Electromagnetic Theory, Vol. 2 (1899) — part 22 of 31

§ 184. Sa 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 indcterniiuateness 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-

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t^nce, 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 nou-oonducting dielectric goes out of range and is vasted.

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- -aeqnent 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 waT^ passing them, instead of geuerating more disturbances. In the absence of this demoniacal possession of the region, the •eneigy will remain within it in the eleotromagnetio 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 cpn- <ductor, which is possibly existent in fact at the limiting sero •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.

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Conductors at Low Temperatures.

§ 185. For this reason, a perfectly conducting mass exj)osed 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 thai a metal, if once brought to zero temperature, would remam t'.ere, provided its only source of heat-energy be electromag netic vibrations. At the same time it is ocmoeivable, 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 or air particles, for example, if air be present. Prof. Dewar'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.

♦ FhiL Mag,, February, 1893.

TBBORT OF PLANB BLEGTROMAONETIO WAVES. 331

£amlibrittm of Badiation. Tlie Mean Flux of Energy.

§ 186. As before said, a perfectly oondacting soreen enclos- ing a dieleotrio region supporting electromagnetio distorbanoes, 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 electromagnetio disturbances, when left to themselyes, subject to the obstruct* ing and reflecting action of the screen.

In certain oases the initial state will continue absolutely tmchanged ; for example, when it is the steady state due to- eleotrification, 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. Agaiu, in other cases, although the subse- quent history of an initial state may be one of constant chauge, 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 e.\ami)les, 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 electromaijnetic disturbance, the general eliect of the repeated reflections from the boundary (especially wlien 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 Lorn matter, when it is tolerably

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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 seoond 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 IV unit area, A, fixed in position anywhere within the enclosure^ and consider the flux of energy through it under different oir* oumstances. First of all, if the energy all moved the same way at the same speed v (that of propagation), as in simple pluie 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 JJv 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 duections 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. Kow, if the ray of unit section be turned round so as to make an angle 0 with the normal to A, the effective

THEORY OF PLANS ELEOTBOMAGNSTIC WAV£S. 33$

section of the ray is reduced from 1 to cos 6 ; 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 Uvcos^, due to the ray at iuolination 6. The true flux through the area A is therefore the mean value of Uvcob ^for all directions in space assumed by the ray. Now the mean value of cos 0 for a com- plete sphere is zero, and therefore the mean flux through A is aero. 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 008 0 is then j^. But we are only conoemed with half the total energy, or JU, when we are confined to one hemisphere. Con- sequently we have

W-JUt., (4)

to ezprass the flux of energy W per second each way tbrougb 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 l/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/f») cos $ through A pe^ second. Now sum this up over the whole hemisphere and th» zMult is W. In the limit, when n is infinitely great^ we have

W. r (^^"^JanedHe-iVv, . . (6)

y 0 y 0 47r

as before. The 4ir divisor in the integral is not the unspeak able 4v of the B.A. units. It is the area of the sphere of unit radius, whilst the other fMtor mn$d6d4} 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 defiuiteness of meaujug as possible. Bartoli made it come to ^\Jv (apart from electromagnetic considerations, which

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are, however, only accidentally inyolved). On the other hand, 'Galitsine makes it ^XJv 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 ^\Jv 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 ^, 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^^. This must now be averaged for all directions in space that we may ^ive to the ray, without preference. Now the mean value of

THEORY OF PLANS ELEQTROMAOKBTIO WAVES. S35

■cob^O for the complete sphere is j^. So the pressure, say p, is ^ven by

l>-iU (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 V/n. Let the axes of these rays be defined by the middle points of n small equal areas into which we may divide the surface of a sphere, luul sum up the normal pres- sures on A. We obtain, in the limit,

jl« / / &m 6 d<l>dd = . . . (<)

This result was given by Boltzmann some years ago, and 'Oalitzine confirms it for the special ca^e 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.

§ 188. Tliis pressure p is not only the mean pressure throughout the enclosure, but aho 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 second 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

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body, nvhilst 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 througli the distance A, 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 7>, 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)k Say

H = (U+i>)A (8>

Now we know p in terms of U, bo that we have, by (6),

H = JU/*. (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 dK is an element of the heat lost iiv the cooling process. On the left side put (U+p)h or ^\Jh 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

S'=*^. (">

from which we conclude that U, and, therefore, also p and W,. ▼aiy as the fourth power of the tempoature, the result above- mentioned as applied to the emissivity W.

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THEORY OF PLANE ELECTBOMAGNETIO 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 difidculty in oonnexion with the ether is a very old one.

Internal Obstruction and Superficial Conductioiu

§ 189. The properties of a perfect conductor are derived from those of common conductors by examining what would happen if the resistivity were continuously reduced, and ultimately booame 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 conduo- ton. Bat 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, in substance or in method, and want to propagate it rapidly, you need not expect anything but hindrance from the old practitioner— especially if 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, in a somewhat sneaking manner, not unmixed with bluster, and make believe he knew all about it when he was a little boy ! He sees a prospect of dollars in the distance, that is the

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reason. The perfect obBtmcdon haying foiled, try the per- fect conduction.

Yon should make your converts out of the rising generation and the ooming men. Thus, passing to another matter. Plot Tait says he cannot understand my vectors, though he oan understand much harder things. But men who have no quater- nionic prejudices can understand them, and do. Younger men are bom 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 yon 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 llmv all tlirou<zh 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 10, 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/K, 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

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THEORY OF TLANB ELECTROMAGNETIC WAVES. 339

smaller the resistance the greater the time taken for the <}urrent 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, tlien, 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 weU. 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 externa] 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- elusions 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 comq 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-

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magnetio wares to slip along its surfaoe in a perfectly fre» manner, without waste of energy. Though perfectly obstruo- tive internally, it is perfectly conductive superficially. It merely g^des the wayes, 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 Oonduction of Waves.

§ 190. The conditions at the interface of a perfect conductor and a dielectric are that the electric force in the dielectric haa no tangential component and the magnetio induction no normal component. Or

YKE-iO, 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 <t of the one, and the magnetic force that of the other, say c, thus

o--in>, e-VNH,

in rational units, without any useletis and arbitrary Att 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 iniluence 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 its- 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 forces neither that of the external field nor that of the superficial current. The current sheet itself merely means the abrupt

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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 permittiTity, 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 £u:-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 behayes 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- nl 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^/avH, moying at the speed of light, to strike a perfect conductor Hush — ^that is, all over at the same time, by reason of pandld- 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

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of the eleotrio kind the reflected eheet diffen from the inoideot in having its dieplaoement xevened, but in no other re^>ect. This is perfect reflection with reversal of S. During the act of reflection, whilst the incident and reflected sheets partly coin- cide, E is sero and H is doubled. Both are tangential ; but there can be no tangential E, so the reflector destroys B and initiates the reflected sheets 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 doubletl 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 propag-ited along it without change of type. This case includes an infinitely fine wire, when w^e 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, d^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 momentaiy 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 grasing incidence, and may be considered the limiting case of permanent union of the incident and reflected waves. Betwe^ these two cases we have the general case of inddence and refleo* 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^

THBORT OF FLAMB ILBOTBOMAGNICTIO WAVK8. 34&