Electromagnetic Theory, Vol. 2 (1899) — part 24 of 31

transmitted wayewill be additive, to that it will emerge a piire wave with extra attenuation. But as regards the reflected wave, we have a peculiar result. The action of the magnetic conductnnce is to reverse the inchiction whilyt keeping the displaceiuent straight ; whilst tliuL of the electric conductance is to reverse the displacement and keep the induction straight. The result is that the reflected wave is reduced in magnitude by the addition of magnetic conductance to provioualy existent electric conductance. With a proper proportioning of the two conductances, the reflected wave may be brought nearly to evanescence from a plate of finite conductance. In the limit the compensation is perfect, and the incident wave goes right through without reflection, though it suflers extra attenua- tion. This is the explanation of the distortionless propagation of waves in a dielectric medium possessing duplex conducti- vity, electric and magnetic. Whilst there is no reflection in transit, there is a coutiuuous loss both of displacement and of induction.

The Persistence of Induction in Plane Strata, and in generaL Also in Cores and in Linear Circuits.

§ 194. Now return to the case of electric conductivity alone, and, as described, let it be locally condensed into the conduc- tance of any number of parallel plates. We know that the effect of any one of them on a thin electromagnetic sheet is to split it, as previously described. If we like, therefore, we can follow each of the resulting waves, and observe how they are, in their turn, split by the first plates they meet, giving rise to four waves, to be a little later split into eight, and so on. This process may seem cumbrous, but it is also an instructive one.

Thus, consider what happens to the total induction. We know that it persists in amount and direction when a single split occurs. Now the same property applies to every sucoee- sive split a wave suffers in our dielectric medium containing parallel conducting plates. So the total induction remaius constant. It is redistributed and spreads out both ways, but without the least loss. There is a small loss of energy at evt ry splits but this does not affect the total induction. This applies when we start from a single pure electromagnetic sheet moving either way. It therefore applies when the initial state oonsiots

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of any number of such sheets, of any strengths, forming a per- fectly arbitrary initial distribution of induction and displa(»- ment in parallel plane layers. There is still persistence of the induction. Finally, the same applies when we split up the conducting plates themselves into plates of smaller con- ductance, and spread them out at uniform distances. TThe ultimate limit of this process is reached when the conduct- ance is quite uniformly spread, so that we have a perfectly homogeneous medium under consideration. It is, fundament- ally, a dielectric propagating disturbances at speed v ; but it is, in addition, a conductor as well, and distorts the waves and dissipates their energy. The speed v is with the proper

values of and c. The conductivity does not interfere with this property of propagation at finite speed. But observe that if we choose to ignore the displacement, then the corresponding speed is infinitely great. We conclude from the above that plaae sheets of induction in electric conductors always preserve the total induction constant in amount, irrespective of the amount of elastic displacement, or whether there is any at alL That is, induction cannot be destroyed by conductance.

If, then, it suffers destruction, this must be due to some other cause. It may be merely a cancellation by the onion of oppositely-directed inductions. This may be termed a vectorial cancellation. It may occur, of course, with plane strata of induction. Thus if, in an infinitely large conductor, the total induction be initially zero, which does not require the induction density to be zero, the final effect will be a complete annihil»> tion of the induction by mutual cancellations. Should, how- ever, the total induction be not zero, it wiU persist. The induction density will tend to zero, but that will be merely on account of its attenuation by spreading, not because there is any destruction by the conductance or resistance when eithet of them is finite. To prevent the attenuation to zero we may interpose infinitely conducting barriers, one on each side, in planes parallel to the sheets of induction. Then the final result will be that the induction will spread itself out uniformly between the barriers and maintain a finite density.

To illustrate this property in a somewhat less abstract man- ner, consider a large ring, say of copper, though iron will do equally well except as regards some complications connected

IB80BT OF PLANS aLBOntOMAONBTIC WAVBS. 359

with ha magnetiBation. Let it be induotised by an envelopuig «oQ-ciixrent so that the indnotion goes along the oore in ft complete circuit. When it is steadily set np, if we remoTe the c(»l-oarrent (and the ooil too, preferably for our present purpose) the induction in the core will, in time, all come out of it. But if we clap an infinitely-conducting skin upon the core, it will not come out. Then we haye a certain flux of induction looked up, as it were, in a conducting material, which has no eflfiect upon it. It can neither be destroyed by the conductance of the oore nor can it get through the perfectly-obstructive ■kin. If the skin is clapped on after the induction has partially escaped (which escape begins on the outside, before the interior is sensibly affected), there is a redistribution of induction, which continues until a new state of equilibrium is veached. During this process there is electric current in the core and some waste of energy. But there is no waste of the induction. The final induction is the mean value of the original induction across the section of the core.

In further illustration, let the core be hollow and be induo- tised circularly — instead of along its length — by means of two currents on its boundaries, inner and outer, oppositely directed, following the length of the core. When this is done, remove the currents and clap on perfectly conducting skins internally and externally. There will be a similar persistence of the Induction, although its tubes now go round the inner boundary circularly. There may be an initial settling down, but the outer skin will not let the induction expand outwardly, and the inner skin will not let it contract inwardly. If the latter could happen we might have cancellation. To get this effect remove the inner skin. Then, whether we fill up the hollow with finitely conducting matter or leave it nonconducting, we allow the induction to spread internally and permit cancellation. The induction will now wliolly disappear, in spite of the external skin. That is to say, there will be a continuous passage of the induc- tion out of the initially inductised region, accompanied by elec- tric current therein, which will continue until the whole of the magnetic energy is wasted as heat in the core.

The same property is exemplified, though in a less easily under- standable manner, with a single closed line or circuit of infinite conductance. If it embrace a certain amount of induction it

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will always do so, in the absence of impressed force to alter tin amount. The induction is locked in and cannot pass through the infinitely conducting circuit to dissipate itself. If the oonductanoe of the circuit be finite, then it can get through. The time-constant varies as the conductance. The dlnppearance of the induction is manifested by the waste of energy in the circuit, the electric current in which is sup- ported by the Toltage of the decreasing induction through it. Bat the current is there all the same (measured magnetically) when the conductance is infinite. The induction is steady^ and there is no voltage in the circuit. But none is needed.

On the other hand, if there is initially no induction through the circuit, there will continue to be none when a magnetic field is created in its neighbourhood. But although the tubes of induo- tion cannot cut through the infinitely conducting circuit so as to make the induction through it be a finite quantity, yet they do pass through a surface bounded by the circuity as much pcsitiyely as negatiyely. The resulting induction distribution is to be got by superimposing the external induction and that due to a cur- rent in the circuit of such strength as to make the total indoctkNi through it be zero. The property is a general one, for if the oir- ouit be moved about in a magnetic field, there is always, in ▼irtait of its impermeability to the magnetic flux, lero total indnotioii through it if its conductance be infinite ; whilst if it be finite but great, there is an approximation to this result so long as th^ motion is kept up, or the external field be kept varying. At the same time, the least amount of resistance in the circuit will be sufficient, if time enough be given, to allow the external induc- tion, when due to a steady cause, to get past it to the full extent, when of course the current in the circuit will cease.

In a similar manner, displacement can be locked up by a circuit of perfect magnetic conductance. There is also per- sistence of displacement in spite of a finite degree of magnetic conductivity in a continuous medium, unless it be electrically conducting as well.

The Laws of Attenuation of Total Displacement and Totil Indnction by Electric and Magnetic Oondnctance.

§ 195. Next consider the effect of a conducting medium upon the total displacement. We know that the latter decreases

TUKORY OF PLANE ELECTUOMAGXETIC WAVES. 361

with the time, and the law of decrease may be readily found from the theory of a single conducting plate. We found that when its conductance exceeded a certain value, the loss of dis- placement exceeded the original. But, in regarding the action of a homogeneous conductor upon a wave as the limit (in the gross) of that of an assemblage of parallel plates in which the conductance is localised (which process may not seem unassail- ably accurate beforehand, but which is justified by the results), it is easy to see that we have merely to deal with plates of such very low conductance that the loss at each is extremely small, so that the above-mentioned difhculty does not enter. Thus, let the loss at one plate be sucli as to reduce the initial displacement D in a wave to iriD, where ?)l is a fraction nearly equal to unity. Here mD is the sum of the displacements in the transmitted and reflected waves, the latter being very small and of the opposite sign to the initial D. As these waves separate, thej reach other plates and are split anew. If these plates have each the same conductance as the first, the total mD is farther attenuated by them to m^D when the two waves become four. Next, when these four waves are split into eight by the next plates that are reached, the total displacement bdcomes m'D ; and so on. These successive displacement totals decrease according to the law of a geometrical series. It fol- lows that, in the limit, we shall have the total displacement xepresented by an exponential function of the time, say by

I> = ^»«-"S (1)

where Dq is the initial value, aud D what it becomes at time t. To find the -value of n, we have merely to examine the form of the fraction m, observe how it depends on the conductance of one plate, and proceed to the limit by making the number of plates infinite, whilst their conductances are infinitely small. The result is that the constant n has the value A/e, where k is the conductivity and e the permittivity of the homogeneous conducting medium.

In the irrational units of the B.A. Committee this quantity is represented by ivk/Cf which is, of course, nonsense, like the qustemionic doctrine about the square of a yector. They are both going to go. The above reasoning applies to any initial distribatioL of displacement in plane layers, instead of merely

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one elementary sheet. Therefore, equation (1) shows that the total displacement subsides according to the time-factor c Now, this represents Maxwell's law of subsidence of

displacement iu a conducting condenser (apart from " absorp- tion" and hysteresis), or of the static distribution of displace- ment associated with electrification in a conducting medium. We see that the law has a far more general meaning. The initial displacement need not be static, but may be accom- panied by magnetic induction, and may consequently move about in the most varied manner, whilst its total amount decreases according to the static law. A homogeneous medium IB prestipposed, and modifications may be introduoed by the action of boundaries.

Passing next to the analogous case of a magnetic conductor, in which the total displacement remains oonstant whilst the total induction subsides, it is unnecessary to repeat the argument, but is sufficient to point out the law according to which the subsidence occurs. If Bg be the initial total mduction, and B what it becomes at time we shall have

B-B,. (2)

where g is the magnetic conductivity and /x the inductivity. The time-constant cjk of the former case has become ixjg.

Returning to the former case, it should be noted that when the initial distribution is of the static nature, unaccompanied by magnetic force, it retains this property during the sub- sidence. For, since the displacement subsides everywhere according to the same time-factor, its distribution does not alter relatively, or it remains similar to itself. Since, then, there is no magnetic force, there is also no true electric current. There is also no flux of energy. That is, the electric energy is converted into heat on the spot

A considerable extension may be given to this property. If there be a conducting dielectric in which the permittivity varies from place to place, containing a static distribution of displacement, then, if the conductivity vary similarly from place to place, so that the time-constant ejh is the same every- where, the displacement will subside everywhere alike, without magnetic force or flux of energy, and with purely local dis-

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THEORY OF PLANE ELECTR0MA6KE11C WAVES. 363

sipatioii of the electric energy. For the Bolution is repre- sented by

where Eq is the initial electric force of the static kind, having no curl, and E that at time t. Both the fundamental circuital laws are satisfied, the first because the true current is the Bum of the conduction and displacement currents, and the second because Eq has no curl and k/c is constant. If it were not constant then, obviously, the property considered would not be true; there would be different rates of subsidence at different placee, and the distribution of displacement would change^ along with magnetic force, electric current and transfer of energy.

The corresponding property in a magnetic conductor requires the constancy of the time-constant fi/g. Then, whether fj. and g are themselves constant or variable from place to place, a static distribution of induction subsides everywhere alike, and without the generation of electric force.

Retuniing again to plane strata of displacement in an deotrioally-cooducting homogeneous dielectric, it may be inquired how the property (1) of the subsidence of the total displacement will be affected by the simultaneous existence of magnetic conductivity. This will undoubtedly affect, the phenomena in detail, but will have no effect on the property in question. Similarly, the law (2) of the subsidence of total mduction will not be affected by the presence of electric con- ductivity. That is, in general, when there are both conduc- tivities present, and both the fluxes displacement and induction present, the total displacement subsides according to one law and the total induction according to the other, without inter- ference. These properties have their parallels in the theory of telegraph circuits, as we shall see later.

It should be remembered that we are dealing always with matter in the gross, and not with molecules at all ; or, equiva- lently, we assume a homogeneous constitution of the elements of volume. Thus, when displacement subsides in an electric conductor without generating magnetic force, the possibility and necessity of which are clearly indicated by the two cir- cuital laws, it may be that if we go in between the molecules

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there is magnetic force. It is, in fact, difl5cult to conceive how displacement in a heterogeneous medium of molecular consti- tution could be done away with without the generation of magnetic force, considering that the energy of the displace- ment is converted into heat energy.

This matter, however, does not belong to the skeleton theory of electromagnetism, but is rather to be considered as a side- matter involving physical hypotheses to account for the influ- ence of matter upon the electromagnetic laws.

The Laws of Attenuation at the Front of a Wave, dae to- Electric and liagnetic Conductance.

§ 196 Besides the above simple laws relating to the subsi dence of the total fluxes (sometimes true for the elementary parts) there are equally simple laws relating to the subsidence of the fluxes at the front of a wave advancing into previously undisturbed parts of the medium, which sometimes admit of extension to the body of the wave. To understand this it may be mentioned first, that the front of a wave in a non-conduct- ing dielectric is always pure ; that is, the electric and magnetic fluxes are in the wave-plane, and are perpendicular and in constant ratio. The body of the wave need not be of this pure type, owing to the change of form of the wave-front and other causes, but the property of purity always characterises the wave-front. This may be disguised in the case of a thin electromagnetic shell, when it is regarded as the front, for the shell itself may be complex. Then the mere front of the shell may be the only quite pure part. But taking cases free from this oomplicatioD, we should next note that the introduction of conductivity into the medium makes no diflerenoe in the form of the wave-front or its position at a given stage of its progress, provided, of course, that the two quantities upon which the speed of propagation depends — the inductivity and permittivity — are not altered. Now, as haa been already explained in connection with the theory of a thin conducting plate, as the wave advances through a continuously conducting medium its successive layers are being continuously subjected to a reflecting process, a minute portion of every layer being thrown back, whilst the bulk is transmitted. In

THEORY OF PLANE ELECTROMAGNETIC WAVE3.

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the body of a wave, therefore, there is a mixed-up state of things. At the very front, on the other hand, there is no such mixture, for the disturbance consists wholly of what has been transmitted of the front layer. We may, therefore, fully expect that the law of its attenuation in transit is of a simple nature.

To find it, locally condense the conductance into that of any number of equal conducting plates. Let any one of these plates attenuate a wave traversing it from E to wE. If initially pure it emerges a pure wave, and passes on to the next plate, where it suffers a second attenuation — viz., to m^E, and again emerges pure. At the third plate it becomes m^E, and so on. The reflected portions we wholly ignore at present. The limit of this process, when the plates are in- finitely closely packed and of infinitely small conductance, so as to become a homogeneous dielectric possessing finite con- ductivity, is that the time-factor of attenuation takes the exponential form. The result is

Eq being the initial, and E the value at time t. The time* constant 2clk is just double that of the subsidence of total displacement. Whilst, for example, the total displacement in a ])l:uie wave attenuates to, say, of its initial value, the disturbance at the wave front has only attenuated to of its original value.

The property (3) applies to the magnetic as well as to the eleotrio force and flux. It does not apply merely to plane waves, but to any waves, because the superficial layer only is involved, and any elementary portion thereof may be regarded as plane. So it comes about that the exponential factor given in (3) makes its appearance in all investigations of waves in electrical conductors when the permittivity is not ignored. It is a more fundamental formula than the previous one with the time-constant e/kf which is the final result of the complex pro- cess of mixture of reflected waves, or is equivalent thereto.

The corresponding property in a magnetic conductor is that the disturbance at the front of a wave is attenuated in time t according tc the timensonstant 2fi/g. Thus,

(3)

(4)

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Here the time-constant is twice that of the subsidenoe of the total induction. Like the former formuln, these modified ones, (3) and (4), have then: representativeB in the theory of a telegraph circuit, in spite of the abeenoe of magnetio con* duotance. It is replaced by something that prodnoes approxi- mately the same result.

In a oonduotor possessing duplex oonduotivity, eleotrio and magnetio, their attenuative actions at the wave-front are independently oamulative^ or additive. The attenuation is expressed by

E-Ai;H-Eo€-(/2«+17/?^)*. ... (5)

It is really the attenuative actions of a single conducting plate that are additive. This applies separately to every successive thin conducting layer through which the front of the wave runs, with the result (5), where the time-factor is the product of the two former time-factors of (3) and (4).

In the theory of coils and condensers, not only do we meet with the time-constants L/R and S/K, the ratios of inductance to resistance and of permittance to conductance, but also with the double values. Their ultimate origin may be traced in the theory of the effect of a thin conducting plate upon a wave.

The exponential time-fisM)torB concerned in (3) and (4), and tlie more complex one in (5), also make their appearance in oonnection with the disturbance in the body of a wave, though Jn a less simple manner. This will be returned to.

The Simple Propagation of Waves in a Distortionless

Oondncting Medium.

§ 197. Coming now to the influence of conductivity on a wave elsewhere than at its extreme front, where we have recognised that the influence is simply attenuative, the easiest way of treating the matter is not to pass from the known to the unknown, but to reverse the process and pick out the cases which theory indicates are most readily understandable. This is to be done by a process of generalisation. The theory of a conductor with duplex conductivity is, in a certain case, far simpler than that of a real electric conductor. We have already mentioned that the reflective actions of two plates, one an electric, the other a magnetic conductor, are of oppo-

THEOHT OF PLANE ELEOTBOMAGNinO WATB. 867

site natures. The fint reverses the displacement, and the seoond reyerses the induction when throwing back a portion of the waye. The joint action of the two plates when coexistent and coincident, or the action of a single plate with duplex conductance, results in a complete disappeaiance of the reflected wave when the conductances are in proper ratio and the plate is infinitely thin. We then have transmission with attenuation but without reflection. This occurs, in a homo- geneous medium, when k/c^g/iJi. Reflex action being abolished, we are reduced to a kind of propagation of unique simplicity.

To see the full meaning of this, start firom any initial distribu- tions of induction and displacement in a non-conducting dielec- tric. Imagine that we have obtained the full solution showing the subsequent history of the disturbances. Now, if we introduce only one kind of couductivity, say electric, we shall, with the same initial state, have a profoundly different subsequent his- tory. Again, with magnetic conductivity alone, we shall have a course of events different from both the previous. But if we add on magnetic conductivity to previously existent electric conductivity, we shall partly counteract the distorting influ- ence of the latter. This counteraction becomes complete when the value of the magnetic conductivity is raised so high as to produce equality of the time-constants of attenuation due to the two conductivities separately. Further increase of the magnetic conduc.tivity will overdo the correction and bring on distortion again, though of a different kind.

Similarly, the distortion due to magnetic conductivity alone is diminished by introducing electric conductivity, and becomes completely abolished when there is enough of the latter to equalise the time-constants. Further increase brings on the distortion again, which is now of the electric kind.

When the state of balance occurs, and the distortion is wholly removed, the course of events following any initial state is precisely the same as in a non-conducting medium, but with a continuous attenuation expressed by equation (5) above spe- cialised to suit the equality of the time-constants. That is, the time-factor of attenuation is now r~^^ This removal of dis- tortion applies to every kind of wave.

This distortionless state in conducting media furnishes a sort of central basis for investigating the more recondite effects

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aooompaiKyiiig diBtortion. NeverthelesB, its oonsideration would pOBsess oiUy a theoretical value, on aooountof the non-ezistenee of the second kind of oondnotivity involved, were it not for the remarkable practical imitation of the distortionleas state of things which is presented in the theory of telephone and other circuits under certain cireumstances. If we abolish the ficti- tious magnetic conductivity throughout the medium traversed by the waves, we should, to have distortionless transmission, also abolish the electric conductivity. This is only to be attained by using wires of no resistance to guide the waves through a non-oonduoting medium. But they have resistance, of greater or lesser importance according to circumstances. Of what nature, then, is the distortion of waves produced by the resistance of a wire along which they runf The answer fa, that it is approximately of the kind due to magnetic con- ductivity in the medium generally. On the other hand, the different kind of distortion due to electric .louduc- tivity in the medium generally remains in actioo, bein^ tlie eHect of the le ikage-coii'luctauce of the i isnlatini.' medium surrounding the wire, or the average eftect of other kinds of leakage at distinct and separate spots along the circuit. Thus we obtain an approximate reproduction of the theory of magnetic conductivity acting to neutralise the distorting effect of electric conductivity. The time-constants fi/g and c/k become L'R and S/K in a telegraph circuit, L being inductance, U resistance, S permittance, and K leakage-conductance. Their equalisation produces the distor- tionless circuit, which may turn up again later on. In the meantime I may remark that if the reader wishes to under- stand these things, he must give up any ancient j-rejudices he may be enamoured of about a " KK law " and the consequent impossibility of telephoning when " KR " is over 10,000. When pointing out, in 1887, the true nature of the telephonic problem and the absurdity of the " KR law " applied thereto generally, I predicted the possibility of telephoning with " KR " several times as great. It has since been done. In America, of course. A short time since, in noticing the KR«b 32,000 reached by the New York-Chicago circuit, 1 further pre- dicted that it would go up a lot more. It did very shortly after. The record is now about 50,000 (Boston-Chicago) for practical

TEBORY OP PLANE BLKCTROMAGNETIC WAVES. 369

work, I believe. It means a good deal more for possible work. But there is no need to stop at 50,000. That can be largely exceeded in an enterprising country.

Tlie Transfonnation by Oonductance of an Elastic Wave to a Wave of DifAudon. Generation of Tails. Distinct Effects of Electric and Magnetic Conductance.

§ 198. We are now prepared to somewhat understand the nature of the changes sutfered by electromagnetic waves in transit through a conducting medium. It being the distortion due to the conductance alone that is in question, we eliminate that due to other causes by choosing plane waves for examina- tion, since these do not suffer any distortion in a homogeneous dielectric wlien it is non-conducting. Imagine, then, a simple electromagnetic plane wave-sheet of small depth to be running through a dielectric at the natural speed conditioned by its inductivity and permittivity. At any stage of its progress, let the medium become slightly electrically conductiiiiif all over, not merely in advance of the wave but behind it as well, for a reason that will presently appear. What happens to the wave now that the fresh influence is in operation ?

A part of the answer we can give at once, by the pre- vious. The wave-sheet will move on just as before, but will attenuate as it goes, according to the time-factor €-*«/2c. Since we suppose the conductivity to be slight, it follows that a great distance may be traversed before there is notable attenuation. We also know that the total induction remains constant. The rest of it — that is, what is not in the sheet at any moment — is therefore left behind. The rejecting process commences the moment the conduotivity is introduced, and continues to act until the plane wave is attenuated to nothing. The rejected portions travel backwards. But th^ are themselves subject to the same laws as the main plane wave, and so get mixed up. The result is that at time t after the introduction of the con- ductivity, the whole region of disturbance extends over the distance 2vi, half to the right and half to the left of the initial posiUon. At the advancing right end we have a strong condensed disturbance, vi&, the original wave attenuated, and behind it a weak diffiised one. We can therefore, without

BB

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misunderstanding refer to them as the head and the tail, with- out any body to complicate matters. Now the nature of the tail is quite different as regards the displacement and the induction. It is therefore convenient to regard one of them alone in the first place, and, of course, we select the induction, on account of the simple property of persistence that it possesses. We can distinguish three or four dififerent stages in its development.

The first stage is when the attenuation of the head is not great, say, whilst the head decreases from 1 to 0-75. Whilst this occurs, the total induction in the tail rises from 0 to 0 25. The tail is long and thin, and tapers to a point at its extreme end, or tip, at distance 2vt behind the head, and is thickest where it joins on to the head.

The second stage roughly belongs to the period during which the head further attenuates from 075 to 05 or 0-4. The total induction in the tail then increases from 025 to 0'5 or 06. During this stage we find that the tail, which has, of course, greatly increased in length, does not go on increasing in thick- ness at the place where it is developed, hut stops increasing and shows a maximum at or near that place.

The third stage occurs during the further attenuation of the head to, say, 01, whilst the total induction in the tail increases to 09. The maximum thickness of the tail is now a long way from the head, and at the end of the stage is nearer to the middle than to the head. Of course, since the head itself is now so small, the additions made to the tail must also hecome smaller.

The fourth stage is when the head practically disappears and all the induction is in the tail. The maximum thickness is now nearly in the middle — on the right side, however — and the tail is nearly symmetrical with respect to its middle, where there is a swelling, beyond which the tail tapers off both ways to its two tips.*

The final state is the con^^ummation of the previous, and is one of perfect symmetry with respect to the middle of the tail, which is situated exactly where the plane wave was when the

• As the diviston into distinct stages is somewhat arbitrary, this deaerip- tion of the transition from an elastic to a diS^tsion wave should be under- stood to be only roughly approximate. It is made up, not from the formula, but by a numerical process of mixture.

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TUKORT OF PLANE ELECTBOMAONETIC WAVE3. 371

spreading began. The spreading now takes place according to the pure ditfusion law, as of heat by conduction.