Electromagnetic Theory, Vol. 1 (1893) — part 26 of 31

difference, according to the way it is reckoned. We may, per- haps, most conveniently reckon it as the circuitation 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 circuitation 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 V + dV/dx. The axial sides give Ex and E2 say, if Ej is the tangential component in the direction of increasing x of the electric force at the boundary of the positive tube, and E2 the same on the negative tube, but reckoned the other way. The complete voltage is then

or + 1 + 2.

dx • It is also equal to - LC, as in § 101. (There is no magnetic

conductance of the dielectric now.) So

.... (16) dx

is the form assumed by the second circuital law.

cc2

388 ELECTROMAGNETIC THEORY. CR. IV.

Observe that we have made no specification of the nature of the leads, so that the equation 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 V 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 E: and E2 as functions of C. Fortunately this can be done, sometimes very simply, and at other 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 Ex is not only the boundary tangential electric force, but is also the axial electric force throughout the substance of the positive tube. Similarly as regards E2 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 force, 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 ^Ex, if ^ is the conductivity of the material; and the total current in the tube is KjEj, where Kj is the axial conductance per unit length, or, which is the same, Ej/R^ if Rx is the resistance per unit length. But this quantity is also the previously investigated quantity C, the circuitation of the magnetic force on the boundary of the tube. So we have the elementary relations

E^R^, E2 = R2C2, . . . (17)

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

(18) ax

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

-^ = RC + LC, ...... (19)

dx

THEORY OF PLANE 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 R 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 R is the resistance of the wires, per unit length axially. What we have to do, there- fore, in order to turn the real problem into one relating to strictly plane waves 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), (14), (15), still hold good, only RC2 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 Ex and the magnetic force, say H1, 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 RjC2.

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 application 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 concerning tubes and wires which do not admit of the compa- ratively simple treatment permissible when V and C are the variables, and the results are exceedingly curious and interest-

390 ELECTROMAGNETIC THEORY. CH. IV.

ing, though quite unlike those at present in question. But these are not practical 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 disturbance, 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 instan- taneous " within the meaning of the Act." It is obviously true for steady currents, when the inertial term disappears and B, is the steady resistance. And if the variations of current 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. Resistance 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 (16), 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

E^B^C, E2 = R"2C. . . . (20)

Is it possible to give a definite meaning to the symbol R," ? Is there a definite connection between C, regarded as a function of the time, and Ex or E2 also regarded in this way ?

THEORY OP PLANE ELECTROMAGNETIC WAVES. 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 force, 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 connection is a definite one. Moreover, if we make the applied force run through the same values increased in a certain ratio, 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 size) 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 is, 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 =/(#), or u is a function of #, we cannot say that C is a function of E. 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, <fec., differential •©efficients up to any order. That is, the symbol R", taken by -,self, is a function of the differentiator d/dt. To illustrate by simple example, suppose

R" = R + LP + ($p)-\ .... (21)

ere R, L, S, are constants, and p stands for the differentiator. 's means that

}0, .... (22)

or E = RC + LC + S/Ccft, .... (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 /Cdt. Consequently the march of

392 ELECTROMAGNETIC THEORY. CH. IV.

E is explicitly known. It is not obvious that when the march of E is given, that of C is known, by the same operator impli- citly. That is, by (22) we have

(24)

and given E as a function of the time, C is known as a function 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, especially by elementary writers) leads to a large amount of unnecessary work, tending to obscure the subject, without helping one on. It would not, perhaps, be going too far to say that such a misuse or inefficient application of analysis often makes rig- marole.

When we say that E = R"C, where IT is the resistance operator (so called because it reduces to the resistance in steady states), we assert a definite connection between E and C, so that when G is fully given as a function of the time, and the operations contained in R" are performed upon it, the func- tion E results, and similarly, when it is E that is given, then the inverse operator (R")"1 (or the conductance operator) act- ing upon it will produce C. There may be an infinite number of differentiators in R", as p, p2, p*, and so on, where pn mean? d^/df1. 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 function of p, depends upon the electrical and geometrical

THEORY OP PLANE ELECTROMAGNETIC WAVES. 393

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

. . . (25) ax

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

As already mentioned, when C is steady, R"j degenerates to Rx and R"2 to R2, 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 varies, 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,

...... (26)

and similarly for R"2. This J/^ is the value of the steady inductance of the wire, ^ being its inductivity. 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.

Simply 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 occurs 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 E and C in the equation E = R"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 frequency

394 ELECTROMAGNETIC THEORY. CH. IV.

is ?&/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 d2/dt2 = - n2. Put, therefore p2 = - n2 in the expressions for the resistance operators, and we reduce them to

B'j-B'j + L'ip, ll"2 = R'2 + L>, . . (27)

where R' and L' are functions of n\ They are, therefore, con- stants at a given frequency, and this is a very valuable property, as it is clear at once that we reduce the equation (25) to the simple form

. (28) ax

or, more briefly and clearly,

_^I = R'C + L'C, .... (29) dx

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 frequency, 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 is, 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 mere skin

THEORY OF PLANE ELECTROMAGNETIC WAVES. 395

penetration. Consequently, the quantity L', the effective in- ductance, goes from the full steady value L + L^ + 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 resistance, 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. The 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 frequency 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. Identity 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 conductive. Also, that by introducing magnetic conductivity into the

396 ELECTROMAGNETIC THEORY. CH. IV

medium we may reduce this distortion, and ultimately abolish it when the magnetic conductivity reaches a certain value. Now observe here that the correcting influence 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 boun- daries, in fact. Nevertheless, 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- ances axially, or parallel to the leads, we virtually assume that the correcting influence of the leads id transmitted laterally outwards instantaneously. We may, therefore, perceive that the wave-length is a matter of importance in determining the applicability of the practical theory. It is of no moment what- ever in the exact theory employing magnetic conductance ; 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 the 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 when 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

THEORY OF PLANE ELECTROMAGNETIC WAVES. 397

distortion should occur in the same way as under the influence of magnetic conductivity 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 thermometric 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 size 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 could 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 effect 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 so short as to come within the scope of the above reservatioDal remarks, or of others of a similar nature. Whether the generation of waves by an oscillator be considered, or their effect on a resonator, or their transmission along leads, their shortness in relation to the apparatus employed may some- times vitiate the results of approximate theories, and render caution necrssary. For example, the plane-wave theory indi-

398 ELECTROMAGNETIC THEORY. CH. IV.

cates that the attenuation factor for waves running along parallel leads is e-Ra;/2Lt> in the distance x, or c-1"/21' in the equivalent time of transit t. This is when R and L are con- 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 his 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 primary 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 distinc- tion between waves in free space and " in wires." This was thoroughly out of harmony with Maxwell's theory, 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 erroneous estimate was made of the permittance of the oscillator 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

THEORY OF PLANE ELECTROMAGNETIC WAVES. 399

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

The Guidance of Waves. Usually Two Guides. One suf- ficient, though with Loss. Possibility of Guidance within 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 measure, 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 second guide. But the energy of such a sheet, even though of finite depth, and containing electric and magnetic forces of finite intensity, would be infinite. The quantity L, the inductance per unit 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 axially 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, the wire should be infinitely fine. 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 ELECTROMAGNETIC THEORY. CH. IV.

spherical wave of which they form a part. Nevertheless, we have the propagation of nearly plane waves of finite energy, 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 displacement 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 convec- tion will not do. Again, if we make the displacement start from one portion gf 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 attention 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 conductive sheets. If the conduction be perfect,

THEORY OP PLANE ELECTROMAGNETIC WAVES. 401

we . shall not interfere with the transmission of the beam within the tube. That is, we have solved the problem stated. We may also notice that the external portion of the wave is not intrvfered with by the tube. But the interposition of the tube in the manner described renders the external and internal waves quite independent of one another, and either of them may be suppressed.

Similarly, if the interior region be made finitely conductive, electrically or magnetically, or both together, we can still in- vestigate the transmission of waves along it in the same way as previously described for complete plane waves in a homo- geneous medium made conductive, and the same applies to the external region, independently of the internal. Going further, we may do away with the diffused conductances, and concen- trate equivalent resistances in the plates bounding the tubes, in the same way as we replace magnetic conductance of the external medium by equivalent electric resistance of the wires in passing from the exact plane-wave theory to the practical theory of wires in terms of V and C. That is, the two plates on which the displacement ends may be made electrically resis- tive, the resistance taking the place of the magnetic conduct- ance in the interior ; whilst the other two plates should be made magnetically resistive, if the interior electric conductance is also to be abolished. We may then express the propagation of waves in the tube in a manner resembling the practical theory of wires, though it will no longer be an exact theory. Nor will the interior and exterior regions be quite independent of one another now that the tube is only finitely conductive.

Interpretation of Intermediate or Terminal Conditions in the Exact Theory.

§ 207. Leaving these somewhat abstruse considerations, re- turn to the practical theory concerning wires and its connection with the exact theory involving magnetic conductance. In the working out of the practical theory (which is not yet, how- ever, the theory of official representatives of practice, though they are decidedly getting on), we have often to consider the effects due to intermediate insertions of resistance in the circuit of the leads, or of shunts across them, and other modifications.

DD

402 ELECTROMAGNETIC THEORY. OH. IV.