Heaviside Electromagnetic Induction And Its Propagation Sec XL 2 XLVII



Although there is more to be said on the subject of induction- balances, I put the matter on the shelf now, on account of the pressure of a load of matter that has come back to me under rather curious circumstances. In the present Section I shall take a brief survey of the question of long-distance telephony and its prospects, and of signalling in general. In a sense, it is an account of some of the investigations to follow.

Sir W. Thomson’s theory of the submarine cable is a splendid thing. His paper on the subject marks a distinct step in the development of electrical theory. Mr. Preece is much to be congratulated upon having assisted at the experiments upon which (so he tells us) Sir W. Thomson based his theory; he should therefore have an unusually complete knowledge of it. But the theory of the eminent scientist does not resemble very closely that of the eminent practician.

But all telegraph circuits are not submarine cables, for one thing; and, even if they were, they would behave very differently according to tlie way they were worked, and especially as regards the rapidity with which electrical waves were sent into them. It is, I believe, a generally admitted fact that the laws of Nature are immutable, and everywhere the same. A consequence of this fact, if it be granted, is that all circuits whatsoever always behave in exactly the same manner. This conclusion, which is perfectly correct when suitably interpreted, appears to contradict a former statement; but further examination will show that they may be reconciled. The mistake made by Mr. Preece was in arguing from the particular to the general. If we wish to be accurate, we must go the other way to work, and branch out from the general to the particular. It is true, to answer a possible objection, that the want of omniscience prevents the literal carrying out of this process; we shall never know the most general theory of anything in Nature; but we may at least take the general theory so far as it is known, and work with that, finding out in special cases whether a more limited theory will not be sufficient, and keeping within bounds accordingly. In any case, the boundaries of the general theory are not unlimited themselves, as our knowledge of N ature only extends through a limited part of a much greater possible range.

Now a telegraph circuit, when reduced to its simplest elements) ignoring all interferences, and some corrections due to the diffusion of current in the wires in time, still has no less than four electrical constants, which may be most conveniently reckoned per unit length of circuit—viz., its resistance, inductance, permittance, or electrostatic capacity, and leakage-conductance. These connect together the two electric variables, the potential-difference and the current, in a certain way, so as to constitute a complete dynamical system, which is, be it remembered, not the real but a simpler one, copying the essential features of the real. The potential-difference and the permittance settle the electric field, the current and the inductance settle the magnetic field, the current and resistance settle the dissipation of energy in, and the leakage-conductance and potential-difference that without the wires. Now, according to the relative values of these four constants it is conceivable, I should think, by the eminent engineer, that the results of the theory, taking all these things into account, will, under different circumstances, take different forms. The greater includes the lesser, but the lesser does not include the greater.

In the case of an Atlantic cable it is only possible (at present) to get a small number of waves through per second, because, first, the attenuation is so great, and next it increases so fast with the frequency, thus leading to a most prodigious distortion in the shape of irregular waves as they travel along. Of course we may send as many waves as we please per second, but they will not be utilisable at the distant end. This distortion is a rather important matter. Mere attenuation, if not carried too far, would not do any harm. Now the distortion and the attenuation, though different things, are intimately connected. The more rapidly the attenuation varies with the frequency, the greater is the distortion of arbitrary waves; and if the attenuation could be the same for all frequencies, there would be no distortion. This can be realised, very nearly, as will appear later.

Now when there are only a very few waves per second, the influence of inertia in altering the shape of received signals becomes small, and this is why the cable-theory of Sir W. Thomson, which wholly ignores inertia, works as a substituted approximate theory. But suppose we shorten the cable continuously, and at the same time raise the frequency. Inertia becomes more and more important; the theory which ignores it will not suffice; and carrying this further, we at length arrive at a state of things in which the old cable-theory gives results which have no resemblance whatever to the real. This is usually the case in telephony, as I have before proved. It is always partly the case, viz., for the very high frequencies, and it may be true, and practically is sometimes, down to the low frequencies also. I have shown that the attenuation tends to constancy as the frequency is raised, except in so far as the resistance of the wire increases, and that at the same time the speed of the waves tends to approximate to the speed of light, or to a speed of the same order of magnitude, which is the only speed which can, I think, be said, even in a restricted sense, to be the “ speed of electricity.” But if the dielectric be solid, there must be some uncertainty about what this speed is, for obvious reasons, with very high frequencies. The speed of the current is never proportional to the square of the length of the line.

Within the limits of approximately constant attenuation the distortion is small. This is what is wanted in telephony, to be good. Lowering the resistance is perhaps the most important thing of all. Other means I will mention later. What the limiting distance of longdistance telephony may be, who can tell ? We must find out by trial. We know that human speech admits of an extraordinary amount of distortion (never mind the attenuation) before it becomes quite unrecognisable. The “perfect articulation,” “even different voices could be distinguished,” etc., etc., mean really a large amount of distortion, of which little may be due to the circuit. There is the transmitter, the receiver, and several transformations between the speaker and the listener, besides the telephone line. What additional amount of distortion is permissible clearly must depend upon what is already existent due to other causes. Even if that be fixed, I see no legitimate way of fixing its amount by theoretical principles; the matter is too involved, and includes too many unknown data, including “ personal equation.” But this is certain, in my opinion—that good telephony is possible through a circuit whose electrostatic time-constant, the product of the total resistance into the total permittance, is several times as big as the recent estimate of Mr. W. H. Preece, and I shall give my reasons for this conclusion.

Increasing the inductance is another way of improving things. Hang your wires wider apart. The longer the circuit, the wider apart they should be; besides this, they may be advantageously raised higher. You can then telephone further, with similar attenuation and distortion. There is a critical value of the inductance for minimum attenuation- ratio. It is from L = Rl/2v to L = Rljv>, according to circumstances to be later explained; L being the inductance and R the resistance per unit length, I the length, and v the speed of waves which are not, or are only slightly dissipated, which is (LS)~t, if S be the permittance per unit length. The resulting attenuation may be an enlargement, as I have before explained, due to to-and-fro reflections. This is to be avoided. I shall explain its laws, and how to prevent it. Bj' this method, carrying it out to an impracticable extent, however, we could make the amplitude of sinusoidal currents received at the distant end of an Atlantic cable greater than the greatest possible steady current from the same impressed force—an unbelievable result. And, without altering the permittance or the resistance, we could make the distortion quite small.

There is some experimental evidence in favour of increasing the inductance (apart from lessening the permittance); though, owing to want of sufficient information, I do not wish to magnify its importance. I refer to the statement that excellent results have been obtained in long-distance telephony with copper-covered steel wires. Here the copper covering practically decides the greatest resistance of the wire; what current penetrates into the steel lowers the resistance and increases the inductance. Clearly, we should magnify this effect, and, electrically speaking, it would seem that a bundle of soft-iron wires with a covering of copper is the thing, as this will allow the current to penetrate more readily, lower the resistance the most, and increase the inductance the most. But it is too complex a matter for hasty decision. We also see that the iron sheathing of a cable may be beneficial.

When we have little distortion, we get into the regions of radiation. The dielectric should be the central object of attention, the wires subsidiary, determining the rate of attenuation. The waves are waves of light, in all save wave-length, which is great, and gradual attenuation as they travel, by dissipation of energy in the wires. There is the electric disturbance and the magnetic disturbance keeping time with it, and perpendicular to it, and both perpendicular to the transfer of energy, which is parallel to the wire, very nearly. A tube of energy- current may be regarded as a ray of light (dark, of course).

It is to such long waves that I attribute the magnetic disturbances that come from the sun occasionally, and simultaneously show themselves all over the world; arising from violent motions of large quantities of matter, giving shocks to the ether, and causing the passage from the sun of waves of enormous length. On such a wave passing the earth, there are immediately induced currents in the sea, earth’s crust, telegraph lines, etc.

But to return to the circuit. The attenuation-ratio per unit length is represented by e~BI2Lv, this being the ratio of the transmitted to the original intensity of the wave. This is when the insulation is perfect. These waves are subject to reflection, refraction, absorption, etc., according to laws I shall give. Of these the simplest cases are reflection by short-circuiting, when the potential-difference is reversed by reflection, but not the current, and in the act of reflection the former is annulled, the latter doubled. Also reflection by insulation, when it is the current that is reversed, and potential-difference unchanged; or, in the act of reflection, the first cancelled, the second doubled. But there are many other cases I have investigated.

I have also examined leakage. This is an old subject with me. An Atlantic cable is worked under the worst conditions (electrical) possible with high insulation; there is the greatest possible distortion. One megohm per mile or less instead of hundreds or thousands would vastly accelerate signalling. The attenuation-factor is now e~R,Uv. cK^vy if K be the leakage-conductance, and S the permittance per unit length. The attenuation is increased, but the distortion is reduced. This has led me to a theoretically perfect arrangement. Make RjL = K/S, and the distortion is annihilated (save corrections for increased resistances, etc.). The solution is so simple I may as well give it now. Let V and C be the potential-difference and current at distance x, subject to

_ RC+ LC, = KV+ SV; dx dx

then, with equality of time-constants as described, the complete solution consists of two oppositely travelling trains of waves, of which we need only write one; thus,

V=f(x - vt)

where f(x) is the state when t — 0. The current is C=V/Lv. The energy is half electric, half magnetic; the dissipation is half in the wire, half outside. Change the sign of v in a negative wave.

There is a perfect correspondence of properties, when this unique state of things is not satisfied, between V solutions with K= 0, and C solutions with R = 0. This perfect system would require very great leakage in an * Atlantic cable, and cause too much attenuation; but this perfect state may be aimed at, and partly reached. Are there really any hopes for Atlantic telegraphy ? Without any desire to be over sanguine, I think we may expect great advances in the future. Thus, without reducing the resistance or reducing the permittance (obvious ways of increasing speed), increase the leakage as far as is consistent with other things, and increase the inductance greatly. One way is with my non-conducting iron, which I have referred to more than once, an insulator impregnated with plenty of iron-dust. Use this to cover the conductor. It will raise the inductance greatly, and so greatly diminish the attenuation ; whilst the insulation-resistance will be lowered, somewhat increasing the attenuation, but assisting to diminish the distortion, which the increased inductance does. The change in the permittance must also be allowed for. But I shall show that we can have practical approximations to almost negligible distortion in telephony, and that it is the reduction of RjL that is most important.

I have also examined the question of apparatus. We must stop the reflection, if possible, to prevent interference. In the perfect system this is also quite easy. The receiver must have resistance Lv and zero inductance. All waves arriving are then wholly absorbed. Similarly, to make the transmitted waves agree with the impressed force, Lv should be the resistance there, (or else zero). Another remarkable property is that if the receiving coil be fixed in size and shape, whilst its resistance varies, then this same Lv is the resistance that makes the magnetic force of the coil a maximum. We cannot imagine anything more perfect. No distortion, and maximum effect. I shall show that these things may be fairly approximated to in telephony. It should be understood that in the perfect system we have nothing to do with what the frequency may be, whilst in telephony it is the high frequency that allows us to approximate to the ideal state.

Then there is the matter of bridges, and the nature of the reflected, transmitted, and absorbed waves. The phenomena formally resemble those due to the insertion of resistance in the main circuit, except that the potential-difference and the current change places. Thus if be an inserted resistance, when there is no leakage and no resistance in the line (1 + RfZLv)'1 is the ratio of transmitted to incident wave. Now let there be no resistance inserted, but a bridge of conductance Kx; then the substitution of Kx for R,, and S for L gives us the corresponding formula. In the first case the reflected current is reversed, in the second case it is the potential-difference of the reflected wave that is reversed. Now let there be both a resistance inserted and a conducting bridge, and choose RJL = KJS; then the reflected wave is abolished. Part of the original wave is absorbed in the bridge, and the rest is transmitted unchanged. This explains the perfect system above described.

I have also examined the changes made when the state is not perfect. The result is that a wave throws out a long slender tail behind it; and whilst the nucleus goes forward at speed % the tail goes backwards at this speed. In time, if the line be long enough, the nucleus, which changes shape as it progresses, diminishes so as to come to be a part of the tail itself. It is then all tail. I will give the equation of the nucleus and tail. It is the mixing up of these tails that causes arbitrary waves to be distorted as they travel from beginning to end of the line. (But I have, in the above, usually referred to distortion as the change in the shape of the curve of current at a single spot.) There is residual reflection due to the self-induction of the receiver, even when the resistance is of the proper amount. The effect of diffusion in the wires is to make a wave with an abrupt front, which would continue abrupt, have a curved front, and thus mitigate that perfection which only exists on paper. I shall also describe graphical methods of following the progress of waves, and of calculating arrival-curves of various kinds, the submarine cable and oscillatory; approximate only, but very easy to follow. Other matters, perhaps more practical, but certainly duller, will find their place, if space allow.



TO explain the word “ permittance ” that I used in the last Section, I may remark that in stating my views in 1885 in several communications to this journal on the subject of a systematic and convenient electrical nomenclature based upon the explicit recognition of the three fluxes, conduction-current, magnetic induction, and electric displacement, proposing several new words, some of which have found partial acceptance, I remarked upon the unadaptable character of the word “capacity.” It must be the capacity of something or other, as of permitting displacement. I did not then go further in connection with the flux displacement than to use “ elastance,” for the reciprocal of electrostatic capacity. The following shows the scheme so far as it is at present developed:—


r, i n , (. Resistance. Conductance. I ™ . .

Conduction-Current j j.csistivity Conductivity. } Electr,c-

Induction j » Inductivity. } Magnetic.

t\' i . I Elastance. Permittance. 1 , .

Displacement -j Elastivity. Permittivity. } Electrlc'

Why elastivity ? Maxwell called the reciprocal of the permittance of a unit cube “ the electric elasticity.” By making it simply elastivity, we first get rid of the qualifying adjective; next, we avoid confusion with any other sort of elasticity; and, thirdly, we produce harmony with the rest of the scheme. There are now only two gaps. “Resistance to lines of force,” or “magnetic resistance,” now used, will not do for permanent employment. Besides the above, there is Impedance, to express the ratio of force to flux in the very important case of sinusoidal current. Impedance is at present known by various names that seem to be founded upon entirely false ideas. The impedance (which, derived from impede, need not be mispronounced) of a coil is the ratio of the amplitude of the impressed force to that of the current. A coil used for impeding may be called an impeder. The same definition obviously applies in any case that admits of reduction to one circuit (even though parts of it may be multiple), e.g., any number of coils in sequence, in sequence with any number in parallel (to be regarded as one), in sequence with a condenser, or arrangement reducible to a condenser. The impedance is always reducible to (R? + L2n2)l, where R is the effective resistance, which is real, and L the effective inductance, or sometimes yuasMnductance. It is not necessary to exclude inductive action on other circuits, although the heat corresponding to R may be partly in them. As for resistance, it is very desirable to confine its use to the established meaning in connection with Joule’s law.

Now let R, L, S and K be the resistance, inductance, permittance and leakage-conductance per unit length of a circuit; and let V and C be the potential-difference (an awkward term) and current at distance sc. We have the following fundamental equations of connection:—

-^=(R+Lp)C, -M=(K+Sp)r, (Id)

p standing for d/dt. Observe that the space-variation of C is related to V in the same manner (formally) as the space-variation of V is related to C, so that we can translate solutions in an obvious manner by exchanging V and C, R and K, L and S, which are reciprocally related, in a manner.

To fix ideas, the circuit may be the common pair of parallel wires. There is one case in which the four constants are all finite that is characterised by such extreme simplicity that it is desirable to begin with it, especially as it casts a flood of light upon all the other cases, which may be simpler in appearance, and yet are immensely more complex in results. Let

EJL = K/S—s. and LSi? = 1 (2d)

The number of circuit-constants is now virtually three, owing to the fixing of the fourth constant. The equation of V is now

^=(s+pfK (3d)

or, which is equivalent,


if V=ue~,t. Since (4d) is the equation of undissipated waves, with constant speed v, whose solution consists of two oppositely travelling arbitrary waves, the complete solution of (3d) consists of such waves attenuated as they progress at the rate s (logarithmic). Thus,

V=f(x - vty~u (5d)

is the complete expression of the positive wave, iff(x) be the state when t = 0. Shift the wave bodily a distance vt to the right, and attenuate it from 1 to e~*e, and we obtain the state at time t. The corresponding current is

C=V/Lv = SvV, (6d)

in every part of the wave. To express a negative wave, change the sign of v in (5d) and (6^). The second form of (6d) says that a charge Q moving at speed v is equivalent to a current Qv.

Since V is an E.M.F., it is convenient to reckon Lv in ohms, as was done before; v is 30 ohms, in air, when it has its greatest value (speed of light, 30 earth-quadrants per second) and L is a convenient numeric. Z = 20 is a common value (copper suspended wires); in this case our “resistance” is 600 ohms. But it is not “ohmic” or “joulic” resistance ; the current and E.M.F. are perpendicular. V is the line-integral of the electric force across the dielectric from wire to wire, and C is the line-integral (-f 47r) of the magnetic force round either wire. The electric and magnetic forces are perpendicular, and so are V and C regarded as vectors, [i.e., their elements E and H are perpendicular]. The product VC is the energy-current; their ratio is the important quantity Lv, the impedance.

In a positive wave V and C are similarly signed, and in a negative wave are oppositely signed. Thus, if the electrification be positive, the direction of the current is the direction of motion of the wave; whilst if it be negative, the current is against the motion of the wave. When oppositely travelling waves meet, the resultant V is the sum of the two Vs, and the resultant C the sum of the two C’s.

Thus, if the waves be so shaped as to fit, then, on coincidence, V is doubled and C is annulled. The energy is then all electric. But if the electrifications be opposite, Vis annulled and C is doubled, on coincidence.

The energy is then all magnetic. On emergence, however, the two waves are unaltered, save in the attenuation that is always going on.

The electric energy is \SV2 per unit length of circuit, and the magnetic energy is \LC2. From this, by (6d) and the second of (2d), we see that the electric and magnetic energies are equal in a solitary wave, either positive or negative. The dissipativity in the wires is IiC2, and outside them KV2, per unit length of circuit. These are also equal, for the same reason. Should the disturbance be given arbitrarily, i.e., V and C any functions of x, the division into the positive wave Vl and the negative wave V2 is effected thus :—

Vx = £( V+ LvC), V2=\(V- LvC) (7 d)

Notice that SVXV2 = - LC\C2, so that the total energy per unit length is always

S(V?+V*) = L(C? + C*) (8 d)

Similarly, the total dissipativity is always

R(C* +Q) = K(Vf+ Vi) (9 d)

Similarly the total energy-current is always

VA + V2C» (lOrf)

since = - V2Cv

If, at a given moment, V=V0 through unit distance anywhere, with no C, this immediately breaks into two equally big waves, one positive, the other negative, which at once separate. If initially there be no V, but only C, the same is true for the current-waves; i.e., the result is two equal but oppositely signed V waves, which at once separate.

What happens when disturbances reach the end of the circuit depends upon the nature of the terminal connections there. At present only one case—the simplest—will be noticed. Let there be a resistance of amount Lv at the distant end B of the circuit. The terminal condition is then V=LvC. But this is the property of a positive wave. Hence all waves travelling towards B are immediately absorbed on reaching B. The electricity is all gobbled up at once, so to speak. Similarly, if there be a resistance Lv at the end A (where x = 0) it imposes the condition V= - LvC, which is the property of a negative wave, so that all disturbances on arrival at A are absorbed immediately. Thus, given the circuit in any state of electrification and current, without impressed force, it is wholly cleared in the time l/v at the most, I being the length of the circuit.

Now, let the circuit be short-circuited at A, and have a resistance Lv at B. Insert an impressed force e at A momentarily, producing V=e through unit distance, say. This will travel towards B at speed v, attenuating as it goes, and on arrival at B, what is left will be at once absorbed. This being true for every momentary impressed force, we see that if it be put on at time t = 0, and kept steadily on thereafter, the full solution is

V=e.€~^\ (lid)

from rc = 0 to x = vt, and zero beyond. Thus the steady state at a given point is instantly assumed the moment the wave-front reaches it. After that, there is still transfer of energy going on there, viz., to supply the waste in the part of the wave that has passed the spot under consideration, and to increase the energy at the front of the wave. The current is V/Lv, as before. On reaching B, the current is

C=±e'mlLv= % x (12d)

Lv , Ml Lv

If we let Rl, the resistance of the circuit, be 3,000 ohms, which is 5 times the before-assumed value of Lv, then the received current is

/■y 8 ® C (\“^d\

~\hOTv~M^O~WRl 1 1 '

The attenuation is such that the current is one-thirtieth part of the full steady current with perfect insulation.

The electrostatic time-constant of the circuit is

BSl* = lx?L; (Ud)

v Lv

or, in our example, five times the time of a journey from A to B. It may have any value we please. If we want it to be *1 second, l/v must be ‘02 second, and therefore I = 6,000 kilometres, which requires R='h ohm per kilom. This is lower than that of any telephone line yet erected. But to make the electrostatic time-constant *05 second, with the same attenuation, it must be 3,000 kilom. at 1 ohm per kilom.

If e vary in any manner at A, the current at B is given by (12d), in which e varies in the same way at a time l/v later. As there is no distortion, it becomes a question of suitable instruments. With proper instruments, no doubt the permissible attenuation could be much greater, and the circuit much longer. Again, if we raise the insulation we lessen the attenuation. We bring on distortion, but a good deal is allowable, so that again we can work further. The insulation-resistance should be

  • 36 megohm per kilom. in the 3,000 kilom. example; the product of the

resistance of any portion of the circuit (wires) into the insulation- resistance of the corresponding part is (Lv)2. In the 6,000 kilom. example it should be -72 megohm per kilom. But if it be not arbitrary waves, but only waves of high frequency that are in question, then we may approximate to the distortionless transmission without attending to the exactly-required leakage.



Although the speed of the current is not quite so fast as the square of the length of the line, yet, on the other hand, it is not quite so slow as the inverse-square of the length, as a writer in a contemporary (Electrical Review, June 17, 1887, p. 569) assures us has been proved by recent researches. However, if we strike a sort of mean, not an arithmetic mean, nor yet a harmonic mean, but wrhat we may call a scienticulistic mean (whatever that may mean), and make the speed of the current altogether independent of the length of the line, we shall probably come as near to the truth as the present state of electromagnetic science will allow us to go. But, apart from this, there is some h priori evidence to be submitted. Is it possible to conceive that the current, when it first sets out to go, say, to Edinburgh, knows where it is going, how long a journey it has to make, and where it has to stop, so that it can adjust its speed (scienticulistic speed) accordingly ? Of course not; it is infinitely more probable that the current has no choice at all in the matter, that it goes just as fast as the laws of Nature, preordained from time immemorial, will let it; and if the circuit be so constructed that the conditions prevailing are constant, there is every reason to expect that the speed will be constant, whether the line be long or short. Q.E.D.

Now, a great and striking thing about the distortionless system, whose elementary properties were discussed in the last Section, is the distinct manner in which it brings the speed of the current into full view. Another and very important thing is this. When the leakage is not so adjusted as to remove the distortion altogether, solutions become difficult of interpretation, owing to the almost necessary employment of Fourier or other transcendental series to express results. But by a proper adjustment of the leakage so as to abolish the tailing, which is the cause of the mathematical difficulties, we are enabled to follow with ease the whole course of events, say, in the setting up of the final state, due to a steady impressed force, without laborious calculations. And, although the state of things supposed to exist in the distortionless system is rather an ideal one, yet it allows us to obtain a very fair idea of what happens when there is distortion, e.g., in the oscillatory establishment of the steady state in a well-insulated circuit.

When we speak of a charge travelling along a wire at speed v, it should be always remembered what this implies. There are two conductors, parallel to one another, and the positive charge on the one is accompanied by its complementary negative charge on the other (corrections due to parallel wires, etc., are ignored here). The two charges move together. More comprehensively, the whole electromagnetic field, of which the charges are a feature only, is moving along at speed i\ in the space between the wires, into which it also penetrates to a greater or less extent. In the distortionless system this penetration is assumed to be perfect and instantaneous, so that the resistance and the inductance are strictly constants; and, by the ratio R/L being made equal to K/S we make any isolated disturbance travel on without spreading out behind. In travelling it attenuates by loss of energy in the conductors and by leakage in such a way that if it attenuate from 1,000 to 900 in the first 50 kilometres, it will attenuate to 810 in the second, to 729 in the third, and so on; multiplying by 9/10 in every 50 kilometres.

In the last Section was considered the uniquely simple case of a short- circuit at A, the beginning of the circuit, wrhere any impressed force is placed, sending any-shaped waves into the circuit, travelling undistorted, with uniform attenuation, and completely absorbed on arrival at the distant end B by a terminal resistance of amount Lv. Of course this complete absorption at B of all waves arriving there is independent of the nature of the terminal arrangements at A. But these will materially influence the magnitude of the waves leaving A. Keeping at present entirely to simple cases, if we insert a resistance Lv at A we can make a safe guess that the current will be just halved, because when there is a short-circuit there, the line itself behaves just as if it were a resistance Lv. That is, the current at A is then ejLv, however e may vary, provided there be a resistance Lv at B ; or, which is equivalent, the circuit be continued indefinitely beyond B unchanged in its properties. This guess may be easily justified. That the current is zero when w*e insulate, or insert an infinite resistance at A, is also evident. In general, the insertion of a resistance R0 at A causes the potential-difFerence V0 there, due to an impressed force e, to be

F0 = eLv(R0 +Lv)-1, (15c?)

and the current to match to be V0)Lv. The transmission to the distant end, and the attenuation are as before.

But if the place of e be shifted along the circuit from A, interferences will result whenever the resistance at A has not the value Lv. Imagine e to be at distance xx from A. When put on, the result is to send a positive wave \e to the right, and a negative wave - \e to the left, both travelling at speed v, and attenuating similarly. Thus the circuit behaves towards e as a resistance 2Lv, half to the right, half to the left. Now, when the negative wave arrives at A, if there be a resistance Lv there to absorb it, there will be no interference with the positive wave, which will go on to B and be absorbed there. The current at B will therefore be

= (16d)

the value of e to be taken at a given moment being that at xv at the time (I - x^/v earlier. But if there be a resistance at A of any other amount than Lv, there will be a reflected wave from A, which will run after the original positive wave, and so make every signal at B have a double or familiar following it after an interval of time 2xjv, which is that required to go from x1 to A, and back again. Now the closer the seat of e is shifted towards A, the more closely will the familiar follow the original positive wave; and when e is at A itself, they will be coincident in front. Now, the current at A corresponding to (1 &d) is

CA = l(e!Lvy^'} (17d)

and (as will be explained in the Section on Reflections) the reflected wave is got by multiplying by pn, where

atpi+u>-i (i&o

Now make ^ = 0, and we shall verify (15d), and, by the union of the positive and the reflected (also positive) wave, show that V at x at time t due to e = f(t), any function of t, at A, is

V=f(t - xjv) x Lv(B0 + Lv)"1 x Rx,L% (19d)

and the current there is V/Lv.

The most simple case after these of complete absorption at B, with complete absorption, or short-circuit, or any resistance at A, is perhaps that in which we short-circuit at both A and B. If a charge be then moving towards B, it is wholly reflected with reversal of electrification. We must have F=0 at B, and this requires every disturbance arriving at B to be at once reversed and sent back again. The same thing happens at the short-circuit at A. Perhaps, however, the easiest way to follow events is to imagine the two charges, positive and negative, which always travel together, to pass through one another when they come to the short-circuit, so as to exchange wires. Thus one charge goes round and round the circuit one way, whilst the other, just opposite, goes round and round the other way. There is the usual attenuation. On this view of the matter, we may imagine the effect of a terminal resistance Lv to be simply to bring the charges to rest against friction. It need scarcely be said, however, that the day has gone by for any such fanciful explanation to be taken seriously.

Since the current in a negative wave (from B to A) is of the opposite sign to the electrification, there is no reversal of current by reflection at a short-circuit. As, therefore, the reflected wave is to be superimposed upon the incident wave, we see that the current is doubled at B from what it would be were the circuit to be continued beyond B, or the critical resistance Lv were inserted in place of the continuation.

The process of setting up the permanent state due to a steady e at A is now this :—First the positive wrave

V1 = e.e-Kxl,% (20d)

if x<vt, which would be the complete solution were there no reflection at B. Now B is reached by Vx in the time l/v, and the value of at B just on arrival is ep, if p = e~m,Lv, which is the attenuation in the circuit. The reflected wave V2 now begins. This is

V2= -ept.cW, (21c?)

which travels towards A at speed v. In the meantime the first wave is still going on, for the battery at A does not know what is going on at B. Thus, from t = l/v to t = 2l/v, the state of the circuit is given by the sum of V1 and V2 so far as V2 has reached, and by Vl alone in the rest. On arrival of V2 at A it is attenuated to - ep2, and reflection then produces a positive wave

Vz = ep2^RliL\ (22d)

which is a copy of Vv only smaller to the extent produced by the multiplication by p2. This wave reaches B when t = 3l/v, and then there commences the reflected wave, V4, given by

-ep*.eRxlL°, (23d)

going from B to A. This is a copy of V2. And so on. Thus we have an infinite series of reflected waves, coming into existence one after the other; the state at any moment is expressed by the sum of the waves already existent; the final state is the sum of them all. Since the sizes of the positive waves form a geometrical series, and also those of th& negative waves, they are easily summed. The positive waves Vv etc., come to

e{\ - (24d)

and the negative come to

- ep\\ - (25c?)

so that the sum of (24d) and (25d) expresses the final V of the circuit. And, since the current is got by dividing by Lv in a positive wave and by - Lv in a negative, the final current is the excess of (24c?) over (25c?), divided by Lv. Notice that whilst it is a process of settling down to the final state of electrification, it is a process of rising up to the final state of current. More strictly, whilst the potential-difference at any spot oscillates about its final value, being alternately above and below it, the excursions getting smaller and smaller as time goes on, the current-increments are all positive, though they get smaller and smaller. Now if the time l/v of a journey be exceedingly small, so that there may be thousands of journeys performed in getting up to say 99 per cent, of the final current, the current will appear to rise continuously, and the potential-difference to have its final value from the first moment, which is in reality its mean value during the oscillatory period. This is the explanation I have before given of how it comes about that there is no sign of oscillation in any purely electromagnetic formulae, such as are universally employed when such short circuits are in question that the current seems to have the same strength (when no leakage) everywhere. It is really rising by little jumps, and differently timed at different places, but the jumps are too small to be perceived, and too rapidly executed. And the electrification at any spot is really (unless the vibrations are specially checked) vibrating about its mean value, which is its final value, though this mean value is assumed (in electromagnetic formulae) to be the actual value. But if the resistance in circuit be great, so that the final current is small, we have an oscillatory settling down of the current, instead of a rise.

The solution (24c?), (25c?) is what we may at once get by considering the differential equation of the steady state and its solution to satisfy the terminal conditions. But our solution gives us the whole history of the establishment of this final state, and allows us to readily follow the oscillatory phenomenon into minute detail. When there is distortion there is difference in detail, which is then difficult to follow; but there is no substantial difference in the general results. We cannot make or break a circuit without a similar action in general. But w'e cannot expect to be able to formularise the results simply when the circuit is of an irregular type, e.g., a laboratory circuit.



If there be a resistance at the end B of a distortionless circuit, its presence imposes the condition V=RlC at B permanently. If, then, there be a wave travelling towards B, we find the nature of the reflected wave from B by applying the above terminal condition to the actual V and C, which are the sum of Vv the potential-differences in corresponding portions of the incident and reflected waves, and of Cv C2 the currents in these portions. Thus we have

Vx = LvCv ^1 + ^2= K V= RA

- V2 = LVC2, Cx+ C2= C,

to represent the full connections. From these we find

VJVx = {Rx - Lv)(Rx + Lv)~' = Pl say, (26d)

giving the reflected in terms of the incident wave. This ratio is positive if jfifj be greater, and negative if it be less than the critical Lv. In the former case there is reversal of current, in the latter of electrification, produced by the reflection. The three most striking cases are when Rx = 0, oo, or Lv, i.e. short-circuit, insulation, and the' critical resistance of complete absorption, making Pl = - 1, + 1, or zero. There is partial absorption and loss of energy whenever Rx is finite, but none whatever in the two extreme cases. The loss of energy is accounted for by the Joule-heat in the terminal resistance.

In a similar manner, if there be a resistance R0 at the near end A, the transforming factor is

PQ = (R0 - Lv)(R0 + Lv)~L (27 d)

If there be given an isolated charge moving towards B at a certain time, it will, after reflection at B, be replaced by another charge moving towards A, which may be of the same or of the opposite kind, according as the reflecting resistance is greater or less than the critical. On arrival at A it is transformed into a third charge moving towards B, and so on. There is the usual attenuation p in each journey, where p = e~mlLv. If there be complete insulation at both ends, there is no other attenuation than this due to the circuit; and, similarly, if the ends be short-circuited; but in all other cases it has to be remembered that the act of reflection attenuates, besides causing a reversal of either the electrification or the current.

The complete history of the establishment of the steady state due to a steady impressed force at A is now expressible in terms of the three constants p0, p, and px; with, of course, x the distance, t the time, and e the impressed force. There is first the positive wave

  • We(l•; (28d)

due, as mentioned in the last Section, to the union of the initial positive wave of half strength and of the positive wave which is the reflection of the initial negative wave of half strength, which latter is rendered visible by shifting the seat of e towards B. The solution (28d) applies to all values of x less than vt, which is the extreme distance reached by the wave at time t after starting. On arrival at B we have to introduce the transforming factor pv above defined. The reflected wave is therefore

^2 = M1 “ PoWi • fRx,L\ (29 d)

which is to be superimposed on the former wave to obtain the real state during the second journey, from B to A. The region over which V2 extends grows at a uniform rate with the time, from B to A. On arrival of V2 at A we must introduce the transforming factor p0 to obtain the third wave, which is

^3 = 2e(l “ PO)P2PIPO • *~RXlL° (30^)

This reaches B at time t = 3Ijr, when th^ fourth wave commences, which is to be found by introducing the transforming factor px; thus

  • 4 = W1 “ PO)P4PIPO • eRXlLV

It is unnecessary to proceed further, as it would only produce repetitions. The positive waves Vx, V3, etc., have the common ratio p2plp()i and are otherwise similar. Their sum is therefore

he(\ - Po)(l - p2plPo)-K ^ (32d)

Similarly the sum of the negative waves is

hK1 ~ Po)P2Pi(1 “ P^PiPo)'1 • c—• (33^)

The final state of V is therefore expressed by the sum of (32d) and (33c?). In all the positive waves the current is from A to B, and in the negative from B to A; hence the excess of (32c?) over (33^), divided by Lv, expresses the final state of current.

The solution of the above problem by means of Fourier-series is extremely difficult. It expresses the whole history of the variable period by a single formula. But this exceedingly remarkable property of comprehensiveness, which is also possessed by an infinite number of other kinds of series, has its disadvantages. The analysis of the formula into its finite representatives, so that during one period of time it shall represent (28c?), then in another period represent the sum of (28d) and (29d), and so on, ad inf., is trying work. And the getting of the formula itself is not child’s play. Considering this, and also the fact that a large number of other cases besides the above can be fully solved by common algebra (with a little common-sense added), the importance of a full study of the distortionless system will, I think, be readily admitted by all who are dissatisfied with official views on the subject of the speed of the current. The important thing is to let in the daylight on a subject which it was difficult to believe could ever be freed from mathematical complications.

There is a rather important remark to be made concerning the two extremes, 11x — 0 and ^ = 00, at the end of the line, in the above solution. Although described as short-circuiting and insulation, they do not really represent the state of things existent when we actually terminate a long circuit of two parallel wires by a thick cross-wire (the short-circuit) or leave the ends disconnected in the air. Every theory that ever was made is more or less a paper theory; we must simplify the real conditions to make a theory workable. Now a theory may very closely represent reality (when pursued into numerical detail) through a wide range, and yet go quite wrong at extremes. The justification for making the constants of the circuit independent of its length is that the length is an enormous multiple of the distance between the wires. But if we terminate the circuit somewhere, it is no longer true that the permittance and the inductance per unit length are constants, near and at the termination. The theory, to be correct, must wholly change its nature, as may be seen at once on thinking of the changed nature of the electromagnetic field as the termination is reached. Now our theory says that when the circuit is insulated at B, every charge arriving there is at once sent back again unchanged; and that during the period of reflection, the potential-difference is doubled and the current annulled. The doubling of the potential-difference is obviously due to there being a double charge with the same assumed permittance. But the permittance is not the same, nor anything like the same, at the termination as it is far away from it. The theory therefore wholly fails to represent the case of insulation, so far as the potential-difference at the termination is concerned, though there does not seem to be any reason to suppose that this will affect matters elsewhere ; for when the reflected wave gets away from the termination, the old state of things is restored. There is a similar want of correspondence between the theory and realitj' when we make a real short- circuit, which we have supposed to be represented by Bl = 0.

Now the question may suggest itself: Since this failure is due to the assumption that the permittance and inductance continue constants right up to the termination, and this assumption being made in all cases, may there not also be a failure when is finite ? The following reasoning will show that this is not to be expected. For if the terminal resistance (although it may be small) be equal to that of a considerable length of the circuit, the influence of this resistance on the course of events must be much greater than that due to the changed nature of the circuit near its end. We therefore swamp the terminal corrections, which become so important themselves when the terminal resistance is quite negligible.

The general principle that may be recognised is this. If the transfer of energy between the circuit and the terminal apparatus (of any kind) be of sensible amount, we may wholly disregard the fact that the circuit changes its nature as the termination is approached. But should it be insensible, then we fail to represent matters correctly at and near the termination.

Again, if the ends of the circuit, supposed insulated, be brought sufficiently close together, there may be a spark or disruptive discharge there when a charge arrives, involving a loss of energy and attenuation. It is scarcely necessary to remark that effects of this kind have no place in the theory.

In the same connection it may be remarked that when we are following the history of an isolated charge, which may, in the theory, be confined to the shortest piece of the circuit imaginable, we should really spread it over a length which is several times as big as the distance between the two wires. This is to make the element of length have the same properties as a great length. Similar assumptions are made (though seldom, if ever, mentioned) in most theories in mathematical physics. An element of volume, for instance, must be large enough to contain such an immense number of molecules as to impart to it the properties of the mass.

Returning now to the study of the properties of the circuit, let us examine the effect of varying the constants. For simplicity, insert the critical resistance at B, and let there be none at A, where the impressed force is. The current at B is then

CB = (e/Lv)c~m,Lv = (e/Bl)ye (34c?)

if y — Bl/Lv. The value of e to be taken in the formula at a given moment should be that at A at the time l/v earlier. Now, with the resistance of the circuit kept constant, vary y to make the current a maximum. We require y= 1, or the critical resistance should equal the resistance of the circuit (without leakage). It then also equals the insulation-resistance (Kl)~l. If the resistance at A be any constant multiple of Lv, we shall have the same property y — 1 to get maximum current. (But should the resistance at A be kept constant, we shall have y2(BJBl)+y = 1, which it is unnecessary to discuss.) The received current is therefore

CB = e (2-718 Bl)~\ (35d)

when no resistance at A; and if there be resistance of amount zLv, we must divide the right side of (3hd) by (1 + z) to obtain the current at B. Thus the result is the same as if the circuit were a mere resistance whose value is a small multiple of the true resistance, with abolition of the leakage, permittance, and inductance, but with a retardation of amount l/v. This is not the electrostatic retardation, of course; it merely means the interval of time that elapses between sending and receiving, whereas electrostatic retardation, as formerly understood, is quite another thing. Neither is it the speed of the current; that is v. But singularly enough, the value of the electrostatic time-constant BSl2 is now l/v itself, proportional to the first power of the length, and inversely proportional to the speed of the current.

Example. 1,200 kilometres at 2 ohms per kilom. Lv should be 2,400. If it be an air-circuit, of copper, with v practically = 30 ohms (the formulae for permittance, inductance, etc., will be given later), we require L = 80. This is much too great. The inductance must be artificially increased, if we are to have so little attenuation as above on a circuit of that length. Or the resistance may be reduced. If 1 ohm per kilom., L = 40 is wanted. If \ ohm per kilom, L = 20.

The shorter the circuit, the smaller is the value of L needed to get the maximum current; and the longer the circuit, the greater L should be. If L could be made large enough, without altering the resistance, the circuit could be of any length we pleased. The lower the resistance of the circuit, the less leakage is needed to prevent distortion, and the less attenuation there is. The higher the resistance, the more leakage is needed, and the greater is the attenuation. We see, by inspection of (34d), that without varying either the resistance or the permittance, but solely by increasing L (remembering that Lv = (L/S)t), we could make Atlantic fast-speed telegraphy possible, with little attenuation and distortion. But the speed of the current would be very low. This I shall return to in connection with the sinusoidal solution.

Section XLIY. ANY Number of Distortionless Circuits RADIATING FROM A



If the ends of the two conductors of a distortionless circuit at its termination at A be caused to have a difference of potential V0, varying in any manner with the time, and if there be an absorbing resistance inserted at the other termination B, we know that the impedance of the circuit to V0 is Lv, a constant, at every moment, so that the current there is VJLv. We also know how the potential-difference and current are transmitted, attenuating to V0p and V0p/Lv on arrival at B.

If there be a second distortionless circuit starting from A, and we simultaneously maintain the same difference of potential V0 on it, we know what happens on it, viz., as above described, merely changing, if necessary, the values of p and Lv. That is, if the circuit be not of the same type as the first one, and of the same length, we require to use different values of p and Lv.

This obviously leads to the working of anj^ number of distortionless circuits in parallel by a common impressed force at A. Call the wires of a circuit the right and the left wires, merely for distinction. Join all the right wires to one terminal A1} and all the left wires to another, A2, and then maintain a difference of potential V0 between Ax and A2. Then, provided every circuit has its proper absorbing resistance at the distant end, we know what happens. The reciprocal of the sum of the reciprocals of the impedances of the various circuits is the effective impedance to V0. Next, V0 divided by the effective impedance (say I) is the total current. Finally the total current divides amongst the circuits in the inverse ratio of their impedances. The current at the distant end B of any circuit is the current entering it at A at the time l/v earlier, multiplied by the attenuation-factor p of the circuit. I do not write out the equations, as the description is fully equivalent.

In order that V0 should be strictly proportional to an impressed force e in the branch joining the two common terminals Alf A2 of the circuits, it is necessary that it should be a mere resistance, which may have any value. Let it be R0; then, B0 added to the previous effective impedance to V0, is the impedance to e; so that the total current is e/(i?0 + 7), and the value of V0 is eI/(R0 + T). In practice, it is not possible to fully realise this simplicity. Suppose, for instance, the secondary of the transformer, in the circuit of whose primary a microphone is placed, is joined across the common terminals of the circuits.

Even if the circuits be distortionless, we see that there must be terminal distortion, or V0 will not vary as it should for the accurate transmission of speech. There are several causes of distortion here. At the distant end, one cause of further distortion will be the inductance of the receiving telephone, and an additional and very important one will be the mechanical troubles that will prevent the disc from copying accurately, in its motion, the magnetic-force variations.

After this example of a complex arrangement of circuits admitting of simple treatment, let us return to a single circuit. Examine the effect of inserting any resistance r intermediately. This should be put half in each wire, if the circuit consist of a pair of equal wires, to prevent interferences. Let there be a wave travelling from left to right towards r. Let Vv Vv V3 be the potential-differences in corresponding portions of the incident, reflected, and transmitted waves, so that, at a certain moment, they are coincident, viz. at r itself, where let V be the actual potential-difference on the left side of r. Then we have

Vl = LvCv V2 ~ ~~LVC2, FS = LVCsA (SQd) Vl+V2=V=rC3+V3, C1 + C2 = C3. f 1 ;

These are the full connections. From them,

Vo T V3 2 Lv

say (37 d)

V1 r + 2Lv Vx r + 2Lv Particularly notice that

^ri=^2+^3» (38d)

as this is an important property. Every element of electrification in the incident wave arriving at the resistance is split into two (without any loss), one part crVx (in terms of potential-difference) is transmitted, the remainder is reflected.

As we have, by (37d),

r = 2Lv (o—1 - 1), (39d)

we see that if 1 per cent, of the incident wave be reflected, and 99 per cent, transmitted, we require r = ~Lv. If 10 per cent, be reflected and 90 per cent, transmitted, then r = %Lv. There is no transmitted wave if r be infinite. Half is transmitted and half reflected when r=2Lv.

There is always a loss of energy by this division of the charge, which is accounted for by the Joule-heat in the resistance. This is rC% per second; and since a wave of unit length takes v~x second to pass, rC£jv is the loss of energy per unit length of the incident wave, which loss, if added to the sum of the energies in the reflected and transmitted waves, makes up the energy per unit length in the incident. Another expression for the loss of energy is given by


There is the greatest possible loss of energy when r=2Lv, making <r = J, and the loss = %SV?. That is, when the intermediate resistance is twice the critical, and the incident wave is consequently half transmitted, half reflected, then half the energy is wasted in the resistance.

As the resistance is further increased, the transmitted wave gets smaller, and when it is infinite, we fall back upon the case already considered of total reflection without reversal of electrification or loss of energy.

If we have the absorbing resistance at A and at B, and any resistance r at an intermediate point C, we have a very simple result when any waves are sent from A to B, or from B to A. Suppose e acts at A, and that pv p2 are the attenuations in the two sections AC and CB. Then V, = e at A becomes Vl = ePl on arriving at C. The reflected wave is r2 = epx(l - cr), where cr is given by (37 d). On arrival (multiplied by pj at A it is absorbed, so there is an end of it. The transmitted wave at C is Vs = cplcr, which attenuates to = ep1<rp2 = co-p on arrival at B, where it is absorbed. The last equation therefore gives the potential- difference at B in terms of that at A at the time IJv earlier. In the first section of the circuit V is the sum of two oppositely travelling waves, and the current is their difference divided by Lv; but in the second section there is but one wave.

We are also able to solve by algebra alone the following problem. Given a distortionless circuit with any terminal resistances and any intermediate resistances at different places, find the effect due to a steady impressed force inserted anywhere in the circuit (half in each wire, pointing oppositely in space, to avoid interferences). For we have the circuit divided into sections, for each of which the attenuation is known (i.e., pl = erRXllLv in a section of length xx); we also know the transforming factors of the terminal resistances (p0 and Pl of the last Section); and we also now know the factors cr and 1 - cr for any intermediate resistance, by which we express how a wave divides there. So, starting when e is first put on, with the initial waves \e to the right, and - \e to the left, we can follow the whole course of events until we arrive (asymptotically) at the steady state. But it is no part of my intention to enter into the details, as nothing new would be contained therein.

But the effect of a great number of equal intermediate resistances equidistantly situated is of importance. Let p1 be the attenuation due to the circuit between two consecutive resistances, and cr the attenuation due to each resistance, that is, the attenuation of the transmitted wave. Let an isolated disturbance go from A to B. If it be initially F0, it becomes V^cr one section further on, F0(Plcr)2 after another section is passed, and so on, becoming F0p?<rn after passing n sections. If these n sections make up the whole circuit, then />" = p, the attenuation in the circuit due to itself only, as before, so that in passing through the circuit, V0 is attenuated to VQpa-n.

Now let the sum of the inserted resistances be nr=Bv Increase n indefinitely, whilst reducing r in the same ratio, thus keeping B1 constant. In the limit the resistance becomes uniformly distributed in the circuit, and the attenuation due to it becomes, by (37d), *

cr” = (1 -f- B1/2Lm)~n, with n — cc ,

= (4 Id)

Observe the presence of the 2. From this we may conclude certainly (as will be shown later), that if this uniformly distributed resistance Blt in addition to the original Bl, be accompanied by uniformly distributed leakage-conductance of total amount Kv such that B1/L = K1/S, the attenuation due to both Bt and K1 together is expressed by the square of (41 d). For what we do is to make the circuit distortionless again, by the additional leakage to compensate the additional resistance of the wires.

But the simplest way of viewing the matter is to start with a perfectly insulated circuit of no resistance. This is a distortionless circuit, of course, since it obeys the law B/L = K/S. The only difference from a real distortionless circuit is that there is no attenuation at all. All the preceding results therefore apply, remembering that p=l, or any waves are transmitted, not merely undistorted, but also unattenuated. They are, in fact, purely plane waves of light (very long waves practically) travelling through a perfectly non-conducting dielectric. They are merely guided through space in a definite manner by the conductors, imagined to have no resistance, so that, to use a very gross simile, the electricity slips along like greased lightning. There is no penetration of the electromagnetic field into the conductors, but purely surface-conduction, where we may use the word in a popular sense (conduct = to lead). Some curious consequences of the absence of resistance I will notice later; at present I may observe that owing to the relative simplicity produced by the absence of attenuation, the imaginary circuit of no resistance is useful for investigating the effect of inserting resistances, bridges, etc., and the action of a real distortionless circuit itself.

Thus, imagine an isolated charge moving from left to right in the circuit of no resistance. Introduce anywhere a resistance r; this will cause an attenuation from 1 to a- in passing the resistance (equation (Sid)), and the remainder 1 -o- will be reflected back. Next let there be a great number of equidistant small equal resistances; every one of these will attenuate in the ratio 1 : o-, and throw back the fraction 1 - cr. The result is that the original isolated charge, as it travels along, becomes a nucleus with a long slender tail behind it; the nucleus travelling forward at speed v and attenuating in the manner described ; the tail stretching out the other way at speed v. If these isolated resistances be packed together very closely, and be each very small, we approximate to the effect of continuously distributed resistance, that is, the resistance of the wires in a real circuit. In the limit, the result is, by (41 d), that the nucleus, if originally represented by V0a, that is, the potential-difference V0 through the very small distance a, with current to match, viz., V0I Lv through the same distance a, and therefore moving entirely to the right at that particular moment, becomes attenuated to

V0ae-w* = roa (42d)

in the time t = x/v, during which it has moved through the distance x to the right, if the resistance per unit length be B.

Since there is here no leakage, the rest of the original charge must be in the tail. The amount of electricity in the tail is therefore

SxF0a{ 1- (43<Z)

when the circuit is perfectly insulated. The length of the tail is 2x, half being to the right and half to the left of the position of the original isolated charge, it being of course supposed that neither the head nor the tail has suffered any extraneous operations, as terminal reflections, etc.

In a similar manner, if initially the isolated charge SV0a be without current, so that it would, were there no resistance, at once divide into equal halves, travelling in opposite directions without attenuation, what will really happen will be an immediate splitting into halves and separation of two nuclei, travelling in opposite directions at speed v, attenuating as they progress according to (42d), and joined by a band, consisting of the two tails superimposed. The equation of this doubletail is

in a finite form (as usually understood, by a convention that a solution in terms of a sine or J0 function, etc., is in a finite form, though it is really an infinite series), true from x = - vt to x = + vt, it being supposed that the origin of x was the original position of the charge. At the ends of this tail the two nuclei, each represented by

V=\V{i*-Rtl*L (45d)

through the very small distance a, must be placed, to make up the complete solution. I shall later illustrate this graphically, and also explain the other kind of tail.



Let a distortionless circuit be bridged across anywhere by a wire whose conductance is 1c, and let us examine its effect on a wave passing along the circuit. In the first place, we may remark that we have already solved one bridge-problem, viz., the result due to an impressed force in the bridge itself, this being made a special case of the first part of the last Section, by limiting the number of radial circuits to two of the same type.

No1w let Vv and Vz be the potential-differences in corresponding parts of an incident, reflected, and transmitted wave; V1 going from left to right on the left side of the bridge, V2 from right to left on the same side, and Vs from left to right on the further side of the bridge.

At a certain moment these are coincident, viz., at the bridge itself. Then, by the properties of positive and negative waves and elementary principles, we have the following full connections :—

l\ = LvC\, V^-LvC,, V,-LrC,\

V^vt~vn c1+c2=ca+kvs. f <46d)

From these we find

r,_ct_ k r,_0,_.‘2Sv

F1 C, f: + 2,SV Ft C, k + 2Sv K '

Particularly notice that

Cx = (\ + Cv (48d)

which, though extremely simple, is not by any means obvious at first sight, whilst it is an extremely important property. It is an example of the persistence of momentum; though this may not be immediately recognised, it will be made plain enough later on.

These equations should be compared with (36d), (37d), the corresponding ones relating to the effect of a resistance r inserted in the circuit. We see that this resistance is replaced by the conductance of the bridge, that L becomes S, and that V and C change places in the expressions for the ratios of the transmitted and reflected waves to the incident.

If we fix our attention upon the current, we see that every element of current, when it arrives at the bridge, is split into two, in the ratio of k to 2Sv, or of \Lv to k-1, half the critical resistance to the resistance of the bridge. The first part is reflected, increasing the current on the left side, and lowering the potential-difference; whilst the other part is transmitted. The electrification in the reflected wave is negative, if that in the incident wave be positive ; and conversely.

It may be as well here to remind the reader that from left to right is the arbitrarily assumed positive direction along the circuit, which is the direction of motion of a positive wave (therefore so-called); whilst a negative wave goes from right to left. Also, that the sign of the current, whether positive or negative, is a quite different thing. That is, the current in a positive wave may be negative, and the current in a negative wave may be positive, or the reverse. What is a possible source of some preliminary confusion is the fact that the vector we term the current, and the vector direction of motion of a wave, are in the same straight line, one way or the other. These connections are all summed up in Vl = LvCv the property of a positive, and V2 = - LvC2, the property of a negative wave. If the first of these relations be true, the wave must move from left to right, whether V and C be both positive or both negative; whilst if the second be true, the wave must move from right to left. I can also recommend the reader to take the advice before given to fix his attention upon the electromagnetic field which is implied by a stated V and a stated C, viz., a field of electric displacement across the dielectric from one conductor to the other, and a field of magnetic induction round the conductors. A very useful purpose may perhaps be served by a careful study of the properties of the distortionless circuit, viz., to assist in abolishing the time-honoured but (in my opinion) essentially vicious practice of associating the electric current in a wire with the motion through the wire of a hypothetical §wzsi-substance, which is a pure invention that may well be dispensed with.

Returning to the effect of a bridge, notice that by the union of (48d) with the last of (46d), we produce

2 C2 = WS (49c?)

That is, the current in the bridge equals twice the current in the reflected wave. The corresponding property when it is a resistance r inserted in the circuit that is in question is, by (36c?), (37d),

2^2 = rCs'> • (50£0

that is, the fall of potential through the resistance equals twice the difference of potential of the reflected wave.

If the bridge have no resistance, making a short-circuit (subject to reservations that need not be repeated), there is no transmitted wave. In fact, the case becomes identical with that of a terminal short-circuit, producing total reflection with reversal of electrification. If, on the other hand, the bridge have no conductance, it does nothing. If the conductance of the bridge be 2Sv, or its resistance be %Lv, the transmitted wave is half the incident, or the attenuation due to the bridge is \. Then, by superimposition, the current on the left side is increased in the ratio 2 to 3, and is therefore made three times the transmitted current.

The current in the bridge being kV3, and the corresponding heat per second divided by v being the heat due to the bridge per unit length of the incident wave, this amounts to

kV/v = 4S2F?kv/(k+2Sv)2, (51d)

by (47d). If k be variable, wre make the quantity in question a maximum when k=2Sv, which is the above case of attenuation The heat in the bridge per unit length of the incident wave is then %SFf, which is half its energy; the other half is equally divided between the transmitted and reflected waves.

If this bridge-wire be a coil of a given size and shape, the variation of k implies a variation of the thickness of the wire and of the number of turns. Whence, in a well-known manner, the magnetic force of the coil varies as the current in it and as the square root of its resistance; in another form, the square of the magnetic force varies as the product of the resistance of the coil into the square of the current, that is, as the heat per second. Hence, by what has just been said, the magnetic force is also a maximum when the resistance of the coil is \Lv. Notice that this is the impedance of the circuit as viewred from the coil itself. A correction is required for the inductance of the coil. It ought not, however, to be a very large correction, if it be a telephone that is in question, and of a really good type, having the smallest possible time- constant consistent with other necessary conditions. We require the magnetic force to be a maximum (i.e., due to the current coming from the circuit) to make the stress-variations the greatest possible, and act most strongly on the disc. [See “Theory of Telephone,” Art. xxxvi., vol. II.] Allowing for the inductance of the coil, if the currents be sinusoidal, we require equality of its impedance to that external to it, which is the general law.

Now let there be any number of bridges at different parts of the circuit, and let the ratio VJVl of a transmitted to an incident wave be denoted by s, its value being given by (47d), separately for each bridge. Let also plt p2, etc., be the attenuations due to the circuit in the different sections into which it is divided by the bridges, and start with an isolated positive wave Vx at A, the beginning of the first section. On arrival at the first bridge, it has attenuated to Vxpv What passes the bridge (not what crosses it) is Vxpxsv which attenuates to Vxpxp^x on arrival at the second bridge. Then there is another sudden attenuation, to F1p1p2s1s2, followed by a gradual attenuation in the third section, to ViPvPiPzhh 5 and so on, to the end of the circuit, at B. The disturbance is then attenuated to V1psls2...sn; where p is the product of all the former p’s, or the attenuation due to the circuit from A to B, and sn is the last s, belonging to the bridge next to B. If the absorbing resistance Lv be put at B, it will at once absorb the wave just described; but after that there will come dribbling in and be absorbed the dregs of the original disturbance at A, arising from the complex system of small reflected waves due to the bridges across the circuit, much attenuated by the many to-and-fro journeys. But if there be but one bridge, and the absorbing resistance be put at A, to get rid of the wave reflected from the bridge, then there is no dribbling in at B.

However many bridges there be, there is, by (48c?), no attenuation of current due to them, when its integral amount is considered, but only a redistribution of current. This exactly corresponds to the absence of any alteration of the total charge by inserting resistances in the circuit. They merely redistribute the charge.

If there be n bridges in the distance x, each of conductance k, the total attenuation produced by them is, by (47d),

sn = {l+kj2Sv}-n. (52 d)

Now place the bridges at equal distances apart, and increase the number n in the distance x indefinitely, keeping the total conductance constant, = Kxx, say. In the limit we shall arrive at a uniform distribution of leakage, Kx being its conductance per unit length, and the attenuation due to it will be the limit of

{1 + KflfiSvn} ~ n, with n = oo, (53 d)

This is therefore the attenuation of the nucleus, when an initially isolated disturbance travels through the distance x, due to the extra leakage Kx per unit length. There is, in addition, the regular attenuation due to the circuit. Disregard this for the present, by letting the circuit have no resistance and no leakage, that is, no leakage before the leakage represented by Kx was introduced. Then we see that if there be initially an isolated disturbance represented by V0 = LvC0, extending through the very small distance a, it becomes, at the time xjv later, removed a distance x to the right, attenuated to (writing K for the leakage-conductance per unit length)

F=LVC=F0€~SxI2Sb, (54d)

extending through the distance a, with a tail of length 2x behind it. This tail is of the negative kind, the electrification being opposite in kind to that in the head, and is such that the line-integral of the current in it amounts to ^ ^ _ t-Kx/2Sv^ because this, when added to the corresponding line-integral for the head, according to (54d), makes up C0a, the initial value of the line-integral. This tail is, as regards current, of the same shape as the corresponding tail due to resistance, as regards electrification, so its equation may be derived from (44d). But I shall consider the tails all together in a later Section.

The property involved in (48d), which leads to the deduction of (55d) from (54d), is worthy of notice. It is the persistence (or conservation) of momentum. If a circuit have no resistance, then, as Maxwell showed, we cannot change its momentum, the amount of induction passing through it. This was a linear circuit, with the current of the same strength all round it. Now our example is a remarkable extension of this property. Our circuit is linear and of no resistance, but it has any number of leaks, or conducting bridges, as well as what is equivalent to a series of condensers. The current in the circuit may be varied indefinitely in its distribution, but we cannot change its momentum. The line-integral of LC expresses the momentum, but since L is here a constant, of course the line-integral of C cannot change either. This property only continues true so long as there is no resistance bounding the magnetic field; therefore, if the circuit be of finite length, we must not insert resistances at the terminals. For instance, short-circuit at A and B, and we can at once say what will ultimately happen due to any initial distribution of current. It will settle down to uniformity of distribution, i.e., making a uniform magnetic field, so that the strength of current will equal the original total momentum divided by the total inductance. There is, of course, a loss of energy in the settling down, due to the leakage. If the circuit be infinitely long, so that the disturbance can spread out infinitely, the total energy will decrease asymptotically to zero, in spite of the persistence of the momentum, which indeed tends to zero in any finite length, but keeps its total amount unchanged.

If the circuit have resistance, the total momentum decreases according to the time-factor z~Rt,L, whatever be the initial distribution, if it be short-circuited at A and B, or be infinitely long. On the other hand, the total charge subsides according to the time-factor €~stls, if the circuit be insulated at A and B, or else be infinitely long. The meaning of termiual short-circuit or of insulation may clearly be extended to various other cases not involving loss of charge in the latter case (e.g. a terminal condenser) or of momentum in the former, with appropriate corresponding changes in the measure of S or L respectively.



Having in Sections XLIV and XLV discussed in some detail the effects due to resistances inserted in, and also those due to conducting bridges across, a distortionless circuit, which are of fundamental importance, and which lead to the development of a positive tail by a continuous distribution of resistance in excess of the distortionless amount, and of a negative tail by an excess of leakage, the full investigation of the case of resistance and leakage combined in any proportions presents no difficulty.

Start with a circuit having no resistance and no leakage, which is therefore both distortionless and conservative (or characterised by the absence of attenuation), and let there be an isolated disturbance going from left to right, defined by V\ = LvCv Also, let there be, at a certain place X, a bridge across the circuit, of conductance k; and, at the same place, a resistance r inserted in the circuit. When our incident wave T\ arrives at X, there result a reflected wave represented by V2 = - LVC2, and a transmitted wave = LvCs.

Now, considering the moment when these are all at X together (corresponding elements, of course), we have the following two equations connecting the three V’s :—

V1+v,= v,{\ +T/LV), ........ (56 d)

C^C^C^l+k/Sv + rk) (57 d)

The first is simply the expression of Ohm’s law applied to the resistance r, and the second expresses the continuity of the current at X. (Remember that Lv and Sv are reciprocal, so that the sum of the second and third terms on the right of (57d) expresses the bridge- current.) The equation (57d) may also be written

Vx - Jr2 = V3(l + kjSv + rk), (58d)

so that, by adding this to (56d) first, and then subtracting it, we obtain the desired ratios. Thus,

VJ V3 = 1 + r/2Lv + Jc/2Sv + %rk> (59 d)

V2/ Vs = rj2Lv - kj2Sv - \rk, (60d)

when written in the simplest manner. Of course the ratio V^JVv if wanted, is the quotient of (60d) by (59d). We see that the reflected wave may be either of the same or of the opposite electrification to the incident; and that, in order to completely abolish the reflected wave, we require, by (60d),

rjLv = kjSv+rk, (6 Id)

and that we then have, by (59c?),

FJ Vz = 1 + rjLv (62d)

simply. The reciprocal VJVl expresses the attenuation suffered by the incident wave in passing X.

The above equations are not in any way altered when we start with a real distortionless circuit instead of an imaginary one of no resistance. But by adopting the latter course we are directed to the nearest approach to a physical explanation of the properties of the real distortionless circuit itself. For, in the case of the circuit of no resistance we are dealing merely with progressive waves in a conservative medium, and we cannot expect to come to anything simpler than this. They simply carry their energy and all their properties forward at speed v unchanged, this speed being (pc)-*, if /x be the inductivity and c the permittivity of the medium; which expression is equivalent to the other, (LS)~i, where L is the inductance and S the permittance, which is more convenient in the practical application concerned. Except in the matter of wave-length, these waves are identical with light-waves, with the peculiarity that the two (supposed) perfect conductors of our circuit prevent the waves from spreading in space generally, by guiding them definitely along the circuit. (The simplest case is that of a tubular dielectric bounded by perfect conductors, say an internal wire and an external sheath.) Now we prove by elementary principles, (Ohm’s law, etc.) that an inserted resistance, causing tangential dissipation of energy, produces a reflected wave of the positive kind, involving a redistribution, without loss, of the electrification on the bounding conductors; and a redistribution, with loss, of the corresponding magnetic quantity, the momentum. On the other hand, we show that a bridge causes a reflected wave of the negative kind, involving a redistribution, without loss, of the momentum ; and a redistribution, with loss, of the electrification. (In speaking of redistribution, the mere translatory motion of waves is disregarded.) And by having both the bridge and the inserted resistance so proportioned as to make the loss of energy in each be. of the same amount (when small enough), we abolish the reflected wave, so that there is no redistribution, but merely attenuation produced by the resistance and bridge. This applies to any number of resistances inserted in the main circuit, each with its corresponding bridge; so that when we pack them infinitely closely together to represent continuously distributed resistance and leakage, we arrive at a real circuit, along which waves are propagated unchanged except in size. Thus any circuit (apart from interferences) may be made distortionless by adding a suitable amount of leakage. This amount is usually too great for practical purposes. Nor is it required. In the very important problem of long-distance telephony, employing circuits of low resistance (which are the only proper things to use), making the well-known ratio RILn of the two components of the electromagnetic impedance small, say \ or which may be easily done without using an extravagant amount of copper, we tend naturally, by bringing the inductance into relative importance, or equivalently, reducing the importance of the factor resistance, to a state of things resembling that which obtains in the truly distortionless circuit (independent of frequency of variations), and approximate to distortionless transmission. These statements may be proved by an inspection of the sinusoidal solutions I have given, but it would enlarge the subject too greatly to discuss them at present. I may, however, repeat that the problem of long-distance telephony is very remote from that of a long submarine cable which can only be worked slowly, unless we should unknowingly create a parallelism by employing quite unsuitable conductors; as, for instance, was done by the Post Office a few years since when they put down conductors having a resistance of 45 ohms per mile of circuit, combined with large permittance and small inductance ; and then, to make the violation of electromagnetic principles more complete, put the intermediate apparatus in sequence, so as to introduce as much additional impedance as possible. The proper place for intermediate apparatus is in bridge, removing all their impedance completely. This method was invented and introduced into the Post Office by Mr. A. W. Heaviside. It makes a wonderful difference in the capabilities of a circuit, as is now pretty well known.

The theory of tails allows us to give an intelligible physical explanation of how it comes to pass that a perfectly insulated circuit violating the distortionless condition completely, will yet tend to behave in a distortionless manner to waves of great frequency, provided the circuit be of a suitable nature, as above described. For let the circuit be so long that we can get several waves into it at once, when telephoning. They divide the circuit into regions of opposite electrification, each of which may (very roughly) represent what I have termed an isolated disturbance. Every one of them has its tail, but as they are alternately of opposite kinds, their residual effect in producing distortion becomes quite small. We can see clearly that the greater the frequency the less is the distortion, unless the increased frequency should bring with it increased resistance, which is very much to be avoided, and is what renders iron wire so unsuitable for long-distance telephony. By this mutual cancelling of the effects of the tails, we simulate the effect of the leakage which would wholly remove distortion, even of the biggest waves, without the disadvantage of the extra attenuation thereby introduced. I am induced to make these remarks rather out of their proper place, as they illustrate the importance of the distortionless circuit from the scientific point of view, in casting light upon the obscurities of dis- tortional circuits.

From (59d) we can get some results relating to the tails of waves in a distortional circuit. Thus, let there be n bridges in the distance x, equidistantly placed, and each of conductance Kx/n, with a corresponding resistance Rxjn in the main circuit. Let a disturbance pass from beginning to end of the length x. If o- be the attenuation at each bridge, the total attenuation of the head of the disturbance produced by all the bridges and resistances is tr". Now make n infinite, keeping R and K finite. The total attenuation becomes, by (59c?),

crn = {1 + Rxj 2Lvn + Kx/2Svn + RKx2j 2?j2} (63c?)

This is therefore the attenuation of the head suffered by every element in traversing the distance x, when R and K are the resistance and the leakage-conductance per unit length in any uniform circuit.

It will now be convenient to introduce a simpler mode of expressing the exponentials. Let

f=R/2L, g = K/2S, h=f-g, q=f+g, (64d)

all four being reciprocals of time-constants. Now (63c?) becomes e~qt simply, if t = x/v be the time of the journey over the length x. If, therefore, we have initially a disturbance V0 = LvC0 extending through the small distance a, possessing the charge SVQa and the momentum LCQO., then, at the time t later, when the disturbance extends over the distance 2x} half on each side of its initial position, being a nucleus of length a and a tail of length 2x, the charge and momentum in the nucleus become

SV0a c~qt and LCQa e-9< (65c?)

We have next to examine to what extent the total charge has attenuated by the leakage, and the total momentum by the resistance. This we can ascertain by (59cZ) and (60c?), applied to find the loss of electrification caused by a single bridge, and of momentum by a single resistance. Those equations give

F2 + V3 _ i + r/2Lv - k/2Sv - rk/2

V1 ~ 1 + rj2Lv + k/2Sv + rkj 2’

C2 + C3 Vs-V2 l+k/2Sv+%rk-r/2Lv Cx — V1 ~ 1 + k/2Sv + \rk + r/2Lv

These fractions, multiplied into the values of the charge and momentum respectively before the splitting, give their total values after the splitting. We can, therefore, apply the previous method of equidistant resistances and bridges, to ascertain the method of subsidence of the total charge and momentum, in the infinitely numerous splittings that occur in a finite time, when we pass to the limit and have uniform R and K. Putting r = Rxjn, etc., as before, and finding the limit of the wth powers of (66c?) and (67c?), we arrive at e~mlL and €-Kt/s respectively.

We thus see that a moving charge, no matter how it redistributes itself, subsides at the same rate as if it were at rest; for, obviously, SjK is the time-constant of the circuit regarded as a condenser, when uniformly charged and insulated at its terminations. It is as if electricity were atomic, so that we could follow the course of every particle. Then,. no matter how it moves about, it shrinks at the same rate as if it were at rest. Similarly as regards the momentum of the moving disturbance. Could we identify its elements, each would shrink in a manner independent of its translatory motions along the circuit. Notice, also, that the attenuation of the total charge equals the square of the attenuation of the nucleus due to leakage alone; whilst the attenuation of the total momentum equals the square of the attenuation of the nucleus due to resistance alone.

Thus, corresponding to (65d), we have

and XC'0«.e_/t(6_/e - ...(68 d)

to express the charge and momentum in the tail; since these, when added to (65d), make up the actual values otherwise found, viz.,

SVQCi.fr-* and LC0a.e~ift.

if f>g, or the resistance be in excess, the current in the tail is from head to tip, if that in the head be positive. But as time goes on, if the circuit be long enough, the head attenuates practically to nothing, leaving the big tail to work with. The region of positive current now extends from the vanishing nucleus a long way towards the middle of the tail; and, in the limit, the disturbance tends to become symmetrically arranged with respect to the origin from which it started as a positive wave, tailing off on both sides, with the current positive on one side and negative on the other.

But if /< g, or the leakage be in excess, a quite anomalous state of affairs occurs, which may be inferred from the preceding by changing V to C, etc. The full solutions of all tail-problems (shape, growth, etc.) are contained in the following four equations. Let a charge SV0a be at the origin at time t = 0, without any current. At time t we shall have, if y = ([h/v)(x2 - vH2)i,

r- J&ir*(»+ (69d)

c- «°*>

to express the double-tail or band connecting the two nuclei at its ends, which are already known. Similarly, if there be initially a current at the origin, of momentum LC0a, without charge, then at time t we shall have <nd) As before, put on the two nuclei at the ends. Since the J0 function is a simple one, viz.,

T (y\ — 1 y* f .

o\47 1 22 2242 224262 "* *

it is quite easy to follow the changes of shape by these formulas, except when t has become large and the nuclei small, when other formulae may be derived from the above which will approximately suit. [For further information, see Part vill. of Art. XL., Part I. of “ Electromagnetic Waves,” and “The General Solution of Maxwell’s Equations.”]



If two distortionless circuits of different types be joined in sequence, a wave passing along one of them will, on arrival at the junction, be usually split into two, a transmitted and a reflected wave. Let, in the former notation, Vv V2, Fs denote the potential-differences in corresponding elements of the incident wave in the first circuit, the reflected wave in the same, and the transmitted wave in the second circuit. The sole conditions at the junction are that V and C shall not change in passing through it. Thus,

v1+v2=v3, c1+c2=c3 (73d)

Now let Lxvx and L2v2 be the impedances of the two circuits, Lx and L2 being the inductances per unit length, and vv v2 the speeds of the current. Put the first of (73cZ) in terms of the currents.


LlVl(Cl ~ Q = L2V2CS (74f0

showing that the momentum of the incident disturbance equals the sum of the momenta of the reflected and transmitted disturbances. Corresponding lengths are compared, of course, proportional to the speed of the current. The condition of continuity of V is therefore identical with that of persistence of momentum. Next, put the second of (73d) in terms of potential-differences. Thus,


which expresses that the electrification suffers no loss by the splitting. The condition of continuity of C is therefore equivalent to that of the persistence of electrification. Multiply the first of (73d) into (75c?); the second of (73d) into (74d); the two members of (73d) together; and (74d) into (75d). The results are

SMK-VZ)=S2v2n L1V1(C?-C2) = L2V2C?, ^A+V2C2=VSC3;

which are equivalent expressions of the fact of persistence of energy, while the last of (76cZ) is the equation of transfer of energy. That it should be equivalent to the others will be understood on remembering that the energy is transferred at speed vx or v2, according to position.

We have, therefore, three things that persist, electrification, momentum, and energy, and these are expressed most simply by the two equations (73d) and by their product. If the continuity of V could be violated at the surface across the dielectric common to the two circuits at their junction, there would be a surface magnetic-current; and if the continuity of C could be violated, there would be a surface electric- current. These statements are implied in the general equations

- curl E = 47rGr, curlH = 47iT, (77d)

where E and H are the electric and magnetic forces, T and G the electric and magnetic currents. That is, tangential continuity of E implies normal continuity of G (or of the induction, since it, like G, can have no divergence); and tangential continuity of H implies normal continuity of T, and therefore, in our special case, of electrification. In fact (73d) express the same facts as (77d) do generally.

Now the continuity of V and C is violated at the boundaries of an isolated disturbance (e.g., V = constant in a certain part of the circuit, and zero before and behind). Then we do have the surface electric and magnetic currents on the front and back of the disturbance. It should, however, be stated that the conception of an isolated disturbance is merely employed for convenience of description and argument. Practically, there cannot be abrupt discontinuities; we must make them gradual. Then the surface-currents become real, with finite volume- densities.

The ratio of the reflected to the incident wave is given by

VliV\I = (Vs ~ LlVl)KL2V2 + Vi)> (78d)

and is positive or negative according as the impedance of the second circuit is greater or less than that of the first. The abolition of reflection is therefore secured by equality of impedances, irrespective of any change of type that does not conflict with this equality. Every element of the transmitted wave therefore carries forward, in passing the junction, its potential-difference, current, electrification, momentum and energy unchanged, but is changed in length in the same ratio (inversely) as the speed of the current is changed.

In a similar manner, we can determine fully what happens when a disturbance travelling along one distortionless circuit is caused to divide between any number of others, of any types. We have merely to ascertain the magnitude of the reflected wave in the first circuit. Let Vx and Ux be the incident and reflected waves. Then, corresponding to (78d), we shall have (79 d)

where I is the resultant impedance of all the other circuits (instead of L2V2, that of one only), viz. the reciprocal of the sum of the reciprocals of their separate impedances. Knowing thus Ul in terms of Vv we know their sum. But this is the common potential-difference in all the transmitted waves, which are therefore known, since by dividing by the impedance of any circuit we find the current. As regards the attenuation as the disturbances travel away from the junction, that must be separately reckoned for each circuit, according to the value of R/L, in the way before described. There will be found to be the previously-mentioned persistences, provided all the waves are counted, including the reflected in the first circuit.

Now put any number of distortionless circuits in sequence. If their impedances be equal, we know, by the above, that a disturbance will travel from end to end without any reflection at the junctions. It will vary in its length and in its speed, and also in the rate at which it attenuates, but there will be no tailing, however many changes there may be in the values of 11 and L. By pushing this to the limit, we arrive at a circuit in which It and L vary in an arbitrary manner (functions of x), whilst K varies in the same way as U, and S in the same way as L. The impedance is a constant, but the rate of attenuation and the speed vary in different parts of the circuit.

If we start an isolated disturbance at one end, it will travel to the other without tailing. But it will be distorted on the journey, owing to the variable speed of its different parts and the variable attenuation. But as regards the reception of the wave, there is no distortion whatever. For, on arrival at the distant end, where we may place the absorbing resistance, every element of the wave has gone through the same ordeal precisely, passing over the same resistances in the same sequence and at the same speed at corresponding places, so as to arrive at the distant end in the same time, attenuated to the same extent. Similarly there is no intermediate distortion as regards the succession of values of V and C at any one spot. There is only distortion when it is the wave as a whole that is looked at, comparing its state at one instant with that at another. And if we should cause this wave to start in a uniform circuit, then pass into an irregular one as just described, and finally emerge in a uniform circuit again, it will then have recovered its original shape, every part being attenuated to the same extent.

As regards the time taken to pass over a distance x in the variable circuit, we have to solve the kinematical problem : given the path of a particle, and its speed at every point, find the time t taken. Thus,


taken between the proper limits, wherein v is to be a function of x. The attenuation suffered in this journey is more easily expressed. Go back to the former case of any number of uniform distortionless circuits of equal impedance joined in sequence. The attenuation produced in passing through any number of them is the product of their separate attenuations, i.e.,

€-X1/LV X x = c-£w» (80d)

where Rv R2, ..., are the resistances of the separate sections, and Lv the common value of the impedances. As this is independent of the number of sections or their closeness, we see that in our variable circuit the attenuation in any distance is expressed by the right member of (80d), wherein 2 R represents the total resistance of the circuit in that distance, or between the proper limits, R being a function of x.

The above-given demonstration of the properties of the variable distortionless circuit, which is rather a curiosity, depends entirely upon our previous proof that the abolition of reflection at the junction of a pair of simple distortionless circuits is obtained by equality of impedances, irrespective of any change that may take place in the resistances. The following is also of some use. Go back to the fundamental equations

-S?V=(B + Lp)C, -VC=(K+Sp)V; (8Id)

-V(LvC) = L(B/L+p)C,

-VC' = S (K/S +p)(LvC).

If our assumption can be justified, these equations must become identical. They do become identical if B/L = KjS, and Lv = constant; becoming

-vVV=(B/L+p)V (83d)

This is for the positive wave. The assumption V= — LrC again makes (6ld) identical under the same conditions, the resulting equation being (83d) with the sign of v changed. The necessary conditions may be written

B/K = L/S = (Lv)2 = constant; (84d)

and since we have made no assumption as to the constancy of B, L, K, and S, we see that B and L are left arbitrary, any functions of x. Or, what comes to the same thing, B/L and v are arbitrary, making the attenuation and the speed variable, but without any tailing.

A third way is to examine what happens when we place a bridge of conductance k across the junction of two distortionless circuits of different types, but of the same impedance, along with a resistance r in the circuit at the same place. The two conditions, using the former notation, are

V1 + V,=(\+r/U)Vsl 1

Vl - F2= {1 + (*/&.)( 1 + r/Lv)} r3; J

from which, VJV3=l+ r/2Lv + (k/2Sv)(l + rjLv), 1

V2/V3 = rj2Lv-(k/2Sv)(l +rjLv), j

which give the ratios of incident and reflected to transmitted wave. We destroy the reflection by

rjLv = kjSv + rk}

and then the attenuation is


due to r and k. An infinite number of these r’s and k's in succession, placed infinitely close together, leads to the expression (80d).

We can also go a little way towards finding what occurs when the only condition is Lv = constant, so that there is tailing. For we then have, at a single junction,

VJ F1 = (1 + r/2Lv) ~!(1 + k/2Sv) ~1; and therefore, when the distribution of r and k is made continuous,

the attenuation of the head of a disturbance in passing through any distance is

x 6-rA-,/asfo

if be the total resistance and Kx the total conductance of the leakage in that part of the circuit. But we cannot similarly estimate to what extent the total charge and momentum have attenuated, as we could when the circuit was uniform, because the attenuation now occurs at a different rate in different parts of the tail, and we are not able to trace the paths followed by the different parts of a charge as it splits up repeatedly. The determination of the exact shape of the tail is of course an infinitely more difficult matter. But an approximation may be obtained by easy numerical calculations, if we concentrate the resistance and leakage in a succession of points.

NOTE (NOV. 30, 1887).—The author much regrets to be unable to continue these articles in fulfilment of Section XL., having been requested to discontinue them.