Electromagnetic Theory, Vol. 1 (1893) — part 20 of 31

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 V0EF 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-IB + B-A-I.AB

302 ELECTROMAGNETIC THEORY. CIT. TIT.

identically. Or,

B = (A-1B + V.VA~1B)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, rt 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 /xcv2 = 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 = 0 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, howevert 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.