increasing the distance 
 through which signalling can be carried on with a cable of given 
 type as regards R and S. But let it not be forgotten in studying 
 K and L, that R and S themselves might perhaps (in some 
 particular scheme, for example) be reduced as cheaply as K and 
 L could be carried out. Be liberal with the copper, for one thing. 
 Look out for some practical way of insulating the wires which 
 shall not make the permittance be so great as it is at present 
 with gutta-percha or india-rubber. Something of this sort has 
 been done for telephone underground cables, though I do not 
 know the practical merits. But, as regards Atlantic cables, it 
 should be something very good and trustworthy, or the reduced 
 permittance may be bought too dearly. 
 
 There has been some difficulty in getting people to grasp the 
 ideas embodied in [K] and [L], especially the latter, of course, 
 on account of its novelty ; whilst [K] was already partially 
 recognised by knowing ones, and is capable of very ready ex- 
 planation. [L] was the trouble. Now, remembering the state 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 417 
 
 of fog that prevailed in the fifties as regards the effects of 
 11 and S alone, there is no reason for wonder that more recon- 
 dite effects, as of L, unaccompanied by prior experience, should 
 not be readily recognised when pointed out, though that was 
 no excuse for burking the matter. [K] and [L], especially the 
 latter, require to be studied to be understood. But if people 
 will not take the trouble to study the matter, how can they 
 expect to understand it 1 Now I find that there has been too 
 much of an idea that I have merely given some very compli- 
 cated formula which no fellow can understand, and have made 
 some dogmatic and paradoxical statements about the conse- 
 quences. But a reference to my " Electrical Papers " will show 
 that I have treated the matter of propagation very fully, in 
 the elementary parts as well as in the advanced, and from 
 many points of view, and that I have not been above pointing 
 the moral by numerical examination of results, whilst at the 
 same time I have been particularly careful in elaborating the 
 theory of the distortionless circuit, on account of its casting 
 so much light upon the subject generally. The notion that 
 has been promulgated, that I cannot make myself understood 
 by physicists, recoils upon its authors with great force. Letting 
 alone the fact that some physicists have understood me, and 
 moreover, the fact that various students have understood me 
 (as I know by my correspondence), there is the very notable 
 fact that the promulgators of the false idea (which must be 
 most discouraging to students) have evidently not taken the 
 trouble to examine my writings to see what my views really 
 are, but have only made acquaintance with fragments thereof. 
 I have advertised the distortionless circuit so often that I have 
 become ashamed of doing so. Nevertheless, since it is, as I 
 have often pointed out, the royal road to an inner under- 
 standing of the subject in general, I do not hesitate now to 
 add some more remarks in connection therewith, and leading 
 up thereto, in attempted elucidation of former work. We 
 should first consider [K] alone, or in conjunction with [R] and 
 [S], and then [L], and finally both [K] and [L]. 
 
 The Beneficial Effect of Leakage in Submarine Cables. 
 212. What leakage does is to increase the attenuation of 
 waves in transit, but at the same time to lessen the distortion 
 
 EB 
 
418 ELECTROMAGNETIC THEORY. CH. IV. 
 
 they suffer. Now there is, by elementary knowledge, no 
 difficulty in seeing why leakage on the line should diminish the 
 strength of the received current, in spite of the fact that the 
 strength of the current sent is increased by the leakage. Ohm's 
 law applied to a simple circuit with a shunt attached will 
 suffice to give the reason so far as one fault is concerned ; and 
 the principle is the same for any number of faults, or for dis- 
 tributed leakage. 
 
 As regards the lessening of the distortion, that is not quite 
 so elementary. It does not present itself for consideration at 
 all in common telegraph circuits. When, however, a cable is 
 in question, and there is manifest distortion of signals, the 
 influence thereupon of leakage becomes an obvious matter of 
 inquiry. We may do it on the basis of the electrostatic theory. 
 In fact, in his telegraph theory, Lord Kelvin did insert a term 
 in the differential equation to express the effect on the poten- 
 tial of uniform leakage, though it was merely in passing, and 
 I do not think there was any development of the matter in his 
 later papers. The obvious attenuating effect of leakage, and 
 the manifest evils of faults, caused high insulation to be aimed 
 at, and possibly caused the full effect of leakage to be over- 
 looked, and its theory to remain uninvestigated. 
 
 Now a cable is a condenser, or leyden, and a very big one 
 too, although its permittance is so widespread that the per- 
 mittance of a yard or so is very small. The discharge of 
 a condenser against resistance takes time, and so does its 
 charge, and the greater the resistance the greater the time 
 needed. In the extreme, a perfectly insulated charged con- 
 denser keeps its charge unchanged, and so does a well-insulated 
 cable, if also insulated at the ends. Even when it is earthed 
 (or grounded, as the Americans say) at both its ends, only the 
 portions close to the end can discharge at once. Away from 
 the ends, the discharge has to take place through resistance, 
 to a varying extent though in different parts, when we regard 
 the long cylindrical condenser as a collection of small con- 
 densers side by side in parallel, with all the similar poles joined 
 together by resisting wires. This is how it comes about that 
 the resistance acts conjointly with the permittance in causing 
 " retardation " in cables, so that the electrostatic time-constant 
 is proportional to their product, or RS/ 2 . It is obvious, then, 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 410 
 
 that we can facilitate the discharge by lessening the resist- 
 ance to discharge, that is, by leakage in the cable, either dis- 
 tributed or in detached lumps. At the same time the charging 
 is similarly facilitated or expedited. 
 
 Now, in making signals from end to end, the operation of 
 charging the cable is necessarily precedent to that of receiving 
 current at the distant end, which, in fact, represents a part of 
 the charge coming out. Therefore, to expedite changes in the 
 charge in the cable, not necessarily or usually complete charges 
 or discharges, results in expediting changes in the current at 
 the distant end. That is, it lessens the distortion at a given 
 working speed, and so expedites the signalling. There is, in 
 the electrostatic theory, no limit to this process, and by suffi- 
 ciently reducing the insulation resistance, or, which is the same, 
 increasing the leakance (which is, perhaps, an admissible short 
 name for leakage conductance), we might signal through the 
 cable as fast as we pleased, were there no other effect produced. 
 This other effect is the extra attenuation. Whilst the possible 
 speed of signalling can be raised apparently ad lib., there is 
 only an infinitesimal current left to do it with. (Remember 
 that the influence of L is not now in question.) It is important 
 to recognise that it is a consequence or natural extension of the 
 old electrostatic theory, without any appeal to self-induction, 
 that you can signal as fast as you like through an Atlantic 
 cable of the present type (with leakage added, of course) simply 
 by throwing away most of the current, and utilising the small 
 residuum that manages to survive the ordeal. 
 
 Thus, a proper Atlantic cable, suitable for rapid signalling, 
 should not have high insulation resistance, but the very lowest 
 possible consistent with getting enough current through to 
 work with, and other considerations, mechanical and com- 
 mercial, &c. The theory, moreover, is a sufficiently clear one, 
 and does not need elaborate mathematics to make it at least 
 probably accurate, though it may, of course, be desirable to 
 thoroughly verify the consequences mathematically. 
 
 It is now about 23 years since I came to see, substantially 
 as above, the theoretical absurdity of a system of signalling 
 which deliberately and seemingly of purpose places in the 
 way the greatest possibly hindrance to the efficient trans- 
 mission of signals, viz., by the use of the highest insulation 
 
 EE2 
 
420 ELECTROMAGNETIC THEORY. OH. IV. 
 
 possible, causing them to be regularly strangled, choked, and 
 mutilated, to say nothing of their sometimes dying of inanition. 
 But this was mere theory. As a matter of fact, the effect 
 of leakage presented itself to me at that time in practice 
 on real live cables in a surprisingly different manner. The 
 opposition between plausible theory and live practice was so 
 glaring that it will be instructive to briefly describe it next. 
 
 Short History of Leakage Effects on a Cable Circuit. 
 
 213. First, as regards long land lines, of say 400 miles in 
 length (which is a long distance in this small country), leakage 
 presented itself as an unmitigated nuisance. The worst case of 
 all was during a continuous soaking rain all over the country. 
 When this had gone on for a few hours the insulation resistance 
 fell so low, by leakage over the dirty wet insulators mostly, 
 though assisted by contacts with wet foliage, &c., that many of 
 the important circuits were completely invalided, the received 
 current (even with increased battery power) being too small for 
 practical use with the instruments concerned. This was usually 
 the case in the U.K. on any very wet day. But it was also 
 the same in effect in the E. and L, although not so many 
 lines were invalided, nor did they break down so quickly. 
 
 Incidentally, it may be mentioned here that the so-called 
 "earth-currents "(meaning currents in the Line itself), occurring 
 during magnetic storms, such currents being a manifestation 
 thereof, did not seem to mind the leakage much, perhaps 
 because their E.M.F. was not like that of a battery at one 
 end of a line, but acted continuously along the circuit, by a 
 continuous slow change going on in the earth's magnetic force, 
 or, rather, in the magnetic force in the air surrounding the 
 earth. Probably the " earth-current " goes the same way in 
 the line as in the earth (provided the line is not set at right 
 angles to the real earth-current), whereas when a battery is 
 used, the earth and line-currents are opposed. But it is not 
 easy in wet weather to get a good magnetic storm, so that close 
 observations can be made regarding the influence of leakage on 
 earth-currents. We may dismiss them, and sum up that, as 
 regards land-lines, leakage does not seem to have a single 
 redeeming feature. 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 421 
 
 The effects of leakage next presented themselves to me on 
 cable circuits with land-lines attached, and were exceedingly 
 curious. There were two circuits of the kind, but one in par- 
 ticular had a varied history as regards leakage. It was a cable 
 of about 400 or 450 miles running east and west, having a land 
 line of 120 miles at the east end, and one of 20 miles at the 
 west end. The apparatus used was the Wheatstone automatic, 
 that is, an automatic transmitter and a polarised receiver, 
 recording dots and dashes, and the working was done without 
 condensers, by utilising the earth direct to complete the circuit. 
 
 The best results were obtained in fine dry weather with a 
 hard frost the harder the better. The insulation of the long 
 (east) land line was then at its highest, and the signalling could 
 be done up to 30 words per minute or a little more from west 
 to east, though considerably less the other way, for a peculiar 
 reason. But this was quite exceptional, and did not last long. 
 The next best was in any very dry weather. But in ordinary 
 average fair weather, not rainy or damp, the speed was much 
 lower, say 24 or 20 words per minute. The effect of rain was 
 always to lower the speed, perhaps down to 15 words per 
 minute. A leakage fault on the land line or a contact with 
 another wire had a similar effect, only worse, provided it was a 
 bad fault, and the speed would go down to 10 words per minute 
 or less. Now it is possible that the extra good results obtained 
 in a hard dry frost arose partly from the reduced resistance of 
 the east land line (the usual iron wire), but there was no doubt 
 at all that the main cause of the wide variations of speed was 
 leakage on the land lines, especially on the long one, of course. 
 
 Now it is to be noted here that in the former case of long 
 land lines the leakage was prejudicial by the attenuation it 
 caused, whereby the received currents were made too feeble to 
 suit the apparatus. The working speed, though relatively fast, 
 was too slow for any notable distortion of signals in transit by 
 the condenser action so marked in cables. But in the latter 
 case, of a cable with terminal land lines upon which similar 
 leakage occurred, the prejudicial effect was not due to the 
 attenuation. There was plenty of current in fact, a super- 
 abundance but it did not come out at the receiving end 
 properly shaped for making dots and dashes, except at a 
 greatly lowered speed. It should also be understood that the 
 
422 ELECTROMAGNETIC THEORY. CH. IV 
 
 actual speed of working was always pushed up to the greatest 
 possible (the press of business being such as to make 25 hours' 
 work per day all too short), and that the automatic character 
 of the signalling made the fixation of a limiting speed, without 
 complete failure of any marks, quite definite at any particular 
 time. Practically, however, the working speed was pushed a 
 good deal higher than this limit by making regular use of 
 mutilated signals. Now it was proved by daily experience that 
 there was a remarkable sensitiveness to damp, or leakage of 
 any kind, the speed of working, conditioned by the reception 
 of good marks, going up and down like the barometer, though 
 with a far wider range than that instrument. The east land 
 line was so manifestly the offender that it came in for a great 
 deal of blame. Yet it was not a bad line by any means, and 
 its leakage would have passed unnoticed, had there not been 
 such a peculiar lowering effect on the speed connected there- 
 with. 
 
 But how about leakage in the cable itself, seeing that the 
 above results were so out of harmony with what they should 
 have been, by general reasoning regarding retardation ? This 
 question was soon answered by the automatic development of a 
 first-class leakage fault in the cable. Its effect on the speed 
 was substantially the same as that of leakage on the land-line. 
 It lowered the speed, and since it was always on, the effect was 
 permanent. The ordinary fluctuations disappeared in a great 
 measure, as of course did the fine weather high speeds. It was 
 like continuous heavy wet. As the fault got very bad the 
 speed went down to 10 words per minute or less, in spite of 
 increased battery power. 
 
 On describing this prejudicial effect of leakage to the elec- 
 trician of the cable ship which came to remove the fault (it was 
 the late Mr. S. E. Phillips, of Henley's), he assured me I was 
 quite wrong. It was impossible for a leak to act like that. So 
 said the captain, too. It was known, said they, that a leak 
 was an excellent thing. It had been found so on the French 
 Atlantic not long before, and in other cases. I fully believed 
 this, because it seemed theoretically sound, and I had heard 
 similar statements before. But they did not assist me to an 
 understanding of the patent fact that leakage could be highly 
 prejudicial instead of beneficial. In corroboration of this, when 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 423 
 
 the fault was removed, the speed went up again, and the normal 
 state of things was restored. 
 
 The next stage in this history was the arrival of that won- 
 derful instrument, Thomson's recorder, which had been recently 
 brought out. It was the original Atlantic cable pattern, with 
 20 big tray cells for the electromagnets, and a batch of smaller 
 ones for the mousemill and driving gear. This instrument 
 simplified matters greatly, for the current could be considered 
 as and seen to be a continuous function of the time, whereas 
 with the polarised recorder of Wheatstone (an instrument quite 
 unsuitable for cable work) the variations of the current could 
 be only roughly guessed. 
 
 Now, a peculiarity of the siphon recorder was that no definite 
 limiting speed was fixable. If one person thought 40 words 
 per minute fast enough for good marks ; another, more prac- 
 tised, would as easily read them at 50 or 60, and even manage 
 to make them out at 70 words per minute, though the last was 
 undoubtedly rather troublesome. The automatic transmitter 
 being still used, the dashes became big humps, and the dots 
 little ones, on the recorder slip, and at 70 words per minute, 
 although the big humps were plain enough, the little ones were 
 mostly wiped out, though it could be seen where they ought to 
 have been, so that some reading was possible. The absence of 
 a definite limiting speed under given circumstances introduced 
 a fresh complication. Nevertheless, a careful study of the 
 recorder slips taken at various speeds, but particularly in the 
 neighbourhood of the critical speed for the polarised recorder, 
 proved that the plain result of general reasoning that leakage 
 should be beneficial in general was certainly true, and, more- 
 over, furnished a partial explanation of the anomalous behaviour 
 of the polarised recorder. 
 
 The siphon (Thomson) recorder, with its shifting zero, was 
 sublimely indifferent to the vagaries of the current which were 
 so destructive of intelligible marks in a dot and dash instrument, 
 so that ordinary leakage passed quite unobserved as a rule. 
 Nevertheless, it was plainly to be seen that when there was 
 much leakage the result was to improve the distinctness of the 
 marks by reducing the range of variation of the current from 
 the zero line, and improving the dots, or small humps, relatively 
 to the dashes, or big humps, which were reduced in size. 
 
424 ELECTROMAGNETIC T11EOUV. GIL IV. 
 
 But, to clinch the matter, a second leakage fault appeared 
 in the cable whilst the siphon recorder was on, and rapidly 
 developed. Now, it was only at the west end that there was a 
 siphon recorder ; at the east end was the polarised recorder. 
 The result was that the speed from west to east went down 
 greatly as the fault developed, in accordance with prior know- 
 ledge. But the other way, from east to west, receiving with 
 the siphon recorder, the working was greatly improved. The 
 ultimate result was that when the fault got very bad no marks 
 at all could be got at the east end, even by key-sending, owing 
 to the great attenuation, whilst we could receive first rate on 
 the siphon recorder at the west end. On the very last day, 
 when communication from west to east had to be done by a 
 roundabout course, we were able to receive a long batch of, I 
 think, 150 messages at high speed (about 55 or 60 words per 
 minute) by using 80 Leclanche cells at the sending end, and 
 putting on the electromagnets all the tray batteries available, 
 including most of the small ones, to magnify the marks. Now 
 this might seem to be merely a testimony to the sensitiveness 
 of the instrument. But the significant point was that although 
 the marks were very small, in spite of magnification by increased 
 battery power at both ends, they were exceedingly clear. They 
 were the best ever got at the speed. We were practically 
 working past a fault of no resistance, for the cable soon gave 
 up the ghost. The tail end of the big batch finally degenerated 
 to a straight line, indicative of a very dead earth. 
 
 On the removal of this fault, the received current was largely 
 increased of course. But, simultaneously, the distortion of the 
 marks on the siphon recorder became huge at the speed 60. 
 It needed to be much lowered to give conveniently readable 
 marks. 
 
 Explanation of Anomalous Effects. Artificial Leaks. 
 
 214. Now the above facts gave a decisive and cumulative 
 proof that leakage was beneficial even when concentrated at a 
 single spot, provided the current-variations were recorded in a 
 simple manner. But a complete explanation of the very large 
 decrease in the speed possible when the polarised recorder was 
 used, turned out to be rather a complicated matter in detail, 
 although sufficiently simple in the fundamental principle. 
 
THEORY OF I'LANB ELECTROMAGNETIC WAVES. 425 
 
 This effect, so detrimental to an efficient use of the cable, was 
 mixed up with another one, viz., a difference in the working 
 speed according to direction, of say 25 per cent, (with the same 
 instruments), but variable with circumstances. This effect, 
 again, seemed to have two causes, one depending upon the 
 circuit, and the uncentrical position of the cable in it, indepen- 
 dent of the kind of instruments used ; whilst the other was 
 special and peculiar to the automatic transmitter and receiver 
 when combined with the uncentrical position of the cable. It 
 may be readily imagined that the resultant effect on the 
 working speed requires some study to be understood. It will 
 be sufficient here to mention the instrumental peculiarity in 
 its relation to the detrimental effect of leakage. 
 
 The original automatic transmitter of Wheatstone was con- 
 structed to make dots only upon the slip of a special receiver ; 
 there were two rows of dots, one set being for the dots, and the 
 other for the dashes of the Morse code. But when the inventor 
 wanted the E. and I. Co. to take it up, they demanded dots 
 and dashes, as usual. So the instruments had to be modified. 
 Now I have personal reason to know that a few years later the 
 inventor was quite at sea as to the reason of the peculiar effects 
 observed in working his instruments on composite circuits and 
 cables. It is therefore pretty safe to assume that previously, 
 when the conversion was made, the manner of the conversion, 
 from making double dots to dots and dashes, was dictated 
 purely by considerations of mechanical convenience, and 
 this conclusion may be made nearly certain by a comparison of 
 the original with the modified transmitter. But Mr. Stroh 
 knows all about it. At any rate, the peculiar way of making 
 the dash and the long space was the cause of all our trouble 
 with leakage, which cost the Company an immense sum of 
 money, viz., what they might have gained otherwise, for they 
 were blocked with work, and might have had much more. 
 
 To make a dot, a short positive current was sent, immediately 
 followed by a similar short negative current. To make a dash, 
 after sending the first short positive current, there was an 
 interval of no current lasting twice as long as the first current, 
 followed by a short negative current which terminated the 
 dash. Similarly, to make a long space, a short negative current 
 came first, then an interval of no current, and finally a short 
 
426 ELECTROMAGNETIC THEORY. CH. IV. 
 
 positive current. Now in these intervals of no current, the 
 sending end of the line was insulated. When sending current, 
 on the other hand, the line was to earth, through the battery. 
 Two distinct electrical arrangements were therefore operative 
 in turn, viz., a cable with both ends earthed, and a cable with 
 only one end (the receiving end) to earth. If the former sys- 
 tem always prevailed, the manner of transmission of electrical 
 changes would follow one law. If the latter system prevailed, 
 it would follow another law, and would give much greater 
 retardation. In real fact, the two ways were in alternate action 
 irregularly. This has to be considered carefully in examining 
 the influence of the uncentrical position of a cable in a compo- 
 site circuit, and also of leakage. 
 
 Now the practical effect worked out thus. There was a real 
 positive and material advantage in making the dash and long 
 space of the Morse code by a short initial current instead of a 
 long one, as in ordinary working with reversing keys. I esti- 
 mated the gain as amounting to from 100 to 125 per cent, 
 under favourable circumstances. This implies that the limiting 
 speed is really worked up to. But it was vitally necessary 
 to secure this advantage, that the insulation should be very 
 good. A peculiar balance had to be preserved between the 
 positive and the negative currents. The currents were of equal 
 duration and of alternate signs, but necessarily not equidistant 
 in time. Now, with exceedingly good insulation, the short* 
 current sent at the beginning of a dash was sufficiently prolonged 
 in reception as to not merely make a good dash firmly, but to 
 keep the balance for the next currents. But if you put a fault 
 on the line, and let the charge leak out during the dead 
 interval of the dash, then the next following negative current, 
 terminating the dash, would be too strong, so that an imme- 
 diately succeeding positive current (with a negative current after 
 it) to make a dot would miss fire altogether. Similar remarks 
 apply to the space. If the negative current that initiated the 
 space were allowed to die away by the action of a leak, the next 
 following positive current would be too strong, and cause an 
 immediately following negative current to miss fire. Remember 
 that the reception of the signals was with a polarised recorder, 
 the zero of which could not be continuously altered to catch the 
 missing currents except by a very artful person who knew what 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 427 
 
 was coming, and was quick enough to alter the regulation exactly 
 at the right moments. It will, by the above, become roughly 
 intelligible how a leak, actually quickening signals in a remark- 
 able degree, can yet be most prejudicial in a special system of 
 signalling requiring the preservation of a sort of balance. If 
 the siphon recorder be substituted the necessity of the balance 
 disappears ; but the marks would be very hard to read at the 
 same speed. To make them readable the speed should be 
 at least doubled. The dashes will then become well-formed 
 humps. 
 
 The reader who may desire to pursue the subject of leakage 
 effects further may be referred to my "Electrical Papers," 
 Vol. I., especially p. 53 for continuous leakage, p. 61 for more 
 details relating to the above-described case, and p. 71 for the 
 general theory of leaks, which is fully worked out in several 
 cases to show the expediting effect. I also there propose arti- 
 ficial leaks. The following extract (p. 77) is easy to read, and 
 so may be worth quoting : 
 
 " When a natural fault, or local defect in the insulation, is developed in 
 a cable, it tends to get worse a phenomenon, it may be observed, nob 
 confined to cable-faults. Under th action of the current the fault is 
 increased in size and reduced in resistance, and, if not removed in time, 
 ends by stopping the communication entirely. Hence the directors and 
 officials of submarine cable companies do not look upon faults with favour, 
 and a sharp look out is kept by the fault-finders for their detection and 
 subsequent removal. But an artificial fault, or connection by means of a 
 coil of fine wire between the conductor and sheathing, would not have the 
 objectionable features of a natural fault. If properly constructed it would 
 be of constant resistance, or only varying with the temperature, would 
 contain no electromotive force of polarisation, would not deteriorate, and 
 would considerably accelerate the speed of working. The best position for 
 a single fault would be the centre of the line, and perhaps ^ 5 of the line's 
 resistance would not be too low for the fault." 
 
 The effect of one leak is marked, perhaps 40 per cent. What 
 does this represent in sterling ? How many thousands or 
 tens of thousands of pounds' worth of copper and g.p., &c. f 
 would be equivalent ? If one leak would not quite do it, then 
 two would. 
 
 But to have only one or two leaks is to look upon the leak 
 as a dangerous nuisance. Although the best position for a 
 given amount of leakance in a lump is in the middle of the 
 
428 ELECTROMAGNETIC THEORY. CH. IV. 
 
 cable, yet to have the given leakance in a lump is the worst 
 arrangement as regards its accelerative efficiency. The best 
 arrangement is uniformly-spread leakance, as much as possible, 
 with, however, a thoroughly useful residual terminal current 
 to work with, which current could be varied far more rapidly 
 than is possible with a single leak. 
 
 Is it impossible to find an insulator of comparatively low 
 resistance which should be suitable in other respects ? Tele- 
 graph cable manufacturers would probably say, No. That, 
 however, need not be considered conclusive, because, as is well 
 known, when an industry or institution is once established, it 
 always gets into grooves, and has to be moved out of them by 
 external agency, if at all. But should such an insulator be un- 
 attainable, then the only alternative is to have artificial leaks, 
 as many as possible. The theory is, I think, quite plain, and 
 I know that the practice is also, up to a certain point. But 
 here be grooves again. For people working in established 
 ways have their own proper work to do, and have no time to 
 waste upon " fads." So much is this the case that it often 
 happens that they have exceedingly little knowledge of how 
 things would work out if they departed from the ruts they are 
 accustomed to run along. That is to say, the spirit of scientific 
 research, which was to some extent present in the industry in 
 its early stages, has nearly all evaporated, leaving behind 
 regular rules and a hatred of fads. I am no wild enthusiast, 
 having been a practician myself, but it has certainly been a 
 matter of somewhat mild surprise to me that cable electricians, 
 who have had such unexampled opportunities in the last 20 or 
 30 years, say, should have done nothing in the matter of leaks. 
 Perhaps there may turn out to be objections at present un- 
 thought of. The same could be said of anything untried. 
 Several cases of resistance to change of established practice, 
 where the changes ultimately turned out to be of great benefit, 
 have come under my personal observation. The proposed 
 changes were (I believe quite honestly) first scouted as fads, 
 and the innovator was snubbed ; next, under pressure, they 
 were reluctantly tried ; thirdly, adopted as a matter of course, 
 and the innovator snubbed again. But this is the nature of 
 things, and can be seen everywhere around very notably in 
 the political world. 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 429 
 
 Self-induction imparts Momentum to Waves, and that 
 carries them on. Analogy with a Flexible Cord. 
 
 215. We have now to consider in what way self-induc- 
 tion comes in to influence the transmission of waves along 
 wires. The first step in this direction was made by Kirchhoff 
 as long ago as 1858, but it is only in recent years that self- 
 induction has become prominent as an active agent, and its 
 real resultant effect worked out and understood, more or 
 less. This effect is, on first acquaintance, a somewhat sur- 
 prising and paradoxical one. Everyone knows the impeding 
 effect of self-induction in apparatus. How, then, in the name 
 of Faraday, can it be beneficial in a telegraph or telephone 
 circuit, and make signals bigger and clearer ? To answer this, 
 it will be convenient to make use of an analogy in the first 
 place. 
 
 There is, in a great measure, a formal resemblance between 
 the problem of a telephone circuit along which electromagnetic 
 disturbances are being propagated, and the mechanical problem 
 of the transverse motions of a stretched flexible cord. But to 
 make the formal resemblance be also a practical resemblance, 
 several little things have to be attended to, and reservations 
 made. Now there are a great many mechanical problems con- 
 cerning wave-motion subject to friction which make better 
 analogies for the use of the mathematical inquirer ; but the 
 case of a flexible cord is more or less familiar to everyone, and 
 is, up to a certain point, easily realisable. 
 
 The commonest knowledge of the transverse motions of a 
 stretched cord is concerning its normal vibrations, the funda- 
 mental mode and its harmonics. But we do not want them in 
 our analogy. For although we may have something similar 
 with short waves, and especially Hertzian waves, it would not 
 be easy to set up such a state of things on a telegraph circuit, 
 which is usually so long as to prevent the assumption of normal 
 modes by reflection at the ends. For the normal vibrations of 
 a cord are complex affairs, depending not only upon propagation 
 of motion along it, but also on the constraints to which it is 
 terminally or intermediately subjected. We desire the propa- 
 gational effect to show itself distinctly. But if we disturb a 
 stationary stretched wire at one end, the pulses produced run 
 
430 ELECTROMAGNETIC THEORY. CH. IV. 
 
 to and fro too quickly for practical observation. Take, then, 
 a long india-rubber cord, suspended from a height, as is some- 
 times done by lecturers. The speed of transmission of pulses 
 may be made small enough to allow the disturbances to be 
 conveniently watched. Then, if the cord hang vertically at 
 rest to begin with, and its lower end be momentarily jerked, 
 bringing it immediately back to its original position, a hump is 
 generated, which runs along to the fixed end. It is not much 
 changed in transit, and is reflected back, and re-reflected many 
 times. But we only want to consider the passage of a hump 
 one way. Properly speaking, then, we should have so long a 
 cord that reflection is (by frictional loss) insignificant. Per- 
 haps a very long cord laid upon a surface of ice would be more 
 suitable for the purpose. But if the " fixed end " be movable 
 transversely, and its motion be resisted by a force proportional 
 to its velocity, their constant ratio may have such a value 
 given to it (depending upon the mass and tension of the cord), 
 that the arriving disturbances do not produce any reflected 
 waves, but areabsorbed by the terminal resistance. The cord 
 then behaves as if it were infinitely long. This is theory, of 
 course, but I dare say a sufficiently good practical imitation 
 could be got by means of a terminal block sliding upon a wire 
 with friction. An imperfect cancellation of the reflection is 
 easily got. 
 
 Disregarding the slow change of type and the attenuation, 
 so that there is undistorted propagation of a hump, or of a 
 succession of humps making a train of waves going one way 
 only, this case is analogous to the transmission of waves along 
 a uniform circuit of no resistance. The two constants L and 
 S in the electrical case are replaced by another two, viz., the 
 density (linear, or mass divided by length) and tension of the 
 cord. Thus, large permittance, that is, small elastance, means 
 small tension, or a slack cord ; and large inductance means a 
 massive cord. If I said a " heavy " cord, it might be thought 
 that its weight was concerned in the matter in question, but 
 it is not, save as a disturbing factor. The transverse displace- 
 ment of the cord may be compared with the transverse voltage 
 in the electrical circuit. Other comparisons (theoretically 
 better) may be made, but this is perhaps the most convenient 
 if the motion of humps be actually watched. 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 431 
 
 Now to imitate the action of the resistance (assumed to be 
 constant) of an electrical circuit, the transverse motion of the 
 cord must be frictionally resisted, and the frictionality (or 
 coefficient of friction) should be constant. That is, the 
 frictional force should be proportional to the transverse 
 velocity. The motion of the cord is frictionally resisted, both 
 internally and externally. It is very unlikely that this should 
 give rise to a constant frictionality, but there is a similarity 
 insofar as the waves sent along the cord do attenuate and are 
 distorted as they progress. 
 
 But the friction is not anything like enough in the case of a 
 cord in air to imitate the submarine cable ; besides that, the 
 mass is too great, and likewise the tension. Even to copy 
 telephonic waves going along very long copper wires of low 
 resistance, we should increase the friction in the mechanical 
 analogue if we desire to have an equal amount of attenuation 
 in a wave length. The cord should, therefore, be in a viscous 
 medium, and the viscous resistance to the cord's motion should 
 be so great as to produce a marked attenuation during the 
 transit of a hump, though it should not be so great as to 
 prevent its distinct propagation as a hump, without great dis- 
 tortion of shape and spreading out. Under these circumstances 
 we may exemplify low-resistance, long-distance, overland tele- 
 phony, where the inductance is relatively large and the 
 permittance relatively small. 
 
 But to imitate a submarine- cable we should have a slack 
 cord of small mass in a highly viscous medium. We can then 
 send a hump only a little way along it as a hump, for it 
 becomes greatly distorted and diffused. Consequently, if we 
 compared the displacements at a great distance along the cord 
 with those impressed upon it at the free end, we should find a 
 state of things quite different from what occurs in the case of 
 an india-rubber cord in air. The main features of the originated 
 disturbances might be recognised, unless the attenuation were 
 too great, but they would be out of all proportion. In particular, 
 there is a tendency to obliterate small humps, or ripples whilst 
 the big ones persist. The propagation is of the sluggish kind 
 characteristic of diffusion, quite unlike that of purely elastic 
 waves. If the slack cord could have no mass at all, we should 
 then imitate a submarine cable wholly without inductance. 
 
432 ELECTROMAGNETIC THEORY. CH. IV. 
 
 Under these circumstances we could convert the diffusive 
 waves to something like elastic waves by giving mass to the 
 cord. Without changing either its tension or the frictionality, 
 we could, by a continuous increase of density, cause the nature 
 of propagation to pass continuously from the sluggish, diffusive 
 kind to the elastic kind with approximate preservation of form 
 of waves. We might also do the same by -increasing the 
 tension and reducing the frictionality, keeping the density 
 small (like, by reducing the permittance and the resistance 
 in an electrical circuit, bringing the inductance into impor- 
 tance). But these are not immediately in question. By 
 increasing the mass, or equivalently the inertia of the cord, 
 we impart momentum to the waves, which allows them to 
 be carried forward, in spite of the little tension, against the 
 resistance of the surrounding medium. 
 
 So it is in the electrical case. By increasing the inductance 
 we impart momentum to the waves, and that carries them on. 
 This statement is independent of the particular mechanical 
 analogy that may be employed for illustration. It is probably 
 representative of the actual dynamics of the matter, associating 
 L with inertia, and LC with momentum. The soundness of 
 this dynamical explanation (made by me some years since in 
 asserting and proving the fact concerned) was endorsed by Lord 
 Kelvin. And I am proud to add, for the benefit of those who 
 may excuse themselves from understanding me by the plea 
 that I am a Schopenhauer (!), that Lord Kelvin expressed his 
 appreciation of the very practical way in which I had reduced 
 my theory to practice. 
 
 Many other dynamical arrangements might be alluded to, 
 exemplifying the power of inertial momentum to overcome 
 frictional resistance, but it is not easy to get a good one 
 exhibiting elastic waves plainly. 
 
 To illustrate the nature, and to emphasise the possible 
 magnitude of the effect, suppose we have the power of 
 continuously increasing the inductance of an Atlantic cable 
 from a small quantity up to a very large one, without altering 
 its resistance or permittance, and without introducing any 
 other actions than those to be controlled by resistance, 
 permittance, and inductance. Starting with L = 0, and making, 
 say, 1,000 waves per second \t one end, there will be 1,000 
 
THEORY OP PLANE ELECTROMAQNBTIC WAVES. 433 
 
 waves per second of workable magnitude only in a small part of 
 the cable near the sending end ; the amplitude will attenuate 
 to insignificance, not exactly in the shore-end, but before 100 
 miles is traversed. But let waves of the same size (which will 
 need greater battery power to produce) be kept up at the 
 sending end whilst L is continuously increased all along the 
 cable. The region of practical waves will continuously extend 
 itself right through the cable, by a partial conversion to elastic 
 waves, with little attenuation in transit. With very big 
 inductance the waves would reach the distant end nearly in 
 full size. Even then they could be doubled in size by the 
 reflection, though there would be no useful purpose served by 
 that. The amount of inductance required, however, to produce 
 these results would be out of all reason. Nevertheless, the 
 illustration serves its purpose in showing the action of self- 
 induction, which is, as before stated, wholly beneficial, as it 
 increases the amplitude and lessens the distortion. Some 
 remarks on increasing inductance will come later. 
 
 Self-induction combined with Leaks. The Bridge System 
 of Mr. A. W. Heaviside, and suggested Distortionless 
 Circuit. 
 
 216. In the meantime, some few remarks on self-induction 
 combined with leakage should come in here. Self-induction 
 came in with the telephone. That was, originally, a mere 
 house-to-house affair. At any rate, lines were only a few miles 
 long. So condenser action was out of practical question then. 
 On the other hand, the frequency of the currents concerned is 
 very great, and that makes inertia important. So the theory 
 of telephones is a magnetic affair when the lines are short 
 enough, just as if they were lines in a laboratory. 
 
 But the telephone soon got beyond that, and it became 
 necessary, with the use of longer and longer lines and under- 
 ground lines, to examine the influence of their permittance in 
 conjunction with their inductance. In hardly any case of 
 telephony can we ignore the self-induction, and take the 
 permittance alone. The resistance would need to be so large 
 that the lines would be wholly unsuitable for telephony, except 
 for mere local lines. So permittance and inductance usually 
 
 pp 
 
434 ELECTROMAGNETIC THEORY. CH. IV. 
 
 go together in considering telephony, and if one of them is 
 ignorable it is the permittance rather than the inductance. 
 
 Following the telephone came exchanges. But there was an 
 exchange before telephones. My brother, Mr. A. W. Heaviside, 
 had an intercommunication exchange at Newcastle, using 
 alphabetical indicators, whereof the manipulation resembled 
 that of the hurdy-gurdy. These were beautiful instruments 
 mechanically ; but, of course, the telephone soon stopped their 
 manufacture. The exchange referred to was then turned into 
 a telephone exchange. Now, in certain cases it was needed to 
 have many instruments in one circuit, with local intercommu- 
 nication ; a sort of family arrangement, so to speak ; for 
 example, a number of coal pits and colliery offices belonging 
 to one firm. The custom was to set all the intermediates in 
 sequence (otherwise, but perhaps not so well, called in series) 
 In the circuit. Why was this done? Simply because they 
 knew no better. It was the custom to do so in the telegraphs 
 generally, being an obvious arrangement not so much in 
 England, though, as elsewhere. But when this was done with 
 telephones, or with call-instruments replacing them, it was 
 found that the intermediates had a very deleterious influence, 
 which very soon set a limit to the admissible number of inter- 
 mediates on a circuit and to the length of the circuit, especially 
 when underground wires were included. My brother, who took 
 up the telephone with ardour from the first (making the first 
 one made in England, I believe), found by experiment that it 
 was quite unnecessary to put the instruments in sequence 
 according to old practice, and that an immense improvement 
 was made by taking them out of the circuit, and putting them 
 across it, as shunts or bridges. Like lamps in parallel, in fact. 
 It seems very easy and obvious now. But it is not so easy to 
 conceive the state of mind of old stagers. Anyhow, the pro- 
 position to put the instruments in bridge was pooh-poohed at 
 first, not being understood. As, however, it was demonstrably 
 a wonderful improvement it was adopted, first at Newcastle, 
 and then elsewhere, and became known as the Bridge System. 
 
 When this system was brought to my knowledge, and I was 
 asked for theoretical explanation, it presented itself in a double 
 aspect. First of all, there was the complete removal from 
 the circuit of intermediate impedance in big lumps. This was 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 435 
 
 manifestly the main reason of the increase in working distance 
 possible, and why a far greater number of intermediates co^ld 
 be put on. For they were not put in the circuit, but on it, st> 
 to speak. 
 
 The other aspect of the matter was this. Considering that the 
 bridge system consists electrically of a circuit with a number of 
 leaks on it, and bearing in mind my old investigations concern- 
 ing leaks, I inquired whether the intermediates were not ac- 
 tually beneficial to through communication, independently of 
 the previous effect. Was not the articulation better, not merely 
 than if the intermediates were in series in the circuit, but better 
 than if the shunts or bridges were removed altogether, leaving 
 insulation ? My brother's answer was Yes, certainly, in certain 
 cases, b*ut doubtful in others. It appeared, however, to be only 
 a minor effect, of not much importance compared with the major 
 one. Besides, the bridges caused a weakening of the intensity 
 of the speech received when many bridges were passed. To pre- 
 vent this weakening becoming inconveniently great, the inter- 
 mediate call-instruments in bridge were purposely made to have 
 considerable resistance and inductance, far more than would be 
 needed or desirable for their natural use. This tended to 
 prevent the currents passing along the line from entering 
 the shunts, and especially so as regards the currents of high 
 frequency, and allowed them to be transmitted in greater 
 magnitude. 
 
 It will be observed here that the expediting effect of leakage 
 considered before ( 213, 214) is merely incidental, and is, in 
 fact, partly destroyed by the large impedance of the bridges, 
 in order to mitigate the attenuative evil. The shunts are 
 inductive shunts, intentionally very much inductive. Now, 
 my proposed leaks (214) for submarine cables were non- 
 inductive, and intentionally so, though it may be remarked 
 that it would make very little difference with the slow signalling 
 of Atlantic cables if they were inductive. On the other hand, 
 a recent proposal of Prof. Silvanus Thompson is to use induc- 
 tive shunts on cables. The arrangement is electrically as in 
 the bridge system, but the object is different. This point will 
 be returned to. 
 
 The investigation of this bridge system of telephony suggested 
 to me the distortionless circuit, as I have before acknowledged 
 
 FP2 
 
436 ELECTROMAGNETIC THEORY. CH. IV. 
 
 ("Electrical Papers," Vol. II., p. 402). It may be of interest 
 to state how it came about, since it is not presented to one by the 
 above-described experiences with the bridge system. The ques- 
 tion was, What was the effect of a bridge, or of a succession of 
 bridges, when the self-induction of the line was not negligible ? 
 Now the influence of the bridges can be very readily stated for 
 steady currents, of course. There is no difficulty, either, in 
 finding the full formulae required to express the voltage and 
 current and their variations in any part of the system when 
 the inductance and permittance are operative, as well as a set 
 of leaks, when one knows how to do it. But the general 
 formulae thus obtained are altogether too complicated to be 
 readily interpretable. I was, therefore, led to examine the 
 effect produced by a leak on a wave passing it of the" elastic 
 type. This implies that the resistance of the line is sufficiently 
 low for waves of the frequency concerned to retain that type 
 approximately. (Ln should be greater than R.) Now this was 
 comparatively an easy problem, and the result was to show that 
 a leak (or conductance in bridge) had the opposite effect on a 
 wave passing it along the line to that of a resistance inserted 
 in the circuit at the same place, in this sense : A resistance in 
 the circuit reflects the charge positively and the current 
 negatively, or increases the charge and reduces the current 
 behind the resistance. On the other hand, a leak reflects 
 the current positively and the charge negatively, or in- 
 creases the current and reduces the charge behind the 
 leak. (See also 197, 209.) A resistance and a leak 
 together at the same place therefore tend to counteract one 
 another in producing a reflected wave, leaving only an at- 
 tenuating effect on the transmitted wave. This led to the 
 distortionless circuit. For although the compensation of the 
 resistance in the circuit and the conductance across it is im- 
 perfect when they are finite, it becomes perfect when they are 
 infinitely small. They may then be a part of the circuit itself, 
 namely, the resistance a part of the resistance of the line, and 
 the conductance a part of the leakage. We now have perfect 
 compensation with this one, or any number of successive resist- 
 ances and leaks of the same kind, provided their ratio is pro- 
 perly chosen. This occurs when R and K, the resistance and 
 conductance per unit length of circuit, of the line and of the 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 437 
 
 leakage respectively, are taken in the same ratio as the two 
 other constants, L and S. Or, when L/R = S/K, indicating 
 equality of the electric and magnetic time-constants. There is 
 now an undistorted transmission of any kind of signals. And 
 since every circuit has R, L, and S, it follows that it can be 
 made distortionless by the addition of leakage alone of a certain 
 amount, or distortionless to the extent permitted by the ap- 
 proximate constancy of R, L, and S. (See " Electrical Papers," 
 Vol. II., pp. 119 to 168, for detailed theory of the distortionless 
 circuit.) 
 
 In 213 we saw that even in the electrostatic theory leakage 
 would allow us to signal as fast as we liked provided we could 
 do it with an infinitesimal current. This was not because the 
 circuit was made distortionless, but on account of the infinitely 
 rapid charge and discharge. Now the inclusion of the in- 
 fluence of self-induction shows that we can do the same with a 
 finite leakage conductance, and have finite distortionless sig- 
 nals, due to waves of the purely elastic type. Given R and S 
 then, if L should be so small that the introduction of K of the 
 proper amount to reach the distortionless condition should 
 produce unreasonable attenuation, then one way of curing this 
 would be to increase L. For then a smaller amount of leakage 
 will serve. The bigger L, the smaller need K be, and when it 
 is very big, then very little K is needed. This brings us round 
 to the case at the end of 215, where we had elastic waves 
 with little attenuation on an Atlantic cable got without any 
 leakage, and since we see now that very little leakage is needed 
 to bring about the distortionless condition, we conclude that it 
 is approximately distortionless without the leakage. 
 
 Evidence in Favour of Self-induction. Condition of First-Class 
 Telephony. Importance of the Magnetic Reactance. 
 
 217. Let us now ask what is the evidence in favour of the 
 beneficial action of self-induction. There was plenty in 1887; 
 it is overwhelming now. But there are two sorts of evidence, 
 the theoretical and the experiential. The two are not funda- 
 mentally different, however ; only the former is more indirect 
 than the latter. We have certain electrical laws established, 
 mainly by laboratory work followed by mathematical inves- 
 
138 BLBCTROMAGi^ExiC THEORY. CH. IV. 
 
 tigation, and there is no reason whatever to suppose that they 
 do not hold good out of doors as well as inside. The important 
 thing is to correctly recognise the prevailing conditions and 
 determine their results. Now, if you can be pretty sure that 
 you have done this, and have confidence in your investigations, 
 then you may consider that you prove a thing, even though 
 there may be no direct evidence. Discoveries are frequently 
 made in this way. " A foundation of experimental fact there 
 must be ; but upon this a great structure of theoretical de- 
 duction can be based, all rigidly connected together by pure 
 reasoning, and all necessarily as true as the premises, provided 
 no mistake is made. To guard against the possibility of mis- 
 take and oversight, especially oversight, all conclusions must 
 sooner or later be brought to the test of experiment ; and if 
 disagreeing therewith, the theory itself must be re-examined, and 
 the flaw discovered, or else the theory must be abandoned." 
 (Oliver Lodge.) But to convince other people is quite another 
 matter. They may not be competent to understand the 
 evidence. Or they may be fully competent, but not have suffi- 
 cient acquaintance with the subject ; or not have time to 
 examine it ; or not have the energy ; or have no interest in it. 
 Then, as Elijah said to the priests of Baal, you must " Call him 
 louder!" 
 
 Now, the foundation of fact is the experimental facts of elec- 
 tricity and the laws deduced therefrom, though these laws are 
 sometimes more comprehensive than the facts upon which they 
 are based. But there was, in 1887, a certain amount of direct 
 evidence as well, quite enough to make the theoretical con- 
 clusions practically certain, by the confirmation they afforded. 
 Assuming the electrostatic theory to be true in telephony (for 
 which, however, there waa never any sufficient warrant) we can 
 calculate its consequences. They indicate that the product 
 RSJ 2 of the total resistance and the total permittance limits 
 the distance I through which telephony is possible. How big 
 it may be is questionable, on account of the very complicated 
 nature of the problem when the human is taken in as a part 
 of the mechanism. We cannot say certainly beforehand how 
 much distortion is permissible before the human will fail to 
 recognise certain sounds as speech. But the electrostatic 
 theory shows such very considerable distortion even when 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 439 
 
 RSI 2 is as low as 5,000 or 7,000 ohms x microfarads, that it 
 is scarcely credible that any sort of telephony could be prac- 
 ticable if it were much bigger, as 10,000 for example. Perhaps 
 5,000 might be taken as the practical limit for rough purposes. 
 But experience showed that good telephony was got up to 
 10,000 and higher, in this country with circuits including 
 underground cables, as well as in America, where long-distance 
 telephony had already made a good beginning. Manifestly the 
 electrostatic theory was erroneous by direct as well as indirect 
 evidence. Now the inclusion of the influence of self-induction 
 in the theory explained these results (and also the failure of 
 iron wires and the success of copper) and pointed out how to 
 improve upon them, viz., by directing us to lower R and 
 increase L in order to increase the value of RS 2 possible. So 
 there was sufficient direct evidence then to satisfy anyone com- 
 petent to judge who considered it fairly without being pre- 
 judiced by the law of the squares. Since that time, the 
 amount of this direct evidence has been greatly multiplied, 
 and RS 2 has got up to 50,000, which may be about ten times 
 as great as the probable value under purely electrostatic con- 
 ditions. But there is nothing critical about 50,000. It might 
 be 500,000 if the circuit were long enough. We may need to 
 reduce R when we increase Z, but not in proportion to the 
 inverse square of Z, or anything like it. 
 
 Any long circuit may be made approximately distortionless 
 if the magnetic side of the phenomena can be made important, 
 and without the inconvenience of the leakage needed to remove 
 the distortion, which leakage, however, is essential when the 
 magnetic side cannot be made important. The useful guide in 
 considering telephony is the size of the quantity R/Ln, the 
 ratio of the resistance to the reactance, the two terms appear- 
 ing in (R 2 + L 2 tt 2 )*, the impedance of unit length of circuit, 
 apart from the permittance. 
 
 The term " reactance " was lately proposed in France, and 
 seems to me to be a practical word. It may be generalised to 
 signify the value of Lrc, positive or negative, of any combina- 
 tion of coils and condensers, L being then the effective induct- 
 ance. When L and the reactance are positive, the magnetic 
 side prevails and the current lags ; and when it is negative, 
 the electric side prevails and the current leads. 
 
440 ELECTROMAGNETIC THEORY. CH. IV. 
 
 At present, however, R and L are as before. Now the ratio 
 R/L/7& may be very large. If so, the influence of L is small, 
 and we shall have something like the electrostatic theory, 
 provided the range of n be such as to make the ratio always 
 large. But this state of things is quite unsuitable for long- 
 distance telephony. The most suitable state is when R/Lw- is 
 a proper fraction throughout the whole range of n ; that is, 
 when R is less than the smallest value of Im. We shall then 
 have approximately distortionless propagation of the type 
 
 as may be seen by examining the solution for simply periodic 
 waves, and observing that when R/Lw- is small we have nearly 
 the same attenuation in a given distance at all frequencies (not 
 too low), so that there is little distortion, and the element of 
 frequency tends to disappear altogether, and we reduce the 
 formula to the above type. 
 
 Practice shows, however, that this state of things is unneces- 
 sarily good for commercial telephony, which admits of a 
 considerable amount of distortion. Then R/Lw, is not less 
 than 1 throughout the whole range of n, but is greater at the 
 low frequencies and less at the higher.* For example, if the 
 lowest value of n be 625, and L = 20 (centim. per centim., usual 
 units), the least value of IM is 12,500. If also the resistance 
 is 5 ohms per kilometre, then R is 50,000. The value of 
 R/L is then 4 at the lowest frequency, and 2, 1, J, , J at 
 the five successive octaves, the last being at n= 20,000. Here 
 we have an excellent state of things from n= 2,500 up to 
 20,000, and only at the low frequencies does R prevail over 
 Lra. If R be halved, or L be doubled (for it is indifferent 
 which is done), then R/Lra is less than 1 from = 1,250 
 upwards. Divide n by 2?r to obtain he frequency. This 2?r 
 has nothing to do with the ridiculous 4?r which the B.A. 
 Committee want to consecrate. The significant point in calcu- 
 lations of this kind is the position of n in the scale of frequency 
 which produces equality of R and Tun. When it is low we 
 have good quality ; when high, then relatively bad. Perhaps 
 the American electricians may be able from their extended 
 
 * See my " Electrical Papers " for details ; especially App. C, p. 339, 
 Vol. II. 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 441 
 
 experience to give practical information on this and other 
 points concerned in telephony, and construct an empirical 
 formula of a sound character for determining the limiting dis- 
 tance roughly in terms of the electrical data, with given kinds 
 of instruments for transmission and reception. It should be 
 understood that the above remarks do not apply to short lines, 
 at least in general. We may, indeed, so arrange matters that 
 the above type is still preserved (with R/Lw, less than 1) by 
 practical annihilation of reflected waves, but otherwise serious 
 modification may be needed. And with very short lines we 
 may apply a purely magnetic theory, of course. 
 
 Various Ways, good and bad, of increasing the Inductance 
 of Circuits. 
 
 218. Supposing, now, that practicians have come to discard 
 the supposed limitations so persistently asserted by official 
 electricians in England, and to open their eyes to the part 
 played by self-induction, they will naturally recognise the wide 
 possibilities that are suggested in future developments, and 
 want to know whether such possibilities can be converted into 
 actualities, and whether the conversion is practicable. For 
 myself, I am not much concerned in this part of the question. 
 It is for practicians to find out practical ways of doing things 
 that theory proves to be possible, or not to find them if they 
 should be impracticable. Nevertheless, since I am responsible 
 for these views concerning the parts played by self-induction 
 and leakage, I add a few remarks on the applicational side. 
 
 The ideal is the distortionless circuit, requiring both self- 
 induction and leakage. If we utilise the present existent 
 inductance only, then the needed leakage may be too great by 
 far in many cases. On the other hand, if the existent induct- 
 ance is big enough to make R/Lw. small, in the sense described 
 in 217, we do not want the leakage particularly, or not at all 
 practically, it may be. It is clear, then, that we should aim 
 at making L/R, the magnetic time- constant, as big as possible. 
 
 Now, here is one way of doing it. If, a few years ago, an old 
 practitioner was asked to put up a long-distance telephone circuit, 
 he, being under the belief that R should be high, or at least 
 that it need not be small, and that L should be as small as 
 
442 ELECTROMAGNETIC THEORY. x CH. IV. 
 
 possible, in order to prevent the deleterious action of self- 
 induction, would probably carry out his principles by putting 
 up a circuit, several hundred miles long, consisting of a pair of 
 fine wires, put quite close together. Or, if he was free from the 
 error as regards R, he would use larger wires. But still close 
 together, for that makes L the smallest. Also, he would not 
 consider this to be bad as regards S, the permittance, being 
 under the belief that the permittance of the circuit of two wires 
 was just one half that of either with respect to the earth, so 
 that their proximity did no harm in increasing condenser action, 
 whilst it destroyed the deleterious self-induction. Now, such 
 a circuit might do very well for local telephony, but for the 
 purpose required there could not be a worse arrangement. 
 The supposed case is not an extravagant one, since it involves 
 the carrying out of official views thoroughly. The cure would 
 be quite easy. Increase the inductance by separating the wires 
 as widely as possible or convenient. If necessary, reduce R as 
 well. The permittance will also be reduced in about the same 
 ratio as the inductance is increased. By this simple process L 
 may be largely multiplied, say from 2 to 20 (cm. per cm.), a 
 tenfold magnification, without separating the wires so much as 
 to require a double line of poles. This way of doing it is the 
 secret of the success of the long-distance telephony which arose in 
 America when iron was discarded. L is made important, and 
 R is kept low. But although L increases very fast on initial 
 separation of wires, it does not continue to increase fast on 
 further separation, so that practical limitations are soon set to 
 the process. The above example shows how a very bad line 
 may be turned into a very good one by increasing the inductance. 
 Another way is suggested, but only to be immediately 
 rejected. It is, nevertheless, instructive to study failures. 
 We can increase L to any extent by using finer and finer wires. 
 This would be admirable if it were all. But, unfortunately, 
 this process alters R, and faster still ; so that R/L increases 
 instead of diminishing. Consider, for simplicity, merely a wire 
 with a concentric tube for return. Then R/L is nearly propor- 
 tional to RS. Now Lord Kelvin gave a formula long ago, 
 showing that this quantity was made a minimum when the 
 radii of the tube and wire bore a certain ratio. Then L/R is 
 a maximum, approximately, and on decreasing the radius of 
 
O^^ 
 
 THEORY OF PLANE ELECTROMAGNETIC WAVES. 443 
 
 the wire it falls off, ultimately very rapidly, which is the 
 reverse of what is wanted. 
 
 A third way must also be immediately rejected. By using 
 iron wires we can increase L largely with steady currents. 
 But then the skin effect, or imperfect penetration, becomes 
 important with telephonic currents in iron wires. It does two 
 things, both harmful. It lowers L and increases R. So we 
 do not get the large increase of L suggested at first, but only 
 a comparatively small one with currents of great frequency, 
 and we do get a lot of extra resistance that should particularly 
 be avoided. Iron wires are failures for long-distance telephony, 
 as was found long ago. 
 
 The case of copper-coated iron or steel wires is doubtful. 
 Such wires were used in America, and it was said that they 
 were successful for long-distance telephony. But here, evidently, 
 the copper is very important in giving conductance. As regards 
 the interior iron core, it must be considered very questionable 
 tvhether it can be so good as an amount of copper of the 
 same resistance (steady). It is not well placed for increasing 
 the inductance, and the imperfect penetration will operate 
 injuriously. 
 
 If we put the iron outside the copper the iron will be better 
 situated for increasing L for steady currents (which is no good), 
 but will act injuriously with rapidly oscillating currents, and 
 partly prevent the utilisation of the conductance of the copper, 
 if it be thick enough to make the increased L important. 
 
 The above considerations show that if iron be introduced to 
 increase the inductance it should not be in the main circuit, 
 but external to it. There are, then, two principal ways sug- 
 gested. First, to load the dielectric itself with finely-divided 
 iron, and plenty of it. This is very attractive from the 
 theoretical point of view, as it results in the production of a 
 strongly magnetic insulator, which is practically homogeneous 
 in bulk, being, therefore, a sort of non-conducting iron of low 
 permeability. In this way L, as compared with the same with- 
 out the iron, may be multiplied greatly. There is a partially 
 counteractive influence, however, due to increased permittance. 
 This plan is interesting, in view of the wave theory. It is like 
 multiplying the p in ether, and lowering the speed of pro- 
 pagation, but at the same time allowing waves to keep up 
 
444 ELECTROMAGNETIC THEORY. CH. IV. 
 
 against the resistance of wires they may be running along. But 
 as regards the practicability of this plan I say nothing, except 
 that I have often smiled at this ironic insulator (in more 
 than one sense ironic), and the idea of telephoning through a 
 core resembling a poker with a copper wire run through the 
 middle. 
 
 The other way is to put the divided iron outside the working 
 dielectric. Then it had better not be divided into particles, 
 but in the way well known to dynamo and transformer people, 
 cut up so as to facilitate the flux of induction, whilst being 
 electrically non-conducting transversely thereto. 
 
 Now it may be said that submarine cables of the present type, 
 having cores surrounded by a large quantity of iron, have large 
 inductance already. But it does not follow. If the iron was in 
 a solid tube, then undoubtedly L would be much magnified 
 with steady currents. But it is not a solid tube, as it consists 
 of spirally laid separate stout iron wires. The magnetic con- 
 tinuity is interrupted, and the permeance is much reduced, as 
 Prof. Hughes pointed out (equivalently) some years ago. But 
 besides that, when the current is not steady, we do not utilise 
 the iron even in the above imperfect manner. This is par- 
 ticularly the case with telephonic currents, when imperfect 
 penetration will still further reduce the effective inductance. 
 So the increased L due to the sheath must be far less than 
 large quantity of iron might suggest. Again, there is the the 
 increased resistance due to the sheath not being properly 
 divided to prevent it. The case is therefore a considerably 
 mixed one. Whilst we can say with confidence that with the 
 very slow signalling which obtains through an Atlantic cable 
 the sheath cannot make much difference, it is not easy to 
 decide whether its action is beneficial with telephonic currents to 
 any great extent, though it would be if there were no increased 
 resistance. It is not even easy to say what is the L of a sub- 
 marine cable under the circumstances. A least value is readily 
 obtained, but the magnitude of the increase due to the iron and 
 the outside return current is rather speculative. It may be 
 well deserving of attention to change the type of the iron 
 sheathing, so as to increase L and keep down R. Remember 
 here, that although there is plenty of evidence of the beneficial 
 (and very important) influence of this principle when it is 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 445 
 
 carried out in a direct and natural manner, there is, so far as I 
 am aware, no experimental evidence yet published of the equi- 
 valent success of indirect ways involving the use of iron. 
 
 Another indirect way is this. Instead of trying to get large 
 uniformly spread inductance, try to get a large average induc- 
 tance. Or, combine the two, and have large distributed 
 inductance together with inductance in isolated lumps. This 
 means the insertion of inductance coils at intervals in the 
 main circuit. That is to say, just as the effect of uniform 
 leakage may be imitated by leakage concentrated at distinct 
 points, so we should try to imitate the inertial effect of uniform 
 inductance by concentrating the inductance at distinct points. 
 The more points the better, of course. Say m coils in the 
 length /, or 2m coils of the same total inductance, and 
 therefore each of half the inductance ; or mn coils each of 
 one 7i th the inductance of the first. The electrical difficulty 
 here is that inductance coils have resistance as well, and if this 
 be too great the remedy is worse than the disease. But it would 
 seem to be sufficient if the effect of the extra resistance be 
 of minor importance compared with the effect of the increased 
 inductance. This means using coils of low resistance and the 
 largest possible time-constants. For suppose 4 ohms per kilom. 
 is the natural resistance, and there be one coil per kilom. 
 having a resistance of 1 ohm. This will raise the average 
 resistance to 5 ohms per kilom. ; and if the time-constant be 
 big enough, the extra inductance may far more than nullify 
 the resistance evil. The same reasoning applies to coils at 
 greater intervals, only of course in a more imperfect manner. 
 To get large inductance with small resistance, or, more generally, 
 to make coils having large time-constants, requires the use of 
 plenty of copper to get the conductance, and plenty of iron to 
 get the inductance, employing a properly closed magnetic circuit 
 properly divided to prevent extra resistance and cancellation 
 of the increased inductance. This plan does not belong to the 
 category of those mentioned before, which a moment's con- 
 sideration showed to be worse than useless. It is a straight- 
 forward way of increasing L largely without too much increase 
 of resistance, and may be worth working out and development. 
 But I should add that there is, so far, no direct evidence of 
 the beneficial action of inductance brought in in this way. 
 
446 ELECTROMAGNETIC THEORY. CH. IV. 
 
 The combination with leaks does not need any particular 
 mention in detail after what has been already said relating 
 to the distortionless circuit, which is the theoretical ideal, 
 any approach to which is desirable if it be done without too 
 much loss ; bearing in mind, too, that we should increase L 
 first in preference to putting on leaks first.* I have confined 
 myself so far entirely to my theory of 1886-87 and results 
 thereof. Inductive shunts involve some other considerations, 
 and a modified theory. 
 
 Effective Resistance and Inductance of a Combination when 
 regarded as a Coil, and Effective Conductance and Per- 
 mittance when regarded as a Condenser. 
 
 219. In any electromagnetic combination the effects of 
 electric and magnetic energy are antagonistic in some respects. 
 Or we may say that magnetic inductances tend to neutralise 
 electric permittances. Or, more generally, that the effects of 
 elastic compliance and of inertia tend to neutralise one another. 
 Thus, when a coil is under the action of an impressed simply 
 periodic voltage, the current lags, owing to the self-induction. 
 But if a condenser be introduced in sequence with the coil, the 
 lag is diminished, and may be reversed or converted into a lead. 
 The result on the current is the same as if we substituted for 
 the coil another of the same resistance and of reduced in- 
 ductance, or even of negative inductance. Thus, the resistance 
 operator of the coil being R + L^, where R and L are its 
 resistance and inductance, and p is the differentiator djdt ; and 
 that of the condenser being (Sp)" 1 , where S is its permittance, 
 when the two are in sequence the resistance operator of the 
 combination is 
 
 Z = R + Lp + (Sp)-i, (1) 
 
 which, in a simply periodic state of frequency n/'2ir t making 
 ni, becomes 
 
 R + /L-AV . . 
 
 * But remember the influence of the frequency in conjunction with the 
 inductance. Leakage can be very beneficial when self-induction does next 
 to nothing, as before described. 
 
THEORY OP PLANE ELECTROMAGNETIC WAVES. 447 
 
 This shows that the effective inductance of the condenser, 
 which is - (S/& 2 )" 1 , is additive with real inductance when in 
 sequence therewith, and being negative, reduces the effective 
 inductance of the combination from L to L - (S^ 2 )" 1 . 
 
 But when the condenser and coil are in parallel, it is more 
 convenient to use the conductance operators. Thus 
 
 Y = (R + L^)- 1 + Sp ..... (3) 
 
 is the conductance operator of the coil and condenser in parallel, 
 being the sum of the conductance operators of the coil and con- 
 denser taken separately. And when p = ni, we convert Y to 
 
 R 
 
 Y- 
 
 from which we see that the combination is equivalent to a 
 condenser of conductance R (R 2 + L 2 ^ 2 )" 1 and of permittance 
 S - L (R 2 + L 2 ^ 2 )" 1 . The steady conductance being R" 1 , we see 
 that the effective conductance is reduced by the self-induction. 
 At the same time the effective permittance is reduced, and may 
 become negative. Remember that in (3) and (4) the standard 
 of comparison is a condenser, not a coil ; whereas in the former 
 case (1), (2) the standard of comparison is a coil, not a con- 
 denser. For further information regarding resistance and con- 
 ductance operators sec " Electrical Papers," Vol. II., p. 355, and 
 elsewhere. The present remarks are merely definitive and 
 introductory to the theory of waves sent along a circuit, either 
 naturally, or partly controlled by subsidiary arrangements. The 
 comparison with a coil is, I believe, more generally useful. 
 Then we imagine any combination to be replaced by a simple 
 coil whose resistance and inductance are those (effectively) of 
 the combination. But there are cases when the other way is 
 preferable. Then we imagine the combination replaced by 
 a condenser whose conductance and permittance are those 
 (effectively) of the combination. And in the following it will be 
 convenient to use both ways in the same investigation. 
 
 Inductive Leaks applied to Submarine Gables. 
 220. Now imagine the permittance of a submarine cable to 
 be concentrated in lumps at a number of points. Let S be the 
 
448 ELECTROMAGNETIC THEORY. CH. IV. 
 
 permittance of one of the equivalent condensers. If it be 
 shunted by a coil of such R and L that S = L (R 2 + lAi 2 )" 1 , we 
 know by the above that the state of the cable, when simply 
 periodic, will be the same as if the condenser and coil were 
 removed and replaced by a simple leak of conductance 
 R (R 2 + L 2 ?! 2 )"" 1 . This reasoning applies to every condenser, if 
 it have its appropriate coil, or to any selected group we please. 
 But if the frequency be changed the permittance will come into 
 sight again, of the real or positive kind, or of the fictitious 
 negative kind, as the case may be. We have, therefore, the 
 power of reducing the effective permittance of a cable under 
 simply periodic forces, by means of numerous auxiliary induc- 
 tive shunts or leaks, and of practically cancelling it at a par- 
 ticular frequency. How will this work out in the transmission 
 of telephonic currents through the cable ? 
 
 From the purely theoretical point of view one may be some- 
 what prejudiced against the system at first. For, obviously, 
 it is not a distortionless arrangement. That is got by balancing 
 the self-induction of the main circuit against the lateral permit- 
 tance, which can, subject to practical limitations, be done per- 
 fectly. That is, there is a balance at any frequency, or the 
 idea of periodic frequency does not enter at all. I doubt 
 whether there can be any other distortionless circuit than that 
 which (in the absence of another) I always refer to as the dis- 
 tortionless circuit. With inductive shunts we produce a par- 
 tial neutralisation of the permittance, the amount of which may 
 vary very widely during the transmission of telephonic currents. 
 Remember, too, that during the change from one frequency 
 to another there are other phenomena than those concerned 
 during the maintenance of a simply periodic state of variation. 
 
 Nevertheless, we should be careful not to be prejudiced against 
 an imperfect plan merely because it is so manifestly imperfect. 
 It may perhaps be more easily ap d than a theoretically 
 perfect system, and may possess practu d advantages of import- 
 ance. Now this is a matter for practical experiment and expe- 
 rience. But it is just here that there is an almost complete 
 dearth of information. For although Pnff, Silvanus Thompson* 
 
 has mentioned that he has made manv"experiments, yet he has 
 
 fl> 
 
 * " Ocean Telephony," Chicago Congress, 1893. 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 449 
 
 only described one case in which an inductive shunt was found 
 good. It was an artificial cable of 7,000 ohms and 10 micro- 
 farads. Presuming that there was a fairly good spreading of 
 the permittance, we see that the permittance is that of about 
 40 kilometres of a submarine cable, whilst the resistance is 
 more like that of 1,000 or 2,000 kilometres. The inductance 
 would probably be very small. We see that the example does 
 not well represent a submarine cable. We should rather com- 
 pare it with a very long overland circuit of very much smaller 
 permittance per kilometre than the cable. Only, such a circuit, 
 if properly put up, would have a large inductance, so that there 
 is a failure here. However, it is mentioned that one shunt of 
 312 ohms and a time-constant of 0'005 sec. rendered tele- 
 phonic transmission possible except for shrill sounds. I cannot 
 help thinking that Prof. Thompson has been much too reticent 
 on the experimental side. He will perhaps furnish us with 
 fuller information later on. At present there is little to 
 judge by. 
 
 General Theory of Transmission of Waves along a Circuit 
 with or without Auxiliary Devices. 
 
 221. In the meantime the following theory and formulae 
 may perhaps be useful to those who may desire to go into the 
 matter. I put the theory into such a form that it may be 
 applied to various other cases of auxiliary arrangements. Let 
 V and C be the transverse voltage and the current, at distance 
 # at time t. Let their connections be 
 
 where Z is a resistance operator, and Y a conductance operator, 
 as described in 219, only now belonging to unit length of 
 circuit. In the simplest p*-p Z reduces to a resistance, and Y 
 to a conductance, per up length. But in general Z and Y 
 may have various forms, in unlimited number, being then 
 functions of electrical constants and of the time differentiator 
 p. When there are no ai xiliary devices, and the natural resist- 
 ance R, inductance L, leakance K, and permittance S (all per 
 unit length) are constants, t .en we have 
 
 .. (6) 
 
 GO 
 
450 ELECTROMAGNETIC THEORY. CH. IV. 
 
 as before, 2014. But whatever the forms of Y and Z may 
 be, in finite terms or transcendental, we may reduce them to the 
 simple standard forms (6) when the simply periodic state of 
 vibration is maintained, by employing the transformation 
 p = ni, or p 2 = - n 2 . We see from this that we can practically 
 examine cases of simple periodicity which might at first seem 
 beyond all bounds of practicability. Note, however, that 
 "forces" and "fluxes" should be in constant ratios, so that 
 such phenomena as hysteresis, magnetic or electric, are ex- 
 cluded. 
 
 From equations (5) we see that the characteristic equation 
 of Vis 
 
 ^ = YZV = ^V,say, .... (7) 
 dx 2 
 
 provided Y and Z are independent of x. This restriction is not 
 a necessary one, but, of course, it makes an important practical 
 simplification to have uniformity along the circuit. In (7) q 2 
 represents the operator YZ, and in passing we may notice an 
 interesting matter connected with partial differential equations 
 of the type (7), or of the more general type 
 
 V 2 V = 2 2 V, (8) 
 
 appropriate to three dimensions in space. This type of equa- 
 tion occurs in all sorts of physical problems, V being some 
 variable which is propagated through a medium in a manner 
 depending upon the nature of the operator q 2 , which involves 
 the time-differentiator. It is usually a very simple rational 
 function of p, such, for example, as to bring in only the first 
 and second derivatives of V with respect to the time, and, so 
 far as I know, no physical problems have presented themselves 
 which involve an irrational partial differential equation for 
 characteristic, as, for example, 
 
 V 2 V = (a + 6p + cp 2 )V. .-. . . (9) 
 
 It might, indeed, seem at first sight that a characteristic of 
 this form was physically impossible,, being meaningless. 
 Nevertheless, it is quite easy to see by the way equation (7) was 
 constructed out of the components (5), that real physical 
 problems may involve irrational characteristic equations, such 
 as are exemplified by (9). For if either Y or Z be irrational, 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 451 
 
 so is YZ. And we can easily (in imagination, not so easily in 
 execution) choose auxiliary devices making Y or Z irrational. 
 Similar remarks apply to (8). It results from the union of 
 two distinct equations involving two variables, one of which is 
 then eliminated to make the characteristic. If either of the 
 component equations involves an irrational operator, the cha- 
 racteristic resultant will be irrational. It must not be imagined 
 that the solutions of such equations obtained by physical 
 reasoning are impossible or beyond the range of mathematics, 
 or that the results are physically meaningless. Some partial 
 characteristics of the form (8), with irrational right members, 
 I have examined and solved. I should imagine that there will 
 probably be a field for equations of the kind in the future study 
 of the complicated influence of matter upon phenomena which 
 occur in their simplest manner in the ether away from matter, 
 as in the theory of dispersion of light, for example, of electric 
 absorption, and in similar subjects. 
 
 Returning to equation (7), the general solution involves two 
 arbitrary functions of the time, as in the form 
 
 V = cosh<^.V + !^^ZC ..... (10) 
 
 But it is preferable to take the case of disturbances propagated 
 from a source into an infinitely long cable, in order to eliminate 
 complications due to reflections and terminal apparatus. Then 
 
 V = -*V ...... (11) 
 
 is the solution representing V at x, t due to V given as a func- 
 tion of the time at x = Q, the origin. And the current is 
 given by 
 
 by using the first of (5). 
 
 To see the final state due to the continued action of constant 
 V , give q, Y, and Z in (11), (12), the values they assume when 
 p is put = in their general expressions, viz., for Y the steady 
 leakance and for Z the steady resistance per unit length, and 
 for q their geometric mean. To find how this state is arrived 
 at, and, more generally, to interpret (11), (12) when V is any 
 given function of the time, demands the conversion of these 
 
 GG2 
 
452 ELECTROMAGNETIC THEORY. CH. IV. 
 
 equations to proper algebraical form by execution of the analy- 
 tical operations implied in q, Y, and Z. This we are not con- 
 cerned with here, except as regards the case arising when V is 
 a simply periodic function. Then put p = ni, and reduce Z to 
 R + L#>, and Y to K + Sp, as before stated, equations (6), 
 making (11), (12) become 
 
 ........ (13) 
 
 where 2 = (R + I#)(K + Sp) ..... (15) 
 
 These expressions have next to be reduced to the form 
 (A + Bp)V , which is a matter of common algebra. When 
 done we shall find that if 
 
 P or Q = (J) [(R 2 + L 2 /* 2 )* (K 2 + S 2 ^ 2 )* (RK - LSrc 2 )]*, . (16) 
 then the V solution is 
 
 V = <J- p *sm(7i*-Qa>), ..... (17) 
 due to V = e sin nt at the origin. From this, by the use of the 
 first of (5), giving 
 
 R-L dV 
 
 we may derive the C solution. Or we may derive it by deve- 
 loping (14). The result is 
 
 C = j^ 8 [(KP + SnQ) sin + (SnP - KQ) coB](n - QOJ), (19) 
 where the value of P 2 + Q 2 is given by 
 
 . . (20) 
 
 It may be only necessary to consider the amplitudes of V 
 and C, and, of course, C is most important. That of V is 
 obviously e e" Pa! , by (17). That of C may be got from (19), and 
 is expressed by 
 
 The wave speed is n/Q, the wave length 27T/Q, and the 
 periodicity n/2ir. Another convenient form of (21) is 
 
 (22) 
 
 v 
 
 where v = (LS)-*. 
 
THEORY OF PLANE ELECTROMAGNETIC WAVES. 453 
 
 Application of above Theory to Inductive Leakance. 
 
 222. The above simply periodic solution is discussed in 
 my "Electrical Papers," Vol. II. (especially p. 396 and p. 339), 
 when R, L are the natural effective resistance and inductance, 
 and K, S the leakance and permittance per unit length, the 
 latter pair being supposed to be the same at all frequencies. 
 (A paper by Prof. Perry, Phil. Mag., August, 1893, may also 
 be consulted, but I do not think he is right in some of his 
 conclusions.) But the same formulae are applicable when 
 auxiliary devices are employed, if they are numerous enough, 
 by a process analogous to the obvious one of representing 
 large numbers of separate leaks by uniform leakance. 
 
 Thus, in the case of inductive shunts, if we treat them as a 
 set of separately located shunts, the full theory is very com- 
 plicated, because it requires a separate formula for every section 
 of the line. The best course to take is to distribute the 
 inductive leakance uniformly, in imagination, of course. The 
 result will represent the fullest possible carrying out of the 
 principle concerned, and more. Thus, let /> be the resistance, 
 and A, the inductance of the leak belonging to the unit length 
 of cable. The reciprocal of p is of course the steady leakance 
 per unit length. But it does not operate fully when the state 
 is changing, owing to A, We shall now have 
 
 (23) 
 
 where (/> + A-p)" 1 is the leakance operator, taking the place of 
 p- 1 , whilst o- is the steady permittance per unit length. We 
 have to use this special Y in the above formulae to represent 
 the effect of inductive leakage. In the simply periodic case 
 Y develops to 
 
 The first term is the effective leakance, and the coefficient of p 
 is the effective permittance per unit length. 
 
 In the developed formulae (16), (17), (19) to (22), therefore, 
 we must give K and S the values 
 
 < 25 > 
 
454 ELECTROMAGNETIC THEORY. OH. IV. 
 
 to suit the case of inductive leakage, whilst R and L remain 
 as before, the effective resistance and inductance per unit 
 length of line. The solutions should be examined through a 
 wide range of frequency in order to arrive at a good know- 
 ledge of how the circuit would behave to telephonic currents, 
 with different values of A and p. It is, of course, necessary 
 that p should not be so small as to produce too great a loss in 
 transit. 
 
 Notice that A and p are immensely big when the unit length 
 is a centimetre. Remember that m equal coils in parallel 
 behave like one coil of one wi th the resistance and inductance 
 of the individual coils. Thus, when ten shunts at the rate of 
 one per kilometre take the place of one shunt per ten kilo- 
 metres, each of the ten should have ten times the resistance and 
 inductance of the one. This leads to physically monstrous 
 results when we do as was done above, and make the action 
 uniform, as it implies infinitely large inductance of an infinitely 
 small section of the leakance. But that need not cause any 
 alarm when we are employing the ideal case for calculating 
 purposes. What is more important is that a stop would soon 
 be set to the multiplication of the shunts. 
 
 At the particular frequency making the effective permittance 
 vanish, the solution may be obtained simply from the funda- 
 mental formulae, since Y reduces to a constant, viz., the value 
 of K in (25), S being zero ; and only the rationalisation of Z 
 has to be attended to. If, further, we assume that L = 0, then 
 Z is also constant, viz., R, and the solutions are simply 
 
 (26) 
 
 V. .... (27) 
 
 (I)' 
 
 These represent a very remarkable state of affairs throughout 
 the cable. But it is thoroughly deceptive, because even if the 
 inductance of the line were quite unimportant under normal 
 conditions, the effective cancellation of the permittance would 
 make it important under the present circumstances ; that is, L 
 cannot be neglected. Including it, when S = 0, the value of P 
 reduces to 
 
 )l}J> (28) 
 
GRAVITATIONAL ANALOGY. 455 
 
 and the current amplitude, according to (21), becomes 
 
 /ri\ _ P ~ * e 
 
 in which, of course, p and Xn are not independent, having a 
 relation fixed between them by the vanishing of S in (25). By 
 giving a suitable value to <r, for instance, J microfarad per 
 kilometre (which is a suitable unit of length to employ), and 
 also fixing the frequency, we may readily apply the formulae to 
 estimate the attenuation with different values of the insulation 
 resistance or its equivalent. 
 
 APPENDIX B. 
 
 A GRAVITATIONAL AND ELECTROMAGNETIC 
 
 ANALOGY. 
 
 Part I. 
 
 To form any notion at all of the flux of gravitational 
 energy, we must first localise the energy. In this respect it 
 resembles the legendary hare in the cookery book. Whether 
 the notion will turn out to be a useful one is a matter for 
 subsequent discovery. For this, also, there is a well-known 
 gastronomical analogy. 
 
 Now, bearing in mind the successful manner in which Max- 
 well's localisation of electric and magnetic energy in his ether 
 lends itself to theoretical reasoning, the suggestion is very 
 natural that we should attempt to localise gravitational energy 
 in a similar manner, its density to depend upon the square of 
 the intensity of the force, especially because the law of the 
 inverse squares is involved throughout. 
 
 Certain portions of space are supposed to be occupied by 
 matter, and its amount is supposed to be invariable. Further- 
 more, it is assumed to have personal identity, so that the posi- 
 tion and motion of a definite particle of matter are definite, at 
 any rate relative to an assumed fixed space. Matter is recog- 
 
456 ELECTROMAGNETIC THEORY. APP. B. 
 
 nised by the property of inertia, whereby it tends to persist in 
 the state of motion it possesses ; and any change in the motion 
 is ascribed to the action of force, of which the proper measure 
 is, therefore, the rate of change of quantity of motion, or 
 momentum. 
 
 Let p be the density of matter, and e the intensity of force, 
 or the force per unit matter, then 
 
 r=e/> ....... (i) 
 
 expresses the moving force on p, which has its equivalent in 
 increase of the momentum. There are so many forces nowa- 
 days of a generalised nature, that perhaps the expression 
 " moving force " may be permitted for distinctness, although it 
 may have been formerly abused and afterwards tabooed. 
 
 Now the force F, or the intensity e, may have many origins, 
 but the only one we are concerned with here is the gravitational 
 force. This appears to depend solely upon the distribution of 
 the matter, independently of other circumstances, and its opera- 
 tion is concisely expressed by Newton's law, that there is a 
 mutual attraction between any two particles of matter, which 
 varies as the product of their masses and inversely as the 
 square of their distance. Let e now be the intensity of gravi- 
 tational force, and F the resultant moving force, due to all the 
 matter. Then e s the space-variation of a potential, say, 
 
 e = VP, ....... (2) 
 
 and the potential is found from the distribution of matter by* 
 
 where c is a constant. This implies that the speed of propa- 
 gation of the gravitative influence is infinitely great, f 
 
 Now when matter is allowed to fall together from any con- 
 figuration to a closer one, the work done by the gravitational 
 forcive is expressed by the increase made in the quantity 
 2JP/9. This is identically the same as the quantity 2Jce 2 
 summed through all space. If, for example, the matter be given 
 
 * See 133. 
 
 t The density is expressed in terms of the intensity of force by 
 p = convce. This is also the case in the extended analogy later, when 
 the lines of e are slightly shifted. 
 
GRAVITATIONAL ANALOGY. 457 
 
 initially in a state of infinitely fine division, infinitely widely 
 separated, then the work done by the gravitational forcive in 
 passing to any other configuration is 2JP/> or 2Jce 2 , which 
 therefore expresses the " exhaustion of potential energy."* 
 We may therefore assume that Jce 2 expresses the exhaustion of 
 potential energy per unit volume of the medium. The equiva- 
 lent of the exhaustion of potential energy is, of course, the gain 
 of kinetic energy, if no other forces have been in action. 
 
 We can now express the flux of energy. We may compare 
 the present problem with that of the motion of electrification. 
 If moved about slowly in a dielectric, the electric force is 
 appreciably the static distribution. Nevertheless, the flux 
 of energy depends upon the magnetic force as well. It may, 
 indeed, be represented in another way, without introducing the 
 magnetic force, but then the formula would not be sufficiently 
 comprehensive to suit other cases. Now what is there analj- 
 gous to magnetic force in the gravitational case? And if it 
 have its analogue, what is there to correspond with electric 
 current ? At first glance it might seem that the whole of the 
 magnetic side of electromagnetism was absent in the gravita- 
 tional analogy. But this is not true. 
 
 Thus, if u is the velocity of p, then /ou is the density of a 
 current (or flux) of matter. It is analogous to a convective 
 current of electrification. Also, when the matter /> enters any 
 region through its boundary, there is a simultaneous conver- 
 gence of gravitational force into that region proportional to p. 
 This is expressed by saying that iff 
 
 C = /3U-ce, . (4) 
 
 then is a circuital flux. It is the analogue of Maxwell's true 
 current; for although Maxwell did not include the convective 
 term />u, yet it would be against his principles to ignore it. 
 Being a circuital flux, it is the curl of a vector, say 
 
 curl h = /ou - ce (5) 
 
 This defines h except as regards its divergence, which is arbi- 
 trary, and may be made zero. Then h is the analogue of mag- 
 
 * " Thomson and Tait," Vol. I., Part II., 549. 
 
 t Observe that it is -e that is the analogue of electric force, and 
 -ce of displacement current. 
 
458 ELECTROMAGNETIC THEORY. APP. B- 
 
 netic force, for it bears the same relation to flux of matter as 
 magnetic force does to convective current. We have* 
 
 h = curl pot 0, (6) 
 
 = curl A, 
 
 if A = pot 0. But, since instantaneous action is here involved, 
 we may equally well take 
 
 A = potpu, (7) 
 
 and its curl will be h. Thus, whilst the ordinary potential P 
 is the potential of the matter, the new potential A is that of 
 its flux. 
 
 Now if we multiply (5) by e, we obtain 
 
 e curl h = e/ou - ece, (8) 
 
 or, which is the same,f 
 
 conv Veh = Fu-U, (9) 
 
 
 
 if U = |ce 2 . But U represents the rate of exhaustion of potential 
 energy, so - U represents its rate of increase, whilst Fu repre- 
 sents the activity of the force on p, increasing its kinetic 
 energy. Consequently, the vector Ven expresses the flux of 
 gravitational energy. More strictly, any circuital flux what- 
 ever may be added. This Ven is analogous to the electromag- 
 netic VEH found by Poynting and myself, f But there is a 
 reversal of direction. Thus, comparing a single moving particle 
 of matter with a similarly-moving electric charge, describe a 
 sphere round each. Let the direction of motion be the axis, 
 the positive pole being at the forward end. Then in the 
 electrical case the magnetic force follows the lines of latitude 
 with positive rotation about the axis, and the flux of energy 
 coincides with the lines of longitude from the negative pole to 
 the positive. But in the gravitational case, although h still 
 follows the lines of latitude positively, yet since the radial e is 
 directed to instead of from the centre, the flux of energy is 
 along the lines of longitude from the positive pole to the 
 
 * Sec 134, page 208. 
 
 t By the transformation (178), 132. 
 
 t See 70, equation (13). Take e = = h . 
 
GRAVITATIONAL ANALOGY. 459 
 
 negative. This reversal arises from all matter being alike and 
 attractive, whereas like electrifications repel one another. 
 
 The electromagnetic analogy may be pushed further. It is 
 as incredible now as it was in Newton's time"* that gravitative 
 influence can be exerted without a medium ; and, granting a 
 medium, we may as well consider that it propagates in time, 
 although immensely fast. Suppose, then, instead of instanta- 
 neous action, which involves! 
 
 curle = 0, ...... (10) 
 
 we assert that the gravitational force e in ether is propagated 
 at a single finite speed v. This requires that 
 
 v 2 V 2 e = e*, ....... (11) 
 
 for thisj is the general characteristic of undissipated propagation 
 at finite speed. Now,|| 
 
 V 2 = V div - curl 2 , 
 so in space free from matter we have 
 
 -v 2 curPe = e ...... (12) 
 
 But we also have, by (5), 
 
 -curlli = ce, ..... (13) 
 
 away from matter. This gives a second value to e, when we 
 differentiate (13) to the time, say 
 
 g= --curlh ..... . (14) 
 
 So, by (12) and (14), and remembering that we have already 
 chosen h circuital, we derive 
 
 (15) 
 
 * To Newton himself, as shown in his of ten -quoted letter to Bentley. 
 
 t This equation is equivalent to (2) above, and it implies that the 
 gravitational force is exactly dependent on the configuration of the 
 matter. 
 
 J As shown by Poisson, the value of a quantity at a given place and 
 time, when controlled by this equation, depends solely on the state of 
 things at distance vt and time t earlier. That is, disturbances travel at 
 speed v. But it is much simpler to understand this property through 
 plane waves. 
 
 See (193), 132. 
 
460 ELECTROMAGNETIC THEORY. APP. B. 
 
 Or, if p is a new constant, such that 
 
 ,W = 1, ...... (16) 
 
 then (15) may be written in the form 
 
 curle = /*h (17) 
 
 To sum up, the first circuital law (5), or 
 
 curlh = /ou-ce, .... (5) bis. 
 
 leads to a second one, namely (17), if we introduce the hypo- 
 thesis of propagation at finite speed.* This, of course, might 
 be inferred from the electromagnetic case. 
 
 In order that the speed v should be not less than any value 
 that may be settled upon as the least possible, we have merely 
 to make /x be of the necessary smallness. The equation of 
 activity becomes, instead of (9),f 
 
 convVeh-Fu-U-f, .... (18) 
 
 if T = J/xh 2 . The negative sign before the time-increase of this 
 quantity points to exhaustion of energy, as before. If so, we 
 should still represent the flux of energy by Veh. But, of 
 course, T is an almost vanishing quantity when ft is small 
 enough, or v big enough. Note that h is not a negligible 
 quantity, though the product /xh is. Thus results will be 
 sensibly as in the common theory of instantaneous action, 
 although expressed in terms of wave-propagation. Results 
 showing signs of wave-propagation would require an inordi- 
 nately large velocity of matter through the ether. It may be 
 worth while to point out that the lines of gravitational force 
 connected with a particle of matter will no longer converge to 
 it uniformly from all directions when the velocity v is finite, but 
 will show a tendency to lateral concentration, though only to a 
 sensible extent when the velocity of the matter is not an in- 
 sensible fraction of v. 
 
 The gravitational-electromagnetic analogy may be further 
 extended if we allow that the ether which supports and pro- 
 pagates the gravitational influence can have a translationalj 
 
 * These equations are analogous to (4) and (5), 35. 
 t This is analogous to (12), 70, with the impressed forces made zero. 
 J This does not exclude rotational motion, which is, in fact, a differential 
 effect in a special kind of translational motion. 
 
GRAVITATIONAL ANALOGY. 461 
 
 motion of its own, thus carrying about and distorting the lines 
 of force. Making allowance for this convection of e by the 
 medium, with the concomitant convection of h, requires us to 
 turn the circuital laws (17), (18) to 
 
 curl(e + /AVah) = /ih, . ... (19) 
 curl (h + cVea) = /)U-ce, . . . (20) 
 
 where q. is the velocity of the medium itself.* 
 
 It is needless to go into detail, because the matter may be 
 regarded as a special and simplified case of my investigation of 
 the forces in the electromagnetic field, with changed meanings 
 of the symbols. It is sufficient to point out that the stress in 
 the field now becomes prominent as a working agent. It is of 
 two sorts, one depending upon e and the other upon h, analo- 
 gous to the electric and magnetic stresses. The one depending 
 upon h is, of course, insignificant. The other consists of a 
 pressure parallel to e combined with a lateral tension all round 
 it, both of magnitude ^ce 2 . This was equivalently suggested 
 by Maxwell. Thus two bodies which appear to attract are 
 pushed together. The case of two large parallel material 
 planes exhibits this in a marked manner, for e is very small 
 between them, and relatively large on their further sides. 
 
 But the above analogy, though interesting in its way, and 
 serving to emphasise the non-necessity of the assumption of 
 instantaneous or direct action of matter upon matter, does not 
 enlighten us in the least about the ultimate nature of gravi- 
 tational energy. It serves, in fact, to further illustrate the 
 mystery. For it must be confessed that the exhaustion of 
 potential energy from a universal medium is a very unintelligi- 
 ble and mysterious matter. When matter is infinitely widely 
 separated, and the forces are least, the potential energy is at its 
 greatest, and when the potential energy is most exhausted, the 
 forces are most energetic ! 
 
 Now there is a magnetic problem in which we have a kind of 
 similarity of behaviour, viz., when currents in material circuits 
 are allowed to attract one another. Let, for completeness, the 
 initial state be one of infinitely wide separation of infinitely 
 
 * The additional terms are analogous to the motional electric and mag- 
 netic forces of 44 and 66, &c. See also (d), 132. 
 
462 ELECTROMAGNETIC THEORY. APP. B. 
 
 small filamentary currents in closed circuits. Then, on con- 
 centration to any- other state, the work done by the attractive 
 forces is represented by 2J/AH 2 , where p is the inductivity 
 and H the magnetic force. This has its equivalent in the 
 energy of motion of the circuits, or may be imagined to be so 
 converted, or else wasted by friction, if we like. But, over and 
 above this energy, the same amount, 2J/xH 2 , represents the 
 energy of the magnetic field, which can be got out of it in work. 
 It was zero at the beginning. Now, as Lord Kelvin showed, 
 this double work is accounted for by extra work in the batteries 
 or other sources required to maintain the currents constant. 
 (I have omitted reference to the waste of energy due to elec- 
 trical resistance, to avoid complications.) In the gravitational 
 case there is a partial analogy, but the matter is all along 
 assumed to be incapable of variation, and not to require any 
 supply of energy to keep it constant. If we asserted that 
 |ce 2 was stored energy, then its double would be the work done 
 per unit volume by letting bodies attract from infinity, with- 
 out any apparent source. But it is merely the exhaustion of 
 potential energy of unknown amount and distribution.* 
 
 Potential energy, when regarded merely as expressive of the 
 work that can be done by forces depending upon configuration, 
 does not admit of much argument. It is little more than a 
 mathematical idea, for there is scarcely any physics in it. It 
 explains nothing. But in the consideration of physics in 
 general, it is scarcely possible to avoid the idea that potential 
 energy should be capable of localisation equally as well as 
 kinetic. That the potential energy may be itself ultimately 
 kinetic is a separate question. Perhaps the best definition of 
 the former is contained in these words : Potential energy is 
 energy that is not known to be kinetic. But, however this 
 be, there is a practical distinction between them which it is 
 found useful to carry out. Now, when energy can be distinctly 
 localised, its flux can also be traced (subject to circuital inde- 
 terminateness, however). Also, this flux of energy forms a 
 useful working idea when action at a distance is denied (even 
 though the speed of transmission be infinitely great, or be 
 
 * It would appear that we must go to the ether to find the source of all 
 energy. 
 
GRAVITATIONAL ANALOGY. 463 
 
 assumed to be so). Any distinct and practical localisation of 
 energy is therefore a useful step, wholly apart from the 
 debatable question of the identity of energy advocated by 
 Prof. Lodge. 
 
 From this point of view, then, we ought to localise gravita- 
 tional energy as a preliminary to a better understanding of 
 that mysterious agency. It cannot be said that the theory of 
 the potential energy of gravitation exhausts the subject. The 
 flux of gravitational energy in the form above given is, per- 
 haps, somewhat more distinct, since it considers the flux only 
 and the changes in the amount localised, without any state- 
 ment of the gross amount. Perhaps the above analogy may be 
 useful, and suggest something better. 
 
 Part II. 
 
 In the foregoing I partly assumed a knowledge on the part 
 of the reader of my theory of convective currents of electrifi- 
 cation* ("Electrical Papers," Vol. II., p. 495 and after), and only 
 very briefly mentioned the modified law of the inverse squares 
 which is involved, viz., with a lateral concentration of the lines 
 of force. The remarks of the Editor f and of Prof. Lodge J on 
 gravitational aberration, lead me to point out now some of the 
 consequences of the modified law which arises when we assume 
 that the ether is the working agent in gravitational effects, and 
 that it propagates disturbances at speed v in the manner sup- 
 posed in Part I. There is, so far as I can see at present, no 
 aberrational effect, but only a slight alteration in the intensity 
 of force in different directions round a moving body considered 
 as an attractor. 
 
 Thus, take the case of a big Sun and small Earth, of masses 
 S and E, at distance r apart. Let / be the unmodified force 
 of S on E, thus 
 
 using rational units in order to harmonise with the electro- 
 magnetic laws when rationally expressed. Also, let F be the 
 
 * See also 52 to 62, and 163, 4. 
 
 t The Electrician, July 14, p. 277, and July 28, p. 340. 
 
 t The Electrician, July 28, p. 347. 
 
464 ELECTROMAGNETIC THEORY. APP. B. 
 
 modified force when the Sun is in motion at speed u through 
 the ether. Then* 
 
 where s is the small quantity ^ 2 /v 2 , and is the angle between 
 r and the line of motion. (" Electrical Papers," Vol. II., pp. 
 495, 499). 
 
 Therefore, if the Sun is at rest, there is no disturbance of the 
 Newtonian law, because its " field of force " is stationary. But 
 if it has a motion through space, there is a slight weakening of 
 the force in the line of motion, and a slight strengthening 
 equatorially. The direction is still radial. 
 
 To show the size of the effect, let 
 
 u = 3 x 10 7 centim. per sec. ) 
 * = 3xlO / 
 
 This value of u is not very different from the speed attri- 
 buted to fast stars, and the value of v is the speed of light itself. 
 So we have 
 
 i.e., one-millionth. All perturbing forces of the first order are, 
 therefore, of the order of magnitude of only one-millionth of 
 the full force, even when the speed of propagation is as small 
 as that of light. 
 
 The simplest case is when the common motion of the Sun 
 and Earth is perpendicular to the plane of the orbit. Then 
 = \TT all round the orbit, and 
 
 F-/(1 + M ....... (5) 
 
 showing an increase in the force of attraction of S on E of one 
 two-millionth part, without alteration of direction or variation 
 in the orbit, f 
 
 * Tliis is the case of steady motion. There is no simple formula when 
 the motion is unsteady. Equation (2), above, is immediately derivable 
 from equation (40), 163. 
 
 f But Prof. Lodge tells me that our own particular Sun is considered to 
 move only 10*9 miles per second. This is stupendously slow. The size of 
 is reduced to about 1/360 part of that in the text, and the same applies 
 to the corrections depending upon it. 
 
GRAVITATIONAL ANALOGY. 465 
 
 But when the common motion of the Sun and Earth is in 
 their plane, varies from to 2r in a revolution, so that the 
 attraction on E, whilst towards the Sun's centre always, under- 
 goes a periodic variation from 
 
 F =/(!-) (6) 
 
 when = 0, to F=/(1 + J), (7) 
 
 when 6 = \v. The extreme variation is, therefore, f-*/, accord- 
 ing to the data used. The result is a slight change in the 
 shape of the orbit. 
 
 But, to be consistent, having made v finite by certain supposi- 
 tions, we should carry out the consequences more fully, and allow 
 not merely for the change in the Newtonian law, as above, but 
 for the force brought in by the finiteness of v which is analogous 
 to the " electromagnetic force." This is very small truly, but 
 so is the above change in the Newtonian law, and since they 
 are of the same order of magnitude, we should also count the 
 auxiliary force. Call it Cr. Then* 
 
 , .... (8) 
 
 where F is as before, in (2) above, q is the actual speed of the 
 Earth (not the same as u\ and in the third vectorial factor, 
 QJ, Up and r t are unit vectors drawn parallel to the direction of 
 the Earth's motion, of the Sun's motion, and from the Sun to 
 the Earth. We see at once that the order of magnitude cannot 
 be greater than that of the departure of F from /, before con- 
 sidered, because u and q will be of the same order, at least 
 when u is big. As for x, it is simply a numerical factor, which 
 cannot exceed 1, and is probably f . 
 
 The simplest case is when the motion of the Sun is per- 
 pendicular to the orbit of the Earth. Then 
 
 G = Fx#s (9) 
 
 gives the tensor or size of the auxiliary force. It is radial, 
 but outwards, so that the result is merely to reduce the size of 
 
 * See 46, equation (9), and 82 and 85. To apply to the present case, 
 we have to note that G Is the " electromagnetic force " of the Sun's "mag- 
 netic induction " on the Earth's " electric current," or rather, the analogue 
 thereof ; and also that the direction must be reversed, just as the ordinary 
 "electric force" is taken reversed. 
 
 HH 
 
466 ELECTROMAGNETIC THEORY., A 
 
 the previous correction, viz., the difference of F from / i 
 same motional circumstances. 
 
 But when the line of motion of the Sun is in the pla 
 the orbit, the case is much more complicated. The fore 
 neither constant (for the same distance) nor radial, exce 
 four positions, viz., two in the line of motion of the Sun, 
 the auxiliary force vanishes, and two when 0= ^, whe 
 greatest. But this force is still in the plane of the orbit, 
 is an important thing, and is, moreover, periodic, so tha 
 tangential component is as much one way as the othei 
 period. 
 
 All we need expect, then, so far as I can see from the 
 considerations, are small perturbations due to the variat 
 the force of gravity in different directions, and to the aus 
 force. Of course, there will be numerous minor perturbat 
 
 If variations of the force of the size considered above a 
 small to lead to observable perturbations of motion, the 
 striking conclusion is that the speed of gravity may even 1 
 same as that of light. If they are observable, then, if exi 
 they should turn up, but if non-existent then the spc 
 gravity should be greater. Furthermore, it is to be obs 
 that there may be other ways of expressing the propa< 
 of gravity. 
 
 Bat I am mindful of the good old adage about the shoei 
 and his last, and am, therefore, reluctant to make any 
 remarks about perturbations. The question of the ether 
 gravitational aspect must be faced, however, and solved 
 or later, if it be possible. Perhaps, therefore, my sugge 
 may not be wholly useless. 
 
 * The solution for steady rectilinear motion has been employed 
 justification thereof is the smallness of u/v and the large periodi 
 If we allow for the small curvature of path of an attracting body, \\ 
 introduce corrections of the second order of small quantities. 
 
 END OF VOLUME I, 
 

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