Electromagnetic Theory, Vol. 1 (1893) — part 27 of 31

The question now is, what do these insertions represent in the exact theory ?

Consider first a pair of parallel leads of no resistance. This will admit of the application of the exact theory, provided there be either no leakage or the external medium be uniformly con- ductive electrically. Now we know that when the leads have resistance, we make the transformation to the exact theory by substituting uniform magnetic conductivity of the external medium of the proper amount. Plainly, then, if we insert an electrical resistance in a lump in the circuit of the leads at any place, we should, in the exact theory, transfer it equivalently (changed to magnetic conductance) to the whole of the corre- sponding reference plane. That is to say, we must make the reference plane (outside the leads) uniformly conductive mag- netically, so that the total magnetic conductance of the equi- valent plate inserted equals the electric resistance which it replaces. We may then investigate the action of the plate on the waves traversing it in the manner described in a previous paragraph (§ 193 and after).

Similarly, the effect of an electrically conducting bridge across the circuit is equivalent to that of an electrically con- ducting plate at the reference plane. Their conductances are here not only equal, but of the same kind. We merely re- arrange the leakage conductance so that it shall act uniformly, to suit plane waves.

Thus, a terminal short-circuit should be replaced by a terminal perfectly conductive plate, reflecting H positively and E negatively, or with reversal of sign ; or, which is the same, C positively and V negatively. This maintains E and V per- manently zero at the short-circuit, unless there be impressed force there. And a terminal disconnection may be represented, in the exact theory, by a terminal perfectly magnetically con- ductive plate, reflecting E positively and H negatively, or V positively and C negatively, thus maintaining C permanently zero at the disconnection.

A somewhat different kind of transformation is required in connection with impressed forces. Suppose, for instance, we insert an impressed voltage in the circuit of the leads at a given reference plane. How is it to be equivalently transferred to the whole plane, or to that part of it outside the leads, so that

THEORY OF PLANE ELECTROMAGNETIC WAVES. 403

it may generate plane waves of the precise type belonging to the given leads ? This is by no means so easy to follow up as the previous plate substitutions ; but the following may suffi- ciently describe the essence of the transformation. First, when the generation of disturbances is concerned, it is not impressed electric force e, but its curl, that is effective. We must there- fore find the curl of the given impressed force. If the latter is half in each wire, acting opposite ways, and uniform across their sections by the reference plane, then the curl of e, say f, is situated on the boundaries of the wires, and encloses them. Having ascertained the integral amount of f, it must be transferred to the whole reference plane outside the wires. Then the question arises how to distribute it properly. To answer this, it may be mentioned that when a distribution of f starts into action, the first effect is to generate induction following the lines of f. This will be made clear later. If, therefore, we distribute the lines of f over the reference plane in such a way as to exactly copy the natural distribution of the magnetic lines in plane waves having the given leads for cores, we shall obtain what we want, viz., the sources so distributed as to generate plane waves of the required type on the spot.

By means of the above and similar devices, most telephonic and telegraphic problems may be converted to problems in an exact plane-wave theory. But we must always be careful to distinguish between a theory and the application thereof. The advantage of a precise theory is its definiteness. If it be dyna- mically sound, we may elaborate it as far as we please, and be always in contact with a possible state of things. But in making applications it is another matter. It requires the exercise of judgment and knowledge of things as they are, to be able to decide whether this or that influence is negligible or paramount.

The Spreading of Charge and Current in a long Circuit, and their Attenuation.

§ 208. The detailed application of the principles already discussed to the data which occur in practice, or which may occur in later practice, would lead us into technical complica- tions which would be out of place in the present stage of

DD 2

404 ELECTROMAGNETIC THEORY. CH. IV.

development of this work. Only a few general considerations can be taken up here in connection with the theory of plane waves. We have seen that there are four distinct quantities which fundamentally control the propagation of " signals " or disturbances along a circuit, symbolised by R, K, L, and S, the resistance, external conductance, inductance, and per- mittance ; whilst there are two distinct variables, the transverse voltage V and the circuital gaussage C, or the difference of potential (in an extended sense) of the leads in a reference plane, and the current in the leads. The words voltage and gaussage were proposed in § 27 to represent electromotive force and magnetomotive force, and have been employed experi- mentally to see how they would work. They work very well for theoretical purposes, but I observe that they are not generally or universally approved of, perhaps on account of the termination " age," or because of the commencements volt and gauss being unitary names. It is open, of course, to any- one to find names which shall be more suitable or meet with general approval.

But I stick to inductance. Of all the words which I have proposed, that one seems to me to be, more than any other except impedance and conductance, the right word in the right place, and I am bound to think it possible that some of those who prefer the old " coefficient of self-induction," do not fully appreciate the vital significance of inductance in a theory of intermediate action of a medium. The idea of the direct mutual action of current-elements upon one another is played out. And as regards the name of the practical unit of induct- ance, I think the best name is mac, in honour of the man who knew something about self-induction, of course. Many other names have been proposed, but none so good as mac. Some critic has made etymological objections. But what has etymology got to do with it 1 The proper place for etymology is the grammar book. I always hated grammar. The teach- ing of grammar to children is a barbarous practice, and should be abolished. They should be taught to speak correctly by example, not by unutterably dull and stupid and inefficient rules. The science of grammar should come last, as a study for learned men who are inclined to verbal finnicking. Our savage forefathers knew no grammar. But they made far

THEORY OF PLANE ELECTROMAGNETIC WAVES. 405

better words than the learned grammarians. Nothing is more admirable than the simplicity of the old style of short words, as in the A sad lad, A bad dog, of the spelling book. If you transform these to A lugubrious juvenile, A vicious canine, where is the improvement ?

Now, as we have already explained the characteristics of dis- tortion of plane waves by conductance in the medium generally, of two kinds, acting differently on the electric and magnetic fluxes, the corresponding relations in a long circuit of leads need be only briefly mentioned. Consider an infinitely long circuit of parallel and equal resisting leads under uniform external conditions, and governable by the relations laid down. Let it be, at a given moment, wholly free from charge or cur- rent except at one place, where there is merely a charge, say Q0. Or, say Q0 = SV0 on unit length only at the place. Left to itself, this will spread. But it will do so in widely diverse manners under different circumstances, although there are some common characteristics. The speed of propagation of disturb- ances is v = (LS)~*, independent of the values of R and K, the dissipation constants. Thus, at time t after the spreading begins, we are bound to find the charge, or what is left of it, within the region of length 2vt, being vt on each side of the origin. Beyond this region there is no disturbance.

Next as regards the amount of the charge. It cannot in- crease, but it may diminish. If the insulation be perfect, which means that K = 0, the charge remains constant. It is unaffected by the resistance of the leads, although energy is wasted in them. It only decreases by the external conductance, or by equivalent leakage. Then we have

...... (1)

to express the charge Q at time t, decreasing according to the well-known exponential law.

As regards the manner of spreading, either with or without the subsidence, this is represented in the exact theory by the spreading and subsidence of displacement in a medium which is magnetically as well as electrically conductive, although the former property does not affect the total amount of displace- ment. We see, therefore, that the distortionless case, which occurs when the time constants L/R and S/K are equal, forms

406 ELECTROMAGNETIC THEORY. CH. IV.

a natural division between two distinct kinds of resultant distortion. In this neutral case, the charge Q0 splits into two halves which at once separate from one another at relative speed 2v. At time t later they are each attenuated to

KHiQo^"8 (2)

The energy was wholly electric to begin with, but in each of the resulting waves it is half electric and half magnetic, be- cause they are pure waves. As they attenuate, their energy is wasted, and this occurs so that half is wasted by the resistance of the leads, and half by reason of the external conductance. This is necessary in order that a pure wave shall remain pure. The final result is attenuation of the two charges to zero, when they are infinitely widely separated, and with this a complete simultaneous disappearance of the other characteristics, as the magnetic force. In the exceptional case of no resistance and no leakage, there is, of course, an everlasting persistence of the two charges and of the waves of which they form a feature. They go out of range, but not out of existence. In all other cases but the distortionless one, the charge (or the remains of the original charge) is only partly in the plane waves at the two ends of the disturbed region, the rest being diffused between them. But this may happen in two ways. The total charge at time t is expressed by (1) above. But the charge in the terminal waves, half in each, is given by

Q0.€-(B/2L+K/28Xf ..... (3)

reducing to (2) doubled in the distortionless case. Therefore the excess of (1) over (3) is the amount of the diffused charge.

It is

Q0.e-K'/s(l-e-(R/2i--K/2S)*)) ... (4)

and is positive or negative according as R/L is greater or less than K/S.

That is, when the resistance of the leads is in excess, the charge is positive between the waves as well as in them. But when the leakage is in excess the intermediate charge is nega- tive. This happens equivalently when the inductance is in excess, or the permittance in deficit. How this reversal comes about has been already explained in the exact theory. In the present circumstances we may say that an intermediate resist-

THEORY OP PLANE ELECTROMAGNETIC WAVES. 407

ance in the circuit of the leads reflects V positively and C negatively, whilst an intermediate leak reflects C positively and V negatively. The same applies to the elements of the circuit, and from this follows the reversal of V when leakage is in excess. Whether the tailing be of the usual positive or of the negative kind (referred to the charge), the maximum or minimum density is in the middle ; that is, at the origin. The density decreases symmetrically both ways to the two ends, where the plane waves are to be found. When they have at- tenuated to practically nothing, there is left merely the widely diffused charge, unless it has attenuated similarly. With the resistance largely in excess, there may be nearly all the charge left when the terminal waves have become of insensible sig- nificance. But if the leakage be in excess, the disappearance of the terminal waves is accompanied by a similar disappear- ance of the intermediate negative charge, because the total charge is always positive.

So far regarding the spreading of a charge, next consider the spreading of induction. The magnetic analogue of Q is P = LC, the magnetic momentum per unit length of circuit. So, as L is constant, the spreading of P is represented by that of C. Suppose initially there be current C0 in unit length at the origin, with momentum P0 = LC0, and no charge there or else- where. This also splits immediately into two plane waves, which only differ from those arising from a charge in having similar C's and opposite V's instead of similar V's and opposite C's.

The total induction remains constant only if the leads have no resistance. In general it decreases, so that at time t we have

P = P0.€-**0- (5)

At the same time the amount in the terminal waves, half in each, is

P0.c-(*/2L+K/2S)e ...... (6)

The amount between the terminal waves is, therefore, the excess of (5) over (6), which is

po€-R«/L(l_e-(K/2S-R/2LX) ... (7)

These equations (5), (6), (7), may be instructively compared with the set (1), (3), (4), their analogues. Notice that in

408 ELECTROMAGNETIC THEORY. CII. IV.

transforming from one set to the other, we exchange V and C, Q and P, R and K, L and S. In the exact theory the R in these formulae would also be a conductance, as before described.

By (7) we see that in the distortionless case all the induction left is in the terminal waves, which is to be expected. But we require the leakage to be in excess for the intermediate induc- tion to be positive, and conversely the resistance must be in excess for it to be negative. Of course, the initial induction is assumed to be positive. Also observe that leakage by itself has no effect on the total induction except to redistribute it, just as the resistance of the leads has no attenuating effect on the total charge. As before, the maximum or minimum density of induction is at the origin, though, of course, the terminal inductions may have greater or smaller density than this, according to the extent of attenuation at the moment concerned. Remember that the destruction of induction by the resistance of the leads is represented, in the exact theory, by its destruction by the equivalent magnetic conductance externally.

Putting together the preceding results, we may readily see the effect of having both V and C initially at any spot. One case is specially noteworthy. Take V0 = L^C^ initially. This means a pure plane wave sheet, travelling in the positive direction. There is, therefore, no initial splitting, and the wave just goes on. To what extent this will continue depends upon the relation the constants of the circuit bear to the distortionless state. When the latter obtains, the wave is transmitted without spreading out behind, so that at time t the initial state, if existent over unit length, will be found over unit length at distance vt to the right, attenuated to

V0€-K</L, ..... (8)

where for t may be substituted its equivalent x/v. In all other cases there is reflection in transit, and the reflected portions travel back, besides getting mixed together, thus making a tail of length Zvt, half on each side of the origin, but now, of course, without a head at the negative end to balance the one at the positive end.

If the resistance be in excess, the charge in the tail is positive and the current is negative, at least initially ; whilst if the

THEORY OP PLANE ELECTROMAGNETIC WAVES. 409

leakage be in excess, the current is positive and the charge initially negative. See §§ 195 to 199 for more details, which may be readily translated to suit the present case. The total charge subsides according to equation (1) above, and the total induc- tion according to (5). Similarly (3) represents the charge in the one terminal wave (instead of both, as there), and (6) shows the induction in the terminal wave.

If initially V0 = - L^CQ, this means a negative wave (going to the left) to start with. This needs no separate notice in detail.

The Distortionless Circuit. No limiting Distance set by it when the Attenuation is ignored.

§ 209. The distortionless state forms a simple and natural boundary between two diverse kinds of propagation of a com- plicated nature, in each of which there is continuous distortion which is ultimately unlimited. Given time enough, and a circuit of infinite length to work with, the least departure from the distortionless condition would be sufficient to allow all the natural changes to be gone through, with ulti- mate unlimited distortion. The attenuation is a separate matter in this connection, though itself of great importance. If, however, we are only concerned with the distortion produced in a finite interval of time, which may be quite small, then the matter may be differently regarded. There is much or little distortion, generally speaking, according as the ratios R/L and K./S are nearly equal or widely different. But this needs to be understood with caution, for obviously there may be next to no distortion of signals even when the distortionless state is widely departed from, provided the changes of charge and current are made slowly enough. But when this is not the case, and there is marked distortion of waves in transit, then we shall increase it by making the time-constants more unequal than they are, and decrease it by a tendency to equalisation. Furthermore, this process is a continuous one, so that, starting from the dis- tortionless state connected with equal time-constants, we may, by continuously increasing one time-constant, bring on con- tinuously increasing distortion of one kind, or else, by increasing the other, whilst the first is kept constant, bring on con- tinuously increasing distortion of the other kind. And this is

410 ELECTROMAGNETIC THEORY. CH. IV.

true by variation of any one of the four quantities concerned, the resistance, leakage, inductance, and permittance, to prove which we need only remark that with any values given to these constants we can equalise the time-constants and abolish the distortion altogether by altering any one only of the four.

When a circuit has been brought to this state, then arbitrary signals of any size and manner of variation, originated at the beginning of the circuit, will run along it at the speed of light, and in doing so suffer no alteration save a weakening according to the exponential law given above. The voltage and gaussage at any spot will be always in the same phase, and the suc- cession of values will faithfully repeat those at the origin. If the circuit be infinitely long, the train of disturbances will run out to infinity, of course, though it may be still a long way from infinity when the attenuation is so great as to make the disturbances insensible. Therefore, disregarding the attenua- tion, there is no limiting distance of possible signalling. More than that, it is not only possible, but perfect signalling, accord- ing to this theory.

But there is no perfection in this world. Even if a circuit were constructed with constant R, K, L, and S, and with R/L = K/S exactly, so that it should be truly distortionless in this practical theory, it would not be so in reality. There are several disturbing causes, which, though they might not be of much importance in general, would -serve to prevent the attain- ment of the clean-cut perfection of the theory which ignores them. Instead of zero distortion, then, it is practically only a state of minimum distortion that is attainable. But it may be a very good imitation of the theoretical ideal. It is unnecessary to enlarge upon the necessary failure of a professedly approxi- mate theory when pushed to extremes. It is sufficient to say that the ignored disturbing influences alone would serve to set a practical limit, even if the condition R/L = K/S were truly attained, and attenuation were of no importance.

But there is an influence in full action in the practical theory itself which must not be overlooked, namely, the attenuation due to resistance and leakage. Of what use would it be to have a distortionless circuit if it took nearly all the life out of the current in the first 100 miles, when you want it to go 1,000? Now, on paper nothing is easier than to increase the

THEORY OP PLANE ELECTROMAGNETIC WAVES. 411

battery power to any extent you want, till, in fact, the minute fraction which got to the desired distance became magnified up to a recognisable size. But practically there would usually be strong reasons against this process. At any rate, it is im- possible to overlook the attenuation. It is not merely enough that signals should arrive without being distorted too much ; but they must also be big enough to be useful. If an Atlantic cable of the present type were made distortionless by the addition of leakage, there would be no sign of any signals at the distant end with any reasonable battery power. Nor can we say that telephony is possible on a circuit of given type to this or that distance merely because a certain calculable and small amount of distortion occurs at that distance. Nor can we fix any limiting distance by consideration of distortion alone. And even if we could magnify very weak currents, say a thousandfold, at the receiving end, we should simultaneously magnify the foreign interferences. In a normal state of things interferences should be only a small fraction of the principal or working current. But if the latter be too much attenuated, the interferences become relatively important, and a source of very serious distortion. We are, therefore, led to examine the influence of the different circuit constants on the attenuation, as compared with their influence on the distortion.

The two Extreme Kinds of Diffusion in one Theory.

§210. Although the equations connecting the voltage V and gaussage 0, through the line constants R, S, K, and L, are of a symmetrical character, so that, abstractedly considered, a study of the nature of their propagation on either side of the distortionless state is as desirable as on the other side ; yet when we examine the conditions prevailing in practice, we see at once that one side only presents itself actively. The reason is easily to be seen. It is in all cases desired to get plenty of received current, because that is the working agent, and leak- age is detrimental to this consummation. There are, besides, practical inconveniences connected with leakage. Thus it comes about that we are mainly concerned with the state of things existing when K/L is greater than, or at the least equal to, K/S, and scarcely at all with the exceptional case of an excess of

412 ELECTROMAGNETIC THEORY. CH. IV.

leakage, unless it be as a curiosity, and for the sake of its theoretical connection with other electromagnetic problems.

Now this concerns the propagation of V and C through the external dielectric. The influence of R prevails over that of K, and controls matters. It is, however, somewhat remarkable that the other side of the propagation problem does actually present itself actively, when we regard the important secondary effect of the propagation into the guides of the cylindrical waves, which result from the passage over them of the plane waves in the external dielectric. Suppose, for example, there is no external conductance. This gives one extreme form of propagation of V and C, the quantity K/S (or equivalently k/c) being zero, and R/L finite. Now, remember that in the exact plane-wave theory R stands for the magnetic conductance of the external medium, so that R/L is the same as g//j,, and there- fore we are, when considering V and C, also discussing the influence upon the plane waves of the fictitious property of magnetic conductance.

On the other hand, in the guides themselves, which are elec- trical conductors, and perhaps, and most probably, dielectrics as well, it is k/c that is greater than g/fj., since the latter is zero. Here, then, we have the other extreme form of the theory. In the external dielectric k/c is zero and g/jj. (equiva- lent to R/L) finite, whereas in the guides g/p is zero and k/c finite. The distortion of the external plane waves by the resistance of the guides is, therefore, of the opposite kind to that of the cylindrical waves in the guides, for E and H change places when the form of the distortion is discussed.

Now, in the guides, on account of their being very good con- ductors, the permittivity is swamped, and may be ignored in calculating resultant effects. Then we have the theory of pure diffusion (as of heat according to Fourier), controlled by the two constants k and //,, since both g and c are now zero. Similarly, in the external dielectric, it may happen that the in- fluence of fj. (or equivalently of L) is insensible, as in the slow working of long cables ; then the propagation of V and C is controlled by the two constants g and c (or actually and equiva- lently by R and S), whilst the other two, k and ^ are zero. Here, then, we have a curious exchange of active properties between the dielectric and the conductor. In both, the resul-

THEORY OP PLANE ELECTROMAGNETIC WAVES. 413

tant propagation follows the diffusion law, but is controlled by different properties. The propagation of V or E outside is like that of H inside, and the propagation of C or H outside is like that of E inside.

We see by the above illustrations that in spite of the absence of magnetic conductivity in reality, we are, nevertheless, obliged to consider both sides of the wave-theory it involves, even in one and the same actual problem, provided it be treated com- prehensively.

The Effect of varying the Four Line-Constants as regards Distortion and Attenuation.

§211. As, however, we are not immediately concerned with the internal state of the guides, and, so to speak, eliminate it by treating R and L as constants, we may now dismiss its consideration, and return to V and C only, on our understand- ing that the quantity K/S is always less than R/L, or does not exceed it in the limit.

Now we have shown that the distortion in transit depends upon the difference

R K. /n\

— 2L-2S (9)

whilst the attenuation of the front of an advancing wave depends upon the sum

o = A + JL. (10)

2L 2S

Both a- and p should be as small as possible, of course. The four line-constants act upon p and o- in different ways, and it is rather important to understand them. So, remembering that the first term of the right members of (9) and (10) is bigger than the second, observe the effect of varying the line-con- stants, one at a time.

[R]. By increasing R we increase both p and <r. Thus the resistance of the guides is wholly prejudicial, since it not only weakens signals in transit, but distorts them. It is, therefore, a fundamental notion that the resistance of the guides should be reduced if possible, but never increased, if the object be to facilitate signalling.

414 ELECTROMAGNETIC THEORY. CH. IV.

[L]. By increasing L we decrease both p and or. Thus the inductance of the circuit (other things being equal) is wholly beneficial, since it diminishes both the attenuation and the dis- tortion in transit, or makes the received signals both larger and plainer. These are not necessarily insignificant effects, but may be very large when applied to very rapid vibrations on a long circuit.

[K]. Increasing K increases p and reduces or. Thus leakage is both beneficial and prejudicial, according to circumstances, since although it reduces the distortion, it simultaneously in- creases the attenuation in transit. This action is most pro- nounced when the signalling is slow.

[S]. Increasing S increases cr and reduces p. It may be, therefore, both prejudicial and beneficial. But practically the important effect of S, notably on cables, is the prejudicial one of distorting the signals, which is a large effect.*

Now the above statements, though obviously true enough as deductions from (9) and (10), are merely qualitative. The practical import of varying one quantity when three others are constant, depends upon the actual immediate values of all four. The length of the line also comes in as an important factor, and likewise the frequency of the waves, to settle under what circumstances this or that variation of the line constants is important in a special case. Nevertheless, the above properties are important, and should be most usefully borne in mind in general reasoning, as well as in the detailed examination of formulae, for instance, the important solutions for simply periodic waves.

Some of these effects have been known and understood from the earliest times, or, more correctly, since the time when William Thomson taught telegraph-cable engineers the prin- ciples of their business (in its electrical aspects), and, to a great extent, the practice too. They were previously in quite a benighted state, generally speaking, though there were a few, whose names it is needless to mention, who combined enough

• Every rule is said to have an exception. It is, at any rate, possible for the above rules to fail, or appear to fail, when the making and reception of the signals is not of a simple kind. We may then attribute the failure to instrumental peculiarities. An example will occur a little later in which K reduces working speed largely.

THEORY OF PLANE ELECTROMAGNETIC WAVES. 415

electrical science with their practical knowledge to have fairly correct ideas about retardation in submarine cables. I do not mention Faraday in this connection, for that great genius had all sorts of original notions, wrong as well as right, and not being a mathematician, could not effectually discriminate, especially as he had so little practical experience with cables.

A complete history of the subject of the rise of Atlantic telegraphy, including the scientific side, written by a man fully and accurately acquainted with the facts, preferably with per- sonal knowledge, and devoid of personal bias, would be of great value and of permanent utility. It will not be very long before no contemporary will be left to do it. Perhaps, on the other hand, it might be more fairly done at second hand. In the meantime, there is a somewhat bulky volume, the Report of the Submarine Cable Committee of 1859, which serves to show the state of knowledge at that date. But it is the history of the march of knowledge in the several preceding years that is important.

The two effects referred to are that resistance and permit- tance (R and S) act conjointly in producing "retardation," so that both R and S should be reduced to improve the electrical efficiency of a cable. This is embodied in Lord Kelvin's theory of 1855, and gave rise to a rather neat law of the squares, something that practicians could grasp. This law soon became matter of common knowledge amongst cable engineers and electricians generally. But when once grasped they could not let go, and in later times the law has been most grossly abused and misapplied. The fact still remains, however, that R and S are prejudicial. But allowance has to be made for other influences, and it may be very large allowance.

But, very curiously, there has been an exception to the rule. There have not been wanting electricians who, when telephony was extended to circuits containing underground work, sup- posed that high, not low resistance, was good for telephony. This was giving up the law of the squares with a vengeance. But there was really no warrant for such a conclusion, either theoretical or practical. As this view turned out manifestly wrong by practical failure, it was followed by a precipitate re- vulsion, the law of the squares being again seized, and grasped more firmly than ever, and against all reason. The fact that R

416 ELECTROMAGNETIC THEORY. CII. IV.

and S are prejudicial does not make the law of the squares. To have that, R and S must be the only controlling electrical factors.

All the other effects are of later recognition, and were mostly discovered, and their laws investigated, and consequences worked out by myself. They completely remodel the subject. The old views remain right, in their proper application, bub have a very limited application in general, as in modern tele- phony particularly. My extended theory opened out a wide field for possible improvements in signalling over long circuits in advance of what could be possible were R and S the only controlling factors, and so far as it goes, modern telephony already goes far beyond the supposed restrictions of the law of the squares.