Electromagnetic Theory, Vol. 1 (1893) — part 24 of 31

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

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

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

358 ELECTROMAGNETIC THEORY. CH. IV.

of any number of such sheets, of any strengths, forming a per- fectly arbitrary initial distribution of induction and displace- ment in parallel plane layers. There is still persistence of the induction. Finally, the same applies when we split up the conducting plates themselves into plates of smaller con- ductance, and spread them out at uniform distances. The ultimate limit of this process is reached when the conduct- ance is quite uniformly spread, so that we have a perfectly homogeneous medium under consideration. It is, fundament- ally, a dielectric propagating disturbances at speed v ; but it is, in addition, a conductor as well, and distorts the waves and dissipates their energy. The speed v is (MC)"*, with the proper values of //. and c. The conductivity does not interfere with this property of propagation at finite speed. But observe that if we choose to ignore the displacement, then the corresponding speed is infinitely great. We conclude from the above that plane sheets of induction in electric conductors always preserve the total induction constant in amount, irrespective of the amount of elastic displacement, or whether there is any at all. That is, induction cannot be destroyed by conductance.

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

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

THEORY OP PLANE! ELECTROMAGNETIC WAVES. 359

with its magnetisation. Let it be inductised by an enveloping coil-current so that the induction goes along the core in a complete circuit. When it is steadily set up, if we remove the coil- current (and the coil too, preferably for our present purpose) the induction in the core will, in time, all come out of it. But if we clap an infinitely-conducting skin upon the core, it will not come out. Then we have a certain flux of induction locked up, as it were, in a conducting material, which has no effect upon it. It can neither be destroyed by the conductance of the core nor can it get through the perfectly-obstructive skin. If the skin is clapped on after the induction has partially escaped (which escape begins on the outside, before the interior is sensibly affected), there is a redistribution of induction, which continues until a new state of equilibrium is reached. During this process there is electric current in the core and some waste of energy. But there is no waste of the induction. The final induction is the mean value of .the original induction across the section of the core.

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

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

360 ELECTROMAGNETIC THEORY. OH. IV.

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

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

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

The Laws of Attenuation of Total Displacement and Total Induction by Electric and Magnetic Conductance.

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

THEORY OF PLANE ELECTROMAGNETIC WAVES. 361

with the time, and the law of decrease may be readily found from the theory of a single conducting plate. We found that when its conductance exceeded a certain value, the loss of dis- placement exceeded the original. But, in regarding the action of a homogeneous conductor upon a wave as the limit (in the gross) of that of an assemblage of parallel plates in which the conductance is localised (which process may not seem unassail- ably accurate beforehand, but which is justified by the results), it is easy to see that we have merely to deal with plates of such very low conductance that the loss at each is extremely small, so that the above-mentioned difficulty does not enter. Thus, let the loss at one plate be such as to reduce the initial displacement D in a wave to mD, where m is a fraction nearly equal to unity. Here mD is the sum of the displacements in the transmitted and reflected waves, the latter being very small and of the opposite sign to the initial D. As these waves separate, they reach other plates and are split anew. If these plates have each the same conductance as the first, the total mD is further attenuated by them to w2D when the two waves become four. Next, when these four waves are split into eight by the next plates that are reached, the total displacement becomes m3D ; and so on. These successive displacement totals decrease according to the law of a geometrical series. It fol- lows that, in the limit, we shall have the total displacement represented by an exponential function of the time, say by

D = Doe-«<, (1)

where D0 is the initial value, and D what it becomes at time t. To find the value of n, we have merely to examine the form of the fraction m, observe how it depends on the conductance of one plate, and proceed to the limit by making the number of plates infinite, whilst their conductances are infinitely small. The result is that the constant n has the value &/c, where k is the conductivity and c the permittivity of the homogeneous conducting medium.

In the irrational units of the B.A. Committee this quantity is represented by 47r&/c, which is, of course, nonsense, like the quaternionic doctrine about the square of a vector. They are both going to go. The above reasoning applies to any initial distribution of displacement in plane layers, instead of merely

362 ELECTROMAGNETIC THEORY. CH. IV.

one elementary sheet. Therefore, equation (1) shows that the total displacement subsides according to the time-factor e ~ 'c' Now, this represents Maxwell's law of subsidence of displacement in a conducting condenser (apart from " absorp- tion" and hysteresis), or of the static distribution of displace- ment associated with electrification in a conducting medium. We see that the law has a far more general meaning. The initial displacement need not be static, but may be accom- panied by magnetic induction, and may consequently move about in the most varied manner, whilst its total amount decreases according to the static law. A homogeneous medium is presupposed, and modifications may be introduced by the action of boundaries.

Passing next to the analogous case of a magnetic conductor, in which the total displacement remains constant whilst the total induction subsides, it is unnecessary to repeat the argument, but is sufficient to point out the law according to which the subsidence occurs. If B0 be the initial total induction, and B what it becomes at time t, we shall have

(2)

where g is the magnetic conductivity and p. the indue tivity. The time-constant cjk of the former case has become p/g.

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

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

THEORY OP PLANE ELECTROMAGNETIC WAVES. 363

sipatipn of the electric energy. For the solution is repre- sented by

E = E0, -**, H = 0,

where E0 is the initial electric force of the static kind, having no curl, and E that at time t. Both the fundamental circuital laws are satisfied, the first because the true current is the sum of the conduction and displacement currents, and the second because E0 has no curl and k/c is constant. If it were not constant then, obviously, the property considered would not be true ; there would be different rates of subsidence at different places, and the distribution of displacement would change, along with magnetic force, electric current and transfer of energy.

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

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

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

364 ELECTROMAGNETIC THEORY. CH. IV.

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

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

The Laws of Attenuation at the Front of a Wave, due to Electric and Magnetic Conductance.

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

THEORY OP PLANE ELECTROMAGNETIC WAVES. 365

the body of a wave, therefore, there is a mixed-up state of things. At the very front, on the other hand, there is no such mixture, for the disturbance consists wholly of what has been transmitted of the front layer. We may, therefore, fully expect that the law of its attenuation in transit is of a simple nature. To find it, locally condense the conductance into that of any number of equal conducting plates. Let any one of these plates attenuate a wave traversing it from E to wE. If initially pure it emerges a pure wave, and passes on to the next plate, where it suffers a second attenuation — viz., to w2E, and again emerges pure. At the third plate it becomes w3E, and so on. The reflected portions we wholly ignore at present. The limit of this process, when the plates are in- finitely closely packed and of infinitely small conductance, so as to become a homogeneous dielectric possessing finite con- ductivity, is that the time-factor of attenuation takes the exponential form. The result is

E = E0e-*</2<, (3)

E0 being the initial, and E the value at time t. The time- constant 2c/k is just double that of the subsidence of total displacement. Whilst, for example, the total displacement in a plane wave attenuates to, say, y^- of its initial value, the disturbance at the wave front has only attenuated to j1^ of its original value.

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

The corresponding property in a magnetic conductor is that the disturbance at the front of a wave is attenuated in time t according to the time-constant 2p/g. Thus,

(4)

366 ELECTROMAGNETIC THEORY. CH. IV.

Here the time-constant is twice that of the subsidence of the total induction. Like the former formulae, these modified ones, (3) and (4), have their representatives in the theory of a telegraph circuit, in spite of the absence of magnetic con- ductance. It is replaced by something that produces approxi- mately the same result.

In a conductor possessing duplex conductivity, electric and magnetic, their attenuative actions at the wave-front are independently cumulative, or additive. The attenuation is expressed by

: ... (5)

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

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

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

The Simple Propagation of Waves in a Distortionless Conducting Medium.

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

THEORY OP PLANE ELECTROMAGNETIC WAVES. 367

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

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

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

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

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

368 ELECTROMAGNETIC THEORY. CH. IV.

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

THEORY OP PLANE ELECTROMAGNETIC WAVES. 3G9

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

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

A part of the answer we can give at once, by the pre- vious. The wave-sheet will move on just as before, but will attenuate as it goes, according to the time- factor e~ kti2c. Since we suppose the conductivity to be slight, it follows that a great distance may be traversed before there is notable attenuation. We also know that the total induction remains constant. The rest of it — that is, what is not in the sheet at any moment — is therefore left behind. The rejecting process commences the moment the conductivity is introduced, and continues to act until the plane wave is attenuated to nothing. The rejected portions travel backwards. But they are themselves subject to the same laws as the main plane wave, and so get mixed up. The result is that at time t after the introduction of the con- ductivity, the whole region of disturbance extends over the distance 2vt, half to the right and half to the left of the initial position. At the advancing right end we have a strong condensed disturbance, viz., the original wave attenuated, and behind it a weak diffused one. We can therefore, without

BB

370 ELECTROMAGNETIC THEORY. CH. IV.

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

The first stage is when the attenuation of the head is not great, say, whilst the head decreases from 1 to 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 0'75 to 0'5 or 0'4. The total induction in the tail then increases from 0"25 to 0-5 or 0'6. During this stage we find that the tail, which has, of course, greatly increased in length, does not go on increasing in thick- ness at the place where it is developed, but stops increasing and shows a maximum at or near that place.

The third stage occurs during the further attenuation of the head to, say, 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 become smaller.

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

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

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

THEORY OP PLANE ELECTROMAGNETIC WAVES. 371

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

Now as regards the displacement in the head and tail in the different stages. In the first and second stages the displace- ment is wholly negative in the tail, assuming it to be positive in the head, where, it should be remembered, it attenuates in the same manner as the induction which accompanies it. Thus, when the head has fallen to 0'9, the total displacement in the head and tail has fallen to (0'9)2 or 0-81, so that the total nega- tive displacement in the tail is of amount 0'09, which is not much less than the coincident induction. And when the head has attenuated to 08, and the total displacement to (08)2 or 0-64, the negative displacement in the tail amounts to 0'16. But, unlike the induction, the displacement increases in the tail from the head up to not far from the tip, where, of course, it falls to zero. There is no tip at the forward end. But as the tail stretches out further to the left, and has fresh addi- tions made to it on the right side, the decrease of the density of displacement in passing towards the head continues, until somewhere about the end of the second stage, it becomes zero next the head. This node is approximately at the place where the induction has its maximum. When the head has fallen to 04, we have the total displacement attenuated to 016, so that the negative displacement in the tail amounts to 0'24.

In the third stage the displacement is negative from the tip up to somewhere near and beyond the maximum of induction, and increasingly positive in the remainder, up to the head. That is, the region of positive displacement now extends itself from the head a good way into the tail. At the same time the place of maximum negative displacement moves forward.