Electromagnetic Theory, Vol. 2 (1899) — part 26 of 31

external disturbance becomes modified. This occurs in a two- fold manner. By cumulative action, the nature of a wave sent along the leads may be profoundly modified as it progresses. Besides this, there is what we may term a local modification, whereby the wave at any place is no longer strictly a plane wave. It is merely with this effect that we are now concerned. The departure from planarity requires a considerably modified and much more difficult theory, no longer expressible in terras of V and C simply, in order to take it into account. But in the construction of a practical theory, we take advantage of the fact that the departure from planarity is slight, to which we have already referred towards the end of § 199. A line of elec- tric force does not now start quite perpendicularly from the positive lead and end similarly on the negative lead. It has a slight inclination to the perpendicular, and therefore curves out • of the reference plane to a small extent. In the dielectric itself, this peculiarity is of little importance. But the slight amount of tangentiality of the electric force at the surface of the leads, which is conditioned thereby, is of controlling import- ance as regards the leads themselyes, and eventually through them, to the waves in the dielectric, by attenuating and altering the shape of waves (considered axially), as just referred to. We should now consider what form is assumed by the second dr- euital law in terms of V and C, when we admit that there is tangential electric force on the sur&ce of the leads, bat on the assumption that the minor effects in the dielectric itself, which are associated with the presence of the tangential foroe^ are of insensible influence.

We may still regard Y as the transverse voltage in the reference plane. Whether we go straight across from the positive to the negative lead, keeping in the reference plane, or leave it slightly in order to precisely follow a line of force in estimating the voltage, is of no moment, because the results differ so slightly. The quantity C is subject to a similar slight

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THEORY OF FLANB BLBCTROMAGKETIC WAVES. 387

difference^ aooording to the way it is reckoned. We may, per- haps, most conveniently reckon it as the circultation of the magnetic force round either lead upon its surface and in the reference plane, because this way gives the exact value of the conduction current in the lead. But the cirouitation in other paths in the reference plane will not give precisely the same value now that the waves are not quite plane. This difference we also ignore in the practical theory. In truly plane waves the electric current in the dielectric is wholly transverse. There is now really an axial component as well, but being a minute fraction of the transverse current, it is ignored. In short, we have to make believe that the waves are planar when consider- ing their propagation through the dielectric, whilst at the same time we take into account the departure from planarity in considering the influence of the leads and what occurs in them. It is unfortunate to have to refer to small corrections, as it confuses the statement of the vitally important matters. Let us, then, set them aside now.

Construct the second circuital equation in the manner fol- lowed in §201, in elucidating the meaning of equation (6). Consider a rectangle consisting of two transverse Sides in reference planes at unit distance apart, beginning upon the positive and ending upon the negative lead, and of two axial sides of unit length upon the leads themselves. Reckon up the voltage in this rectangle. The transverse sides give V and Y+dV/dx. The axial sides give and say, if E| is the tangential component in the direction of increasing x of the electric force at the boundary of the positive tube, and the same on the negative tube, but reckoned the other way. The complete voltage is then

It is also equal to - LC, as in § 101. (There is no magnetic conductance of the dielectric now.) So

or

5^ Y + E J + £^

dx

— — - = £j + £2 l-'^i • •

(16>

is the form assumed by the second circuital law.

cc2

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BLECTROVAOVSnO THBORT.

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Observe that we have made no specification of the nature of the leads, so that the ecjuation possesses a high degree of generality. As regards the first circuital law, that is unchanged, being expressed by (5) when the dielectric is non-conductive, and by (9) when conductive, since the minor changes alluded to as regards Y and C are not significant.

Being general, the equation (16) needs to be specialised before it can be worked. If our electromagnetic variables are to continue to be V and C, we require to express and E., as functions of C. Fortunately this can be done, sometimes very simply, and at otlier times in a more complicated way. To take the simplest case, let the leads be tubes (or sheets) of so small depth that penetration is practically instantaneous as waves pass along them. Then Ej is not only the boundaiy tangential electric force, but is also the axial electric foroe throoghoat the substance of the positive tube. Similarly as regards for the negative tube. It is true that^ in virtue of the transverse electric current from tube to tube, there is also transverse current in the tubes, and, therefore, transverse electric foroe, but this is to be ignored, because it is a small fraction of the axial. The current density in the positive tube is therefore axial, of strength ^Ep if is the conductivity of the material; and the total current in the tube is K^E^, where is the axial conductance per unit length, or, which is the same, E^/R^, if is the resistance per unit length. But this quantity is also the previously investigated quantity C, the circuitation of the magnetic foroe on the boundary of the tube. So we have the elementary relations

Ei = RiCi, E2 = R,C5p . . . (17)

which are, be it observed, essentially the connections of the electric and magnetic foroes at the boundaries, though brought to a particularly simple form by the instantaneous penetrati<»u The second circuital equation (16) therefore takes the form

-^-(Rj + igC + LO. . . • . (18)

dx

Or, finally, if R is the resistance per unit length of the two leads,

^^-RO + LC, (19)

dx

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THEORY OF FLAME ELECTROMAGNETIC WAVES. 389

which is the practical equation for most purposes. Observe, comparing it with (12), that there is an identity of form, but with a changed meaning of the symbol R. In the exactly stated problem of plane waves running through a medium of duplex conductivity bounded by perfect conductors, the quantity B is the external magnetic conductance per unit length axially. In the present approximately stated problem of very nearly plane waves running through an electrically conducting medium bounded by resisting wires, the quantity E is the resistance of the wires, per unit length axially. What we have to do, there* lore^ in oider to turn the real problem into one relating to •trictly plane wayes admitting of rigorous treatment^ is to abolish the resistance of the leads and substitute equivalent magnetic conductance in the dielectric medium outside. The two materially different properties are nearly equivalent in their effects. Equations (13), (U), (15), still hold good, only BC* is now to be the rate of waste in the leads, instead of in the medium generally due to the (now suppressed) magnetic conductance. The flux of energy is now not quite parallel to the leads everywhere, but has a slight slant towards them. But the usual formula gives the waste correctly. The product EjHj of the tangential electric force Ej and the magnetic force, say H^, at the boundary of the positive lead, is the rate of supply of energy to the lead per unit surface. Therefore, by circuitation, the product EjC is the rate of supply per unit length axially. This is the same as the previous R^C^.

The equivalence of effect of magnetic conductance externally and of electric resistance in the leads, is undoubtedly a some- what mysterious matter, principally because it is hard to see (apart from the mathematics) why it should be so. As regards the above reasoning, however, it is essentially simple, and is a direct appUoation of fundamental electrical principles. It may, therefore, cause no misgivings, except the doubt that may present itself whether the ignored small effects in the dielectric due to departure from planarity of the waves are really ignor« able. As a matter of fact, they are not always of insensible effect, and plenty of problems can be made up and worked out tubes and wires which do not admit of the oompsn vatively simple treatment permissible when V and 0 are the variables, and the results are exceedingly curious and interest-

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ELECTROMAGNETIC THEORY.

CH. IV.

ing, though quite unlike those at present in question. But these are not })rac'tical problems, and have little bearing upon the question of the free propagation of waves along long parallel straight wires. They do not start as full grown plane waves from the source of disturbanoe, but they very soon fit them- selves on to the wires properly, and then follow the laws of plane waves pretty closely.

Although thin tubes for leads were specially mentioned in connection with equation (19), yet any sort of leads will do provided the penetration be sufficiently rapid to be instant taneoos " within the meaning of the Act" It is obviously true lor steady currents, when the inertial term disappears and B Is the steady resistance. And if the variations of cuxxent be not sufficiently rapid to cause a sensible departure from uniformity of distribution of current in the conducting wires, then, of course, (19) may still be safely used. At the same time, it should be mentioned that the inductance L should, under the circumstances, be increased by a (usually) small amount due to induction in the conductors themselves, as will be presently noted more closely. Furthermore, remember that no allow- ance has been made for the influence of parallel conductors, should there be any. The earth does not count if the leads be alike and equidistant from the ground, as its influence can be embodied in the values of L and S, the inductance and per- mittance.

The Second Circuital Equation when Penetration is Not Instantaneous. Besistance Operators, and their Definite Meaning.

§ 203. The next question is what to do when penetration is not instantaneous within the meaning of the Act. We should first go back to (IG), which remains valid, and inquire whether the tangential electric forces cannot be expressed in terms of the current (or conversely) in some other way than by a linear relation. Suppose we say

Ei = R"iC, E2 = ir2C. . . . (20)

Is it possible to give a definite meaning to the symbol B" ? Is there a definite connection between C, regarded as a funotioii of the time, and or also regarded in this way ff

THBORY OF PLANS ELECTBOMAONETIC WAVE8. 391

Imagine a wire to be free from current, and, therefore, elec- trically neutral. Now expose its boundary to tangential electric force, beginning at a certain moment, varying in some particular way with the time later, and then ceasing. The result is that C, the total current in the wire, will run through a particular sequence of values, and then finally cease. If we begin again with the applied foroe^ and make it run through the same values in the same manner, we shall again obtain the same values of C at corresponding moments. So far, then, the oonneotion is a definite one. Moreover, if we make the applied foroe run through the same values increased in a certain ration the same for all, then the current will have its previous values increased in the same ratio. But if we change the nature of the applied force as a function of the time (irrespective of sise) we shall find that C is not merely changed as a function of the time, but also as a function of the applied force. That ia, the mere value of one does not necessitate any particular simulta- neous value of the other. So, if we keep to the usual sense meant when algebraists say u=f{x), or ii is a function of a?, we cannot say that C is a function of £. It is, nevertheless, true that the march of C is strictly connected with that of E, so that when the latter is given, the former is obligatory. To deny this would be equivalent to the denial of there being definite controlling laws in operation. The full connection between E and C, however, involves not merely their values, but also the values of their first, second, third, &c.i differential coefficients up to any order. That is, the symbol R", taken by itself, is a function of the differentiator djdt. To illustrate by a simple example, suppose

R'-R+Lp + (S/))-i, .... (21)

where B, I^ S» are constants, and p stands for the differentiator. This means that

E-{R + l4»+(Sp)-i}C, .... (22)

or £-RC + LC + S/C(ie, .... (23)

in the common notation of integrals. Now imagine the march ' of C to be given. This implies that the march of C is also known, and likewise that of yCeft. Consequently the march ol

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ELBOTBOMAGNRIO THSOBT.

OH. IV,

E is ezplidUy known. It is not obvioas that when the maroh of £ is given, that of C is known, by the same operator impli- citly. That is, by (22) we have

^"R + Lp + (S^)-^ ^^^^

and given E as a function of the time, C is known as a fanction of the time, or if not known, can be found without ambiguity. It is not obvious, because we do not immediately see how the operation indicated in (24) is to be carried out, whereas, in the case of (22), it is visible by inspection. Nevertheless, the fact that the march of E, physically considered, conditions that of C, makes the above equation (24) not only definite, but com- plete. In the usual treatment of the theory of differential equations, there is no such definiteness. Arbitrary constants are brought in to any extent, to be afterwards got rid of. Now this is all very well in the general theory of differential equations, where arbitrary constants form a part of the theory itself, but for the practical purpose of representing and obtaining solutions of physical problems, the use of such arbitrary and roundabout methods (which are too often followed, espeoiaUy by elementary writers) leads to a large amount of nnneoeeBary work, tending to obscure the subject, without helping one on. It would not, perhaps, be going too far to say that sudi a misuse or inefficient application of analysis often makes rig- marole.

When we say that E-R'G, where is the resistanoo operator (so called because it reduces to the resistance in steady states), we assert a definite connection between E and Q so that when 0 is fully given as a function of the time, and the operations contained in R' are performed upon it, the funo- tion E results, and similarly, when it is E that is given, then the inverse operator (R")"^ (or the conductance operator) act- ing upon it will produce C. There may be an infinite number of differentiators in R", as p, p^, p^^ and so on, where means cP*ldt^, But there is not a single arbitrary constant involved in E -• R''C, nor, indeed, anything arbitrary.

Returning to the wires, the form of the resistance operator, as a i unction of p, depends upon the electrical and geometrical

THBOBT OF FLAKE XLBOrBOMAONBTIO WAVES. 393

data. It has been determined for rooiid wires and round tubes, and plane sheets. We may take

^^^{B,\ + 'Br^C + LC, . . . (25) ax

as the form of the seoond oirouital law, and make the determi- nation of the operators a matter of separate calculation.

As already mentioned, when C is steady, B'^^ degenerates to Kj and R% to li^, the steady resistances (per unit length) of the wires, tubes, rods, or cylinders of any shape that may be employed. At the same time the inertial term disappears. Also, when C Taries, it is sometimes sufficient to take into account only the first approximation to the form of the opera- tors. This is, for a solid round wire,

R'l - Bi + i/i,;*. (26)

and similarly for R'g. This Jfi^ is the value of the steady inductance of the wire, /Mj being its itiJiictivity. When not solid, or not round, some other expression is required. Notice that this brings (25) to the elementary form (19), because the inductance of the wires may be included in L itself, which then becomes the complete steady inductance (per unit length axially), including that due to the dielectric and that due to the two wires. This usually means only a small increase in the value of L, unless the wires be of iron.

fiimply Periodic Waves Easily Treated in Case of Imperfect

Penetration.

§ 204. But besides the above simplification, there is an ex- ceedingly important general case in which a similar reduction takes place. This oocnrs when the sources of disturbance vary simple periodically with the time. Then the electric and mag- netic fluxes everywhere vary ultimately according to the same law with the same period, provided the relations of the fluxes to the forces are linear ; that is, when the conductivity, permit- tivity, and inductivity, are constants at any one place. Now, when this comes to pass, both £ and 0 in the equation E^B'^C vary simple periodically with the time. But when the sine or ■cosine is differentiated twice, the result is the same function negatived, and with a factor introduced. Thus, if the i'ret^uency

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ttiBCTROM AGNSnC THBORT.

CH. IV.

is n/27r, so that sin (nt + 0) may be considered to be the time- factor in the expression for either E or C, we have the property (Pjdt' = - n^. Put, therefore = - in the expressions for the resistance operators, and we reduce them to

R'', = R', + L>, R% = R'j + L>, . . (27)

where B' and L' are functioiiB of n\ They are, therefore^ oon- ■taiits at a given frequency, and this is a very valuable property, as it is dear at once that we redoce the equation (25) to the simple form

• ^-(R'j + R,)C + (L + L'i + L,)i?C, . (28)

or, more briefly and clearly,

-4^-R'C + L'C, • . . . (29) ax

which is the same as (19), valid when the penetration is in- stantaneous, but with different values of the constants involved. The steady resistance R is replaced by R', the effective resist- ance (of both tubes per unit length) at the given frequeni^, and L the steady inductance (inclusive of the parts due to the wires) by L', the effective inductance.

Owing to the reduction of the second circuital equation to the primitive form, we are enabled to express the propagation of simply periodic waves along wires by the same formulae, whether there be or be not imperfect penetration. We simply employ changed values of the constants, resistance and induct- ance, which may be independently calculated, or left to the imagination, should the calculation be impracticable. This ia, when it can be effected, the best way of making extensions of theory. Do the work in such a way that harmony is produced with the more rudimentary results, and so that they will work together well, and the new appear as natural extensions of th» old. An appearance of far greater originality may, indeed, be produced by ignoring the form of the elementary results, but the results would be cumbrous^ hard to understand, and un- practical.

The effect of increasing the frequency from zero to a high degree is to first lessen the penetration, and end in mece skin

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TUKOIiY OF PLANE ELECTROMAGNKTIO WAVES. 395

penetration. Consequently, the quantity L', the effective in- ductance, goes from the full steady value L + L'j + L'g, and finishes at L simply, the inductance of the dielectric. But the difference need not be great. In the case of suspended copper telephone wires the inductance of the dielectric is far larger than the rest, so that there is no important variation in the value of the inductance possible, nor would there be were the frequency increased up to that of Hertzian vibrations. But as regards the effective roBistaaoe, the case is different. As the distribution of current in a wire changes from that appropriate to the steady state, the resistance increases, and the increase is not always a negligible matter. If long-distance telephony were carried on along iron wires it would be a very important effect. But the Americans, who were the introducers of long- distance telephony, soon found that iron would not do, and that copper would, ^e reason of the failure is mainly the largely increased resistance of iron. In copper, on the other hand, it is an insensible effect at the lower limit of telephonic frequency of current waves, with the size of wire employed, and is not very important at a fluency three or four times as great. On the other hand, in the numerous experiments with very rapid vibrations of recent years, due to Hertz, Lodge, Tesla, and many others, the increased resistance due to imperfect penetration becomes a very important matter, and is one of the controlling factors that should be constantly borne in mind.

Long Waves and Short Waves. Mentitir of Speed of Free

and Guided Waves.

§ 205. By replacing fictitious magnetic conductance of the medium outside the pair of leads by real electric resistance of the leads themselves, we have obtained the same results as regards the propagation of waves, subject to certain reserva- tions referring to the practical applicability of the theory. It is worth while noticing, in passing, a certain peculiarity show- ing roughly when we may expect the theory to be admissible, and when it should faiL We know that in the transmission of waves along perfectly conductive leads the waves are con- tinuously distorted if the medium be electrically oonduotive. Alio^ that by introducing magnetic conductivity into the

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BLBOTBOMAONKnO THEOBT.

CH. IV

medium we may reduce this distortion, and ultimately aboHsh it wheu the magnetic conductivity reaches a certain vahie. Now observe here that the correcting iufluonee of the magnetic conductivity is exerted precisely where it is required, namely, in the body of the wave itself. The wave is acted upon in every part by two counteracting distorting influences, so that the correction is performed exactly ; in other words, we obtain an exact theory of distortionless transmission.

But, on the other hand, when we employ the resistance of the leads to perform the same functions, the correcting in- fluence is not exerted uniformly throughout the body of the wave, but outside the wave altogether ; at its lateral boim- daries, in fact. Nevertheleas, for reasons before stated, we still treat the waves as if they preserved their planarity under the influence of the resisting leads. Whilst, therefore, we fully recognise and employ the finite speed of propagation of disturb' anoes azially, or parallel to the leads, we virtually assume that the correoting iidSuenoe of the leads is transmitted laterally outwards instantaneously. We may, therefore^ perceive that the wave-length is a matter of importance in determining the applicabilil^ of the practical theory. It is of no moment what- ever in the exact theory employing magnetic condnotanoe ; but, when we remove this property from the medium generally, and (virtually) concentrate it at the leads, we should at the same time keep the wave-length a considerable multiple of the dis- tance between the leads if the practical theory is to be applic- able. To see this, it is sufficient to imagine tiie case of waves whose length is only a small fraction of the distance between the leads, when it is clear that their correcting influence could no longer be assumed to be exerted laterally and instantaneously, as if the small portion of the leads between two close reference planes belonged to and was associated solely with the slice of the medium between the same planes.

We have referred in the above to the action of the resistance of the leads as a correcting one, neutralising the distortion due to another cause. But the same reasoning is applicable wheu the action is not of this nature. Thus, when the external medium is non-conductive, and the leads are non-resistive, we have perfect transmission. Making the leads resistive therefore now brings on distortion. We may now say that in order that this

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THEORY OF PLANE BLBOTBOXAONBTIO WAVES. 397

dlitortion should oooor in the same way tm under tiie influence of magnetic oonduotivity in the external medium, the wave- length should not be too small, as specified above. The prac- tical theory is therefore the theory of long waves, and in the interpretation of the word " long," some judgment may be exercised as regards the leads and other matters, because there is no hard and fast distinction, and what may be a long wave under some circumstances may be a short one under others.

To illustrate this point, imagine that we have got an inge- nious instrument (not yet made), for continuously recording the electromagnetic state of a non-conducting medium, say the air at a certain place (just as we have thermometrio and barometric recorders), and that this instrument is so immensely quick in its action as to take cognisance of changes happen- ing in very short intervals of time, say one thousand-millionth of a second. Now in applying this instrument to register the state of air traversed by electromagnetic waves, it is clear that the sise of the waves must be considered in relation to the size of the instrument If the instrument were one decimetre across, then waves of one kilometre in length could as well be regarded as of infinite length. But if only one metre in length, though it oonld still be used, there would be no longer the same accu- racy of application. And if of only a centimetre in length, then it is plain that several waves would be acting at once on the apparatus in different parts, and the resultant eirect recorded would not represent the history of the waves by any means.

Now as regards waves sent along parallel leads, it is obvious that light waves are totally out of the question, being im- mensely too short. On the other hand, telephonic waves are to be treated as long waves — very long, in fact — though they are short compared with telegraphic waves. But it is somewhat curious that the electromagnetic waves investigated by Hertz are sometimes ho short as to come within the scope of the above reservatiooal remarks, or of others of a similar nature. Whether the generation of waves by an oscillf*^ jr be considered, or their efieot on a resonator, or their ^Ldnsmission along leads, their shortness in relation to tKd apparatus employed may some- times vitiate the resulJU of approximate theories, and render oaution neo^'ssary. y^ot example, the plane-wave theory indi-

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BLBOIROMACanfiTIO THBORT.

CH. IV.

Cates that the attennatioii factor for waves running along parallel leads is c-^''^ in the distance or €'^^1^ in the equivalent time of transit t. This is when R and L are oon- stants. If not, and the waves be simply periodic, then we may use the same formula with the effective values of R and L at the frequency concerned. But if this be true for long waves, we cannot expect it to continue true on shortening the waves to the transverse distance of the leads, more especially if we are ignorant of the precise type of the waves. At the best, we should not expect more than results of a similar kind. But not much has been done yet in the quantitative examination of Hertzian waves, for sufficiently obvious reasons.

In one respect, however, a formerly very strange anomaly has been cleared up satisfactorily. When Hertz opened people's eyes and made them see the reality of Maxwell's ether as a medium propagating electromagnetic disturbances at the speed of light, by showing their transmission across a room and reflection by a metallic screen, the full acceptation of Maxwell's theory was considerably hindered for a time by bis finding that the speed of waves sent along wires was much less than that of free waves. The discrepancy was a large one, and gave support apparently to the old view regarding the function of wires, which made the wires the primacy seat of transmission, and effects outside secondary, due to the wires. And it came to pass that people, whilst admitting the truth of Maxwell's theory, yet made a disUno- tion between waves in free space and " in wires." This was thoroughly out of harmony with Maxwell's theoxy, which makes out that the wires, though of great importance as guides, are nevertheless only secondary. On the other hand, it should be mentioned that Lodge found no such large de- parture from the speed of light in his experiments. But the matter has been explained by the discovery that an erroneoos estimate was made of the permittance of the osciUatw in the experiments which apparently showed that the speed was largely reduced. When corrected, there is not left any notable differ- ence between the speed of a free wave and of one guided by wires. Of course, there is no reason why reference to waves " in wires " should not be dropped, unless the laterally-propa- gated cylindrical waves are meant. These are secondary, and

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THEORT OF PLANK ELBOTROVAONBTIO WAVES. 399

have no essential connection with the propagation of the primary waves throogh the external dielectric, although modi- fying their nature.

The Guidance of Waves. Usually Two Guides. One suf- ficient^ though with Lobs. Possihility of Guidance iRdihin a Single Tube.

§ 206. When waves are left to themselves in ether without

the presence of conductors, they expand and dissipate them- selves. Even if they are initially so constituted as to converge, to centres or axes, they will subsequently expand and dissipate. To prevent this we require conducting guides or leads. Now this usually involves dissipation in the leads ; but the point at present under notice is the property of guidance only. We can stop the expansion in a great mesisure, and cause a wave to travel along wherever we wish it to go. Practically there are two leads, as a pair of parallel wires ; or if but one wire be used, there is the earth, or something equivalent, to make another. But it is still much the same, as regards guidance, when there is but one wire, if we choose to imagine the case of a single infinitely-long straight wire alone by itself in ether. If we make it the core of a plane electromagnetic sheet, this sheet will run along the wire just as well and in the same way as if there were a seoond guide. But the energy of such a sheeti even though of finite depth, and containing electric and magnetic forces of finite intensity, would be infinite. The quantity the inductance per nnit length of guide, is infinite under the circumstances. We could not, therefore, set up such a wave from a finite local source. If we cause an impressed voltage to act azially for a very short interval of time across any section of the guide, say in a reference plane, the result is an approximately spherical wave. (To be perfectly spherical, tiie wire should be infinitely ihie. The case is then that of a spherical wave-sheet with conical boundaries, already referred to, with the angle of the cone made infinitely small.) Its centre is at the origin of the wave, and as it expands, the por- tions of the wave-sheet nearest the wire become approximately parallel plane waves, one going to the right, the other to the left along the wire. But not being pure plane waves they are weakened as they progress, by the continuous expansion of the

400

BLBOTBOMAONBTIO THBORY.

OH. IT*

spherical wave of which they form a part. Nevertheless, we have the propagation of nearly plane waves of finite euerj^y, or of a perfectly plane wave-sheet of infinite energy, along a single guide.

Now, this takes place outside the conducting guide, and the question presents itself whether we cannot transmit an electro- magnetic wave along the interior of a tube, in a manner resem- bling a beam of light ? We can certainly do so if we have a second guide within the first tubular one, for this does not differ substantially from the case of two parallel wires, each out- side the other. But it does not seem possible to do without the inner conductor, for when it is taken away we have nothing left upon which the tubes of displaoement can terminate inter- nally, and along which they can run. A theoretical expedient is to carry the electrification forward at the proper speed. But we want the process to be automatic^ so to speak, hence oonveo- tion will not do. Again, if we make the displaoement start from one portion of an electrically conducting tube and terminate upon the rest^ we must insulate the two portions from one another, and then there will be a division of the charges between the interior and exterior, so that the result will be an external as well as an internal wave, or rather, one wave occu- pying both regions.

It would appear that the only way of completely solving the problem of the automatic transmission of plane waves within a single tube is a theoretical one, employing magnetic as well as electric conductance. To see this, imagine any kind of purely plane wave being transmitted in the normal manner through the ether, and fix attehtion upon a tube of the flux of energy, or a beam, using optical language. This beam cuts perpen- dicularly through the reference planes, in which the lines of electric and magnetic force lie, which again cross one another perpendicularly. The shape of the section of the beam by a reference plane may be arbitrary. But let it be quadrilateral, and so that it is bounded by magnetic lines on two opposite sides, and by electric lines on the other two. Now let the two sides of the tube upon which the displacement terminates perpen- dicularly, be electrically conductive thin sheets, and the other two sides, upon which the induction terminates perpendicularly, be magnetically oonductiye sheets. If the conduction be perfect,

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THSORT OF PIAKB SLBCTROKAONBTIO WAVES. 401