Electromagnetic Theory, Vol. 2 (1899) — part 10 of 31

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 ''V 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 mm ways than one^ and can be used in very many ways. The easiest of all is to assume that the magnetic force is the vdooity of the medium, magnetic induction the momentum, and so on, as is done by Prof. Lodge (Appendix to "Modem Views of Electrici^ 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 tliat 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 typcw This is very convenient.

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But the difficulties in the way of a complete and BatisfiuJtQiT' lepreeentation of eleotramagnetio phenomena by an elaatlo- solid ether are insuperable. Beoognising this, Sir W. Thomson has recently brooght 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 1 shall ignore later), but there is no distorting stress, and ita place is taken by a rotating stress. It gives rise to a tnm»> lational force and a torque. If E be the torque, the stress on any plane V (unit normal) is simply YEN, the Tector product of the torque and the normal Tector.

The force is —curl E. We have therefore the equation of motion ,

-curlE = /AH,

if H is the velocity and fi 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-rigidity.

Now these are the equations connecting electric and magnetic force in a non-conducting dielectric, when /i is the inductivity and c the permittancy. We have a parallelism in detail, not merely in some particulars. The kinetic energy JftH^ repre- sents the magnetic energy, and the potential energy JcE- the electric energy. The vector-flux of energy is YEH, the activity of the stress.

This mode of representation dififers from that of Sir W. Thomson, who represents magnetic force by rotation. This sjrstem 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 trio- tacnally resisted, and let the above equations become

-curl E=^H + fiH,

curlH«^E + cS,

where g is the translational friotionality ; h will be considered later. We have now the equations of electric and magnetic force in a dielectric with duplex conductivity, k being the

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eleotrio and g the magnetic oonduotivity (by analogy with elec- tric force, but a frictionality in our present dynamical analogy).

We have, therefore, still a parallelism in every detail. We have waste of energy by friction ^H^ (tranalational) and (rotational). If ^//x = kjc the propagation of disturbanoes 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 circuity which is most simply taken to be three oo-azial tubes. Let, A, B, and 0 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 fint places 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) tmns- mitted along B, without change of type, at constant speed, and without aflfocting A and C.

This is the analogue of a oonoentrio 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 applieil 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 pt»-eminently a dielectric ; whilst the other two should have very high conductivity, so that they are pre-eminently con ductors. The three constants, fi, c, "k^ may have any value iu the three tubes, but practically k should be in the middle tube a very small fraction of what it is in the others.

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It is remarkable that the guasi-roiationa I resistance in A and € should tend to counteract the distorting effect on waves of the quasi-rotational resistance in B. Bat the two rotations, it should be observed, are practically perpendicular, being aiual or longitudinal (now) in A and C, and transverse or radial in B ; due to the relative smallness of k 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 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 alMorptioui 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 f notional resistance to rotation we get an equation of the form

E = ^a + 6^^curlH,

£ being (as before) the torque, and H tiie velocity. But this is not of the right form, which is (as above)

curlH.

therefore some special arrangement is required (to produce the dissipation of energy k'E^)f which does not obviously present itself in the mechanics of the rotational ether.

On the other band, if we follow up the other system, in which magnetic force is allied with rotation, we may put ^a*0, let - E be the velocity and H the torque ; fi the compliancy, c the density, and k the translational friotronality. This gives

~ curl E = /xH

curl HsAS+dS.

We thus represent a homogeneous conducting dielectric, with a translational resistance to cause the Joulean waste of energy.

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Bat it ii now seemingly imposnble to properly satisfy the con- •ditions of conttnuity 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 ati •imperfect state.

In the general case, the djdt 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 goes 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 «clcctromagnetio comparisons without unnecessazy constants.]-

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CHAPTER IIL

THE ELEMENTS OF VECTORIAL ALGEBRA ANI>

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 Bud that the fundamental entity oonoemed, namely a straight line, when regarded as an entity, cannot be treated simply, as a quantity in the algebraical sense. It has, indeed, size, vis., 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.

Agam, 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.) AIL 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 aud direction, are vector magnitudes, or simply vectors..

ELEMSNTS OF VECTORIAL ALGBBRA AND ANALYSIS. 133

Now, just as there is an algebra and analysis for scalan, 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- iietism.

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 scalaxs. Similarly, in the usual treatment •of physiosl vectors, there is an avoiduice 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 ^hibit useful results and relations in a ready manner. When •we work directly with vectors, we have our attention fixed vpoa them, and on their mutual relations; and these are Qsnally 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 reijuired to find out what it means ; and, even 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 formulse 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 toM as a good joke upon a mathematician that the poor man went mad and mistook his symbols for realities ; as M for 4^6 moon and S for the sun. There is another side to the

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Btory, however. If our object be ultimately physical, rathet than mathematical, then the more closely we can identify the- symbols with their physical representattyes the more usefully can we woric, with aToidance 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 lormulse may become eUive, as it were» Now this is possible, and indeed, comparatively easy, in vectorial analysis ; but is very diflBcult 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.

Abstnislty of Quaternions and CkmqpmtiTtt Simplieity 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 quatemionic old-man-of-the-sea who fastened him- self on my shoulders when reading the only accessible treatise on the subject — Prof. Tait's Quaternions. But I came later to see that, so far as the yeotor analysis I required waa concerned, the quaternion was not only not required, but ws»

KLBMBNTB OF VECTORIAL ALOBBRA AND ANALT8IS. 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 ghoet 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 learu 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 bis 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 quatemionic 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 diflusiou 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 to go to Hamilton's ponderous volumes, or else to Prof. Tatt'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 greali measure from the quatemionic basis.

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Elementary Vector Analysis independent of the

Quaternion*

§ 100. Sappoee a sufficiently competent mathematician desired to find out from the Cartesian mathematics what yector algebra was like, and its laws. He could do so by careful inspection and comparison of the Cartesian formulse. 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 wliich 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 manij)ulation of his vectors. Vector analysis is, in its elements, entirely independent of the exceedingly ditlicult theory of ([uaternions ; that is, when the latter is treated quaternionically initio,

"Quaternion" was, 1 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. Eveiything revolves

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Sr.EUBKT8 OF TBCIOBIiU^ ALGBBRA AND ANALTSIS. 137

round the quaternion. The lawB of vector algebra themseWee are eetablished through quatemiouB, assisted by the imaginary V - 1* 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 neyer 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. Nov 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 qoatemionic is an undesirable way of beginning the subject, and impedes the diflfasion of vectorial analysis in a way which is as vexatious and brain-wasting as it is unnecessary.

Tait Oibbs and Oibbs 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. r>ut I have at heart the spread of a working knowledge of cleinentary 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 wlio 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 t^uaternions too soon" (Third edition, p. 110). This is as characteristic of the sardonic philosopher as a certain heavy kind of " liippancy " 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 qoatemionic progress, in virtue of his pamphlet on Vector

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Analysis, a son of hermaphrodite monster, compounded of the notations of Hamilton and Gruamann." Grassmann, I may observe, eetablished, inter aliti, a oaloulus of yeoton, 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. Gibbe's pamphlet (Nor Publuhbd, Newhaven, 1881-4, pp. 83), is not a quatemionic treatise, but an able and in some respecta 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 Ifatwre) 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. Gibba clearly separates the quatemionic question from the quesUon of a suitable notation, and argues strongly against the quatemionic establishment of vector analysis. I am able (and am happj) 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 tlie Minns Sign of Qaatemions.

§ 102. In Quaternions, the square of a unit vector is - 1. This singular convention is quatemionically 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 (juateruion) ; 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 quatornionist (unless prejudiced) admit- ting the simplicity of establishment and of operation, and the

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of the notstKHn, and its sufficiency for praotioal re- quirements up to a cotain point ; and yet adding the inquiry whether there is not, over and ahove 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-quatemionic 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 practioal utility of the* ex-quatemionic system.

We may, however, if we wish to go further, after ex- quatemionic 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 concemed with scalars and vectors only. The reader should entirely divest his mind of any idea that we are concerned with the imaginary in vector analysis. AlsOi 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 Boman Letters un- suitable. Clarendon Type suitable. Typographical BackslidlTig 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

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desirable that this should be done, in order to fietcilitate the reading of yectorial vrork, 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 niucli alike that a close scrutiny with a glass is needed to distinguish them, unless one is lynx-eyed. This is a fatal objection. lint, 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 Koman characters, which he who runs may read. It is a relic of medieval monkery, and is iiuite 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- gified in the adult, and again transmitted to posterity. German letters must go.

Rejecting Germans and Greeks, I formerly used ordinary Boman letters to mean the same as Maxwell's corresponding

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QmoMUM. They are plain enough, of ooiirae ; but, as before mentioned, are open to objection. Finally, I found salvation in Cnarandioiui, and introduced the use of the kind of type sa called, I believe, for vectors (Phil, Mag., August, 1886), and have found it thoroughly suitable. It is always in stock ; it is veiy neat; it is perfectly legible (sometimes alarmingly so), and is suitable for use in formulsa 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 ;i subject which is important to the community in general. 1 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 tliink the print is too small, or ihat 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 generation since, and now largely in use, differs from the more legible " modem style " (of this page, for instance) in two re- spects. It has certain eccentricitieB 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*

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to drive out the modern style, diicarded. Not that the latter is perfect. A still better style would be arrived at by thicken- ing the fine Unes in it, prodaoing something intermediate between it and the Glaxendon style (which last, of course, would be too much of a good thing). As everyone who has had to mad 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 an print. Thicken the fine lines, and the effect is magical.

ITotation. Tensor and Components of a Vector. Unit

Vectors of Beferenee.

§ 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 «, r, w ; or ^» (i 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, £ denoting any vector, its tensor is E, and its components are Ej, Eg, E3. Simi- larly, the tensor of a is a, and the components are a^, a^. A large stock cf 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 £q 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 (to be) vector. Compositors are very intelligent, read mathe-

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