Electromagnetic Theory, Vol. 2 (1899) — part 30 of 31

The above considerations show that if iron be introduced to increase the inductance it should not be in the main circuit, but external to it. There are, then, two principal ways sug- gested. First, to load the dieleotno itself with finely-divided iron, and plenty of it This is very attractive from the theoretical point of view, as it results in the production of a strongly magnetic insulator, which is practically homogeneous in bulk, being, therefore, a sort of non-conducting iron of low permeability. In this way L, as compared with the same with- out the iron, may be multiplied greatly. There is a partially counteractive influence, however, due to increased permittance. This plan is interesting, in view of the wave theory. It is lika multiplyhig the /* in. ether, and lowering the speed of pro- pagation, but at the same time allowing waves to keep up

BLIOrilOMAOKBnO VHBORT.

OH. nr.

*mgunBt the resiBtaiioe of wires tl^y may be nmniiig along. Bab aa regards the praoticability of this plaoi I say nothing, ezo^ that I have often soiiled at this itonio insulator (in more .than one sense ironic), and the idea of telephoning through a •eore resembling a poker with a copper wire ran through the middle.

The other way is to pat the divided iron outside the working dielectric. Then it had better not be divided into particles, but in the way well known to dynamo and transformer people, cut up so as to facilitate the flux of induction, whilst being -electrically non-conducting transversely thereto.

Now it may be said that submarine cables of the present type, having cores surrounded by a large quantity of iron, have large inductance already. But it does not follow. If the iron was in a solid tube, then undoubtedly L would be much magnified with steady currents. But it is not a solid tube, as it consists of spirally laid separate stout iron wires. The ni.iirnetic con- tinuity is interrupted, and the permeance is much reduced, as Prof. Hughes pointed out (equivalently) some years ago. But besides that, when the current is not steady, we do not utilise the iron even in the above imperfect manner. This is par- tioulaiiy the case with telephonic currents, when imperfeet penetration will still further reduce the effective inductance. So the increased L due to the sheath must be fu less than laige quantity of iron might suggest. Agam, there is the the inoreased resistance due to the sheath not being properly divided to prevent it. The case is therefore a considerably mixed one. Whilst we can say with confidence that with the very slow signalling which obtains through an Atlantic oaUe .the sheath cannot make much difference, it is not easy to decide whether its action is beneficial with telephonic currents to any great extent,. though it would be if there were no increased resistance. It is not even easy to say what is the L of a sub- marine cable under the circumstances. A least value is readily obtained, but the magnitude of the increase due to the iron and the outside return current is rather speculative. It may be well deserving of attention to change the type of the iron sheathing, so as to increase L and keep down R. Remember here, that although there is plenty of evidence of the beneficial (and very important) influence of this principle when it is

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carried out in a direct and natural manner, there is, so far as I am aware, no experimental evidence yet published of the equi> Talent success of indirect ways involving the use of iron.

Another indirect way is this. Instead of trying to get large uniformly spread inductance, try to get a large average induc- tance. Or, combine the two, and have large distributed inductance together with inductance in isolated lumps. This means the insertion of inductance ooils at intervals in the main circuit. That is to say, just as the effect of uniform leakage may be imitated by leakage concentrated at distinct points, so we should try to imitate the inertial effect of unifono inductance by concentrating the inductance at distinct points. The more points the better, of course. Say m coils in the length /, or 2m coils of the same total induetanoeb and therefore each of half the inductance; or nm ooils each of one the inductance of the first The electrical difficulty here is that inductance coils have resistance as well, and if this be too great the remedy is worse than the disease. Bat it would seem to be sufficient if the effect of the extra resistance b» of minor importance compared with the ciffect of itbe inoieaaed inductance. This means using coils of low resistanoe and the- largest possible time-oonstants. For suppose 4 ohms per hllom. is the natural resistance, and there be one 'coil per kilom. haying a resistance of 1 ohm. This will raise the average resistance to 5 ohms per kilom. ; and if the time-constant be big enough, the extra inductance may far more than nullify the resistance evil. The same reasoning applies to coils at greater intervals, only of course in a more imperfect manner. To get large inductance with small resistance, or, more generally, to make coils having large time-constants, requires the use of plenty of copper to get the conductance, and plenty of iron to get the inductance, employing a properly closed magnetic circuit properly divided to prevent extra resistance and cancellation of the increased inductance. This plan does not belong to the category of those mentioned before, which a moment's con- sideration showed to be worse than useless. It is a straight* forward way of increasing L largely without too much increase of resistance, and may be worth working out and development. But I should add that there is, so far, no direct evidence of the beneficial action of inductance bxou^t in in this way.

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The combination with leaks does not need any particular mention in detail after what has been already said relating to the distortionless circuit, which is the theoretical ideal, any approach to which is desirable if it be done without too much loss ; bearing in mind, too, that we should increase L first in preference to putting on leaks first.* I have confined myself so far entirely to my theory of 1886-87 and results thereof. Inductive shunta iavolve some other considerations, and a modified theory.

Effective Resistance and Inductance of a Combination when regarded as a Coil, and Effective Conductance and Per- mittance when regarded as a Condenser.

§ 219. In aoy elfictromagnetic combination the effects of electric and magnetic energy are antagonistic in some respects. Or we may say that magnetic inductances tend to neutralise eleotrio permittances. Or, more generally, that the effects of elastic compliance and of inertia tend to neutralise one another. Thus, when a coil is under the action of an impressed sin^dy periodic voltage, the current lags, owing to the self-induetioiL But if a condenser be introduced in sequence with the coil, the lag is diminished, and may be reversed or converted into a lead. The result on the ourxent is the same as if we substituted for the coil another of the same resistance and of reduced in- ductance, or even of negative inductance. Thus, the resistance operator of the coil being R + Lp, where B and L are its resistance and inductance, and p is the differentiator d/dt ; and that of the condenser being (S^)~^, where S is its permittance, when the two are in sequence the resistance operator of the combination is

• But remember the influence of the frequency in conjunction with the in<luctance. Leakage can be very beneficial when self-mduction does next to nothing, as before described.

THEORY OF PLANE ELKCTaOMAGNKTlC WAVES.

447

This shows that the effective inductance of the condenser, Trhich is - (S?i-)^, is additive with real inductance when in sequence therewith, and being negative, reduces the effective inductance of the combination from L to L - (Sn-)^.

But when the condenser and coil are in parallel, it is more eonTenient to use the oonductanoe operators. Thus

Y = (R + Lp)-i + S/> (3)

is the conductance operator of the coil and condenser in parallel, being the sum of the conductance operators of the coil and con- denser taken separately. And when p ni^ we convert Y to

from which we see that the combination is equivalent to a condenser of conductance R (R^ + L^n,^)-! and of permittance S - L (R-+ \aV)~^, The steady conductance being R"^, we see that the effective conductance is reduced by the self-induction. At the same time the effective permittance is reduced, and may become negative. Remember that in (3) and (4) the standard of comparison is a condenser, not a coil ; whereas in the former case (1), (2) the standard of comparison is a coil, not a con- denser. For further information regarding resistance and con- ductance operators see " Electrical Papers," Vol. II., p. 355, and elsewhere. The present remarks are merely definitive and introdactory to the theory of waves sent along a circuit, either naturally, or partly controlled by subsidiaiy arrangements. The comparison with a ooil is, I belieye^ more generally useful. Then we imagine any combination to be replaced by a simple ooil whose resistance and induotanoe are those (effectively) of the combination. Bat there are oases when the other way is preferable. Then we imagine the combination replaced by a oondenser whose condactanoe and permittance are those (effectively) of the combination. And in the following it will be convenient to nse both ways in the same investigation.

Indnctlve Leaks applied to Submarine Gables.

§220. Now imagine the permittance of a submarine cable to be concentrated in luuipb at a number of points. Let S be the

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permittanoe of one of the equivalent oondensen. If it be ■hunted by a ooil of euch R and L that S-L (R*+L^^)-^ we know by the above that the atate of the cable, when simplj periodic will be the same as if the condenser and coil were removed and replaced by a simple leak of conductance R (R2 + L^n^)-!. This reasoning applies to every condenser, if it have its appropriate coil, or to any selected group we please. But if the frequency be changed the permittance will come into sight again, of the real or positive kind, or of the fictitious negative kind, as the case may be. We have, therefore, the power of reducing the eflfective permittance of a cable under simply periodic forces, by means of numerous auxiliary induc- tive shunts or leaks, and of practically cancelling it at a par- ticular frequency. How will this work out in the transmission of telephonic currents through the cable ?

From the purely theoretical point of view one may be some- what prejudiced against the system at first. For, obviously, it is not a distortionless arrangement. That is got by balancing the self-induction of the main circuit against the lateral permit- tanoe, which can, subject to practical limitations, be done per- fectly. That is, there is a balance at any frequency, or the idea of periodic frequency does not enter at all. I doubt whether there can be any other distortionless circuit than that which (in the absence of another) I always refer to as ^ dis- tortionless circuit. With inductive shunts we produce a par- tial neutralisation of the permittance, the amount of whioh may vaiy very widely during the transmission of telephonlo currents. Remember, too, that during the change from one frequency to another there are other phenomena than those ooncemed during the maintenance of a simply periodic state of variation.

Nevertheless, we should be careful not to be prejudiced against an imperfect plan merely because it is so manifestly imperfect. It may perhaps be more easily applied than a theoretically perfect system, and may possess practical advantages of import- ance. Now this is a matter for practical experiment and expe- rience. But it is just here that there is an almost complete dearth of information. For although Prof. Silvanus Thompson* has mentioned that he has made many experiments, yet he has

• " Ooean Telephony," Chicago CongrsM, 1893.

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only described one case in whioh an induotive shunt was found good. It was an artificial cable of 7,000 ohms and 10 micro- farads. Presuming that there was a fairly good spreading of the permittance, we see that the permittance is that of about 40 kilometres of a submarine cable, whilst the resistance is more like that of 1,000 or 2,000 kilometres. The inductance would probably be very small. We see that the example does not well represent a submarine cable. We should rather com- pare it with a veiy long overland circuit of very much smaller permittance per kilometre than the cable. Only, such a circuit, if properly put up, would have a large inductance, so that there is a failure here. However, it is mentioned that one shunt of 312 ohms and a time-constant of 0 005 sec. rendered tele- phonic transmission possible except for shrill sounds. I cannot help thinking that Prof. Thompson has been much too reticent on the experimental side. He will perhaps furnish us with fuller information later on. At present there is little to judge by.

Qeneral Theory of Transmission of Waves along a Oircuit with or without Auxiliary Devices.

§ 221. In the meantime the following theory and formulsB may perhaps be useful to those who may desire to go into the matter. I put the theory into such a form that it may be applied to various other cases of auxiliary arrangements. Let y and 0 be the transverse voltage and the current, at distance

at time i. Let their connections be

where Z is a resistance operator, and Y a conductance operator, as described in § 219, only now belonging to unit length of oirouit. In the simplest case Z reduces to a resistance, and Y to a conductance, per unit length. But in general Z and Y may have various fonns, in unlimited number, being then functions of electrical constants and of the time difierentiator p. When there are no auxiliary devices, and the natural resist- ance B» inductance leakance K, and permittance S (all per unit length) are constants, then we have

Z = K + Lp, Y = K + Si?, ... (6)

450

KLECmuMAGNETlC TUEORT.

OH. IV.

iis before, § 201 — 4. But whatever the forms of Y and Z may be, in finite terms or transcendental, we may reduce them to the simple standard forms (6) when the simply periodic state of vibration is maintained, by employing the transformation p = nij or p^= -n~. We see from this that we can practically examine cases of simple periodicity which might at first seem beyond all bounds of practicability. Note, however, that "forces" and "fluxes" should be in constant ratios, so that such phenomena as hysteresis, magnetic or electric, are ex- cluded.

From equations (5) we see that the characteristic equation of Vis

^=.YZV = j2V,8ay, .... (7)

provided T and Z are independent of x. .This restriction is not a necessary one, bat, of course, it makes an important practical simplification to have uniformity along the circuit. In (7) ^ represents the operator YZ, and in passing we may notice an interesting matter connected with partial differentiid equations of the type (7), or of the more general type

V^V-jsy, (8)

appropriate to three dimensions in space. This type of equa- tion occurs in all sorts of physical problems, V being some variable which is propagated through a medium in a manner depending upon the nature of the operator if^ which involves the time-differentiator. It is usually a very simple rational function of ^, such, for example, as to bring in only the first and second derivatives of V with respect to the time, and, so far as I know, no physical problems have presented themselves which involve an irrational partial differential equation for characteristic, as, for example,

V2V«(a + 6p+<3i«)»V. .... (9)

It might, indeed, seem at first sight that a characteristic of this form was physically impossible, being meaningless. Nevertheless, it is quite easy to see by the way equation (7) was constructed out of the components (5), that real physical problems may involve irrational characteristic equations, such as are exemplified by (9). For if either Y or Z be irrational

THSOBY OF FLANB SLBOTROMAGMBTIC WAVES. 4dl

80 is YZ. And we can easily (in imagination, not so easily in execution) choose auxiliary devices making Y or Z irrational. Similar remarks apply to (8). It results from the union of two distinct equationB involving two yariables, one of which is then eliminated to make the oharacteristio. If either of the component equations involves an irrational operator, the cha- raoteristio resultant will be irrational. It must not be imagined that the solutions of suoh equations obtained by physical reasonuig are impossible or beyond the range of mathematicsi or that the results are physically meaningless. Some partial characteristics of the form (8), with irrational right members, I have examined and solved. I should imagine that there will probably be a field for equations of the kind in the future study of the complicated influence of matter upon phenomena which occur in their simplest manner in the ether away from matter, as in the theory of dispersion of light, for example, of electric absorption, and in similar subjects.

Returning to equation (7), the general solution inyolTCS two arbitrary functions of the time, as in the form

• V-po»hjx.Vo + !ll^i^ZCo. . . . . (10)

But it is preferable to take the case of disturbances propagated from a source into an infinitely long cable, in order to eliminate complications due to reflections and terminal apparatus. Then

V-€-^Vo (11)

is the solution representing Y n,t t due to Yq given as a func- tion of the time at a;-iO, the origin. And the current is

given by

C-|V-(I)'.--'V, .... (12)

by using the first of (5).

To see the final state due to the continued action of constant Vq, give q, Y, and Z in (11), (12), the values they assume when p is put"-0 in their general expressions, viz., for Y the steady leakance and for Z the steady resistance per unit length, and for q their, geometric mean. To find how this state is arriyed at, and, more generally, to interpret (11), (12) when Vq is any given function of the time^ demands the conversion of these

og2

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■LKJIBOMAOinmO THBOBT.

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equations to proper algebraical form by ezeention of the analy- tical operations implied in Y, and Z. This we are not ooor

cerued with here, except as regards the case arising when ia a simply periodic function. Then put j? = 7ii, and reduce Z to R + Lp, and Y to K + Sp, as before stated, equations (6), making (11), (12) become

V-«-«'V, (13)

Hmy^'' • • • • ^^*>

whete 9 = (R + Lp)»(K + Sp)» (15)

These expressions have next to be reduced to the form (A + Bp)Vo, which is a matter of common algebra. Wben done we shall find that if

P or Q - (i)* [(U- + LW)* (K2 + SV)» ± (IlK - LS»2)]», , (16) then the Y solution is

y^er^Bin{nt'Qx), (17)

due to Yq = € sin iit at the origin. From this, by the use of the first of (6), giving

dV

dx _ R-Jjp dV

(18)

li + Lp li- + L'n^ dx'

we may derive the C solution. Or we may derive it by deve- loping (14). The result is

(KP + SnQ)8in + (S»P-KQ)cos, (19)

where the value of P'+Q' is given by

P« + Q««:(R8 + W)i(K2 + S«»«)». . . (20)

It may be only necessary to consider the amplitudes of V and C, and, of course, C is most important. That of V is obviously e c"^* by (17). That of C may be got from (19), and is expressed by

<f^-' Qim--- ■ ■ ■

The wave speed is n/Q, the wave length 27r/Q, and the periodioity n/2ir. Another convenient form of (21) ia

where »«(LS)^.

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Application of above Theory to Inductive Leakance.

§ 222. The above simply periodic solution is discussed h) my " Electrical Papers," Vol. II. (especially p. 396 and p. 339), when R, L are the natural effective resistance and inductance, and K, S the leakanoe and permittanoe per unit length, the latter pair being supposed to be the same at all frequenoiet. (A paper by Prof. Perry, Phil. Mag., August, 1893, may also be consulted, but I do not think he is right in some of his oenclusions.) Bat the same formnlee are applicable when •oxiliary devices are employed, if they are numerous enough, by a process analogous to the obvious one of representing laige numbers of separate leaks by uniform leakance.

Thus, in the case of inductive shunts, if we treat them as a set of separately located shunts, the full theory is veiy com- plicated, because it requires a separate formula for every section of the line. The best course to take is to distribute the inductive leakance uniformly, in imagination, of course. The result will represent the fullest possible carrying out of the principle concerned, and more. Thus, let p be the resistance, and A the inductance of the leak belonging to the unit length of cable. The reciprocal of p is of course the steady leakance per unit length. But it does not operate fully "when the state is changing, owing to A. We shall now have

Y--ir-+crp, (23)

where {p-^XpY^ is the leakance operator, taking the place of fr\ whilst <r is the steady permittanoe per unit length. We have to use this special T in the above formulsa to represent the effect of inductive leakage. In the simply periodic case Y develops to

The first term is the effective leakance, and the coefficient of |> is the efiective permittance per unit length.

In the developed formulc6 (16), (17), (19) to (22), therefore, we must give K and S the values

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ELECTROilAUNETlC lUliORY,

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to suit the case of inductive leakage, whilst R and L remain as before, the effective resistance and inductance per unit length of line. The solutions should be examined through a wide range of frequency in order to arrive at a good know- ledge of how the circuit would behave to telephonic currents, with different values of A, and p. It is, of course, necessary that p should not be so small as to produoe too great a lost in transit.

Notice that A and p are immensely big when the unit length is a centimetre. Remember that m equal ooils in parallel behave like one coil of one the resistance and induotanot of the individual coils. Thus, when ten ■hunts at the rate of one per kilometre take the plaoe of one shunt per ten kilo- metres, each of the ten should have ten times the resiatanoe and inductance of the one. This lesds to physically monstrous results when we do as was done above, and make the action uniform, as it implies infinitely large inductance of an infinitely small section of the leakance. But that need not cause any alarm when we are employing the ideal case for calculating purposes. What is more important is that a stop would soon be set to the multiplication of the shunts.

At the particular frequency making the effective permittance vanish, the solution may be obtained simply from the funda- mental formulso, since T reduces to a constant, viz., the value of K in (25), S being zero; and only the rationalisation of Z has to be attended to. If, farther, we assume that L = 0, then Z is also constant, viz , R, and the solutions are simply

V-€-«^^»*Vo, (26)

These represent a very remarkable state of affairs throughout the cable. But it is thoroughly deceptive, because even if the inductance of the line were quite unimportant under normal conditions, the effective cancellation of the permittance would make it important under the present circumstances ; that is, L cannot be neglected. Including it^ when S^O, the value of P reduces to

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GRAVITATIONAL ANALOGY.

and the current amplitudei according to (21), becomes

^^^"(p«+A%«)*(R« + L«n«)»' • • • ^^^^

in which, of course, p and An are not independent, having a relation fixed between them by the vanishing of S in (25). By giving a suitable value to o-, for instance, \ microfarad per kilometre (which is a suitable unit of length to employ), and also fixing the frequency, we may readily apply the formulae to estimate the attenuation with diJBSarent values of the insulation mistanoe or its equivalent.

APPE>iDlX B.

A GRAVITATIONAL AND ELECTROMAGNETIC

ANALOGY.

Part L

To form any notion at all of the flux of gravitational energy, we must first localise the energy. In this respect it resembles the legendary hare in the cookery book. Whether the notion will turn out to be a useful one is a matter for subsequent discovery. For this, also, there is a well-known gastronomical analogy.

Now, bearing in mind the successful manner in which Max- well's localisation of electric and magnetic energy in his ether lends itself to theoretical reasoning, the suggestion is very natural that we should attempt to localise gravitational energy in a similar manner, its density to depend upon the square of the intensity of the force, especially because the law of the inverse squares is involved throughout.

Certain portions of spaoe are supposed to be occupied bj matter, and its amount is supposed to be invariable. Furthei^ more, it is assumed to have personal identity, so that the posi- tion and m tion of a definite partiole of matter are definite, at any rate relative to an assumed fixed spaoe. Matter Is reoog-

456

EI.EtrmOMAGNBTiC THBORT.

APF. &

nised by the property of inertia, whereby it tends to persist in the state of motion it possesses ; and any change in the motion is ascribed to the action of force, of which the proper measure is, therefore, the rate of change of quantity of motion, ox momentum.

Let p be the density of matter, and e the intensity of force, or the force per unit matter, then

r-ofi (1)

expresses the moving force on /j, which has itss equivalent in increase of thu momentum. Tiiere are so many forces nowa- days of a generalised nature, that perhaps the expression "moving force " may be permitted for distinctness, although it may have been formerly abused and afterwards tabooed.

Now the force F, or the intensity e, may have many origins, but the only one we are concerned with here is the gravitational force. This appears to depend solely upon the distribution of the matter, independently of other circumstances, and its opera- tion is concisely expressed by Newton's law, that there is a mutual attraction between any two particles of matter, which varies as the product of their masses and inversely as the square of their distance. Let e now be the intensity of gravi- tational force, and F the resultant moving force, due to all the matter. Then e s the space-variation of a potential, say,

e«VP, (2)

and the potential is found from the distribution of matter by*

P«pot^-2_^, c iircr

where c is a oonstant. This implies that the speed of propa- gation of the gravitative influence is infinitely great, f

Now when matter is allowed to fall together from any con- figuration to a closer one, the work done by the gravitational forcive is expressed by the increase made in the quantity ]^/^Pp. This is identically the same as the quantity ^Jce* summed through all space. If, lor example, the matter be given

• &«§133.

t The density is expressed in terms of the intensity of focoe Iff

prrcotivce. This is also the case in the extended ani^ogy later, when the Unes of e are slightly shifted.

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initiallj in a state of infinitely fine division, infinitely widely Beparated, then the Work done by the gravitational forcive in passing to any other configuration is ^^Pp or Di^ce^, which therefore ezpresseB the ''exhaustion of potential energy."* We may therefore assume that ^ce* expresses the exhaustion of potential energy per unit volume of the medium. The equiva- lent of the exhaustion of potential energy is, of course, the gain of kinetic energy, if no other forces have been in action.

We can now express the flux of energy. We may compare the present problem with that of the motion of electrification. If moved about slowly in a dielectric, the electric force ia . appreciably the static distribution. Nevertheless, the flux of energy depends upon the magnetic force as well. It may, indeed, be represented in another way, without introducing the magnetic force, but then the foriuula would not be sufficiently comprehensive to suit other cases. Now what is there au'^lj- gous to magnetic force in the gravitational case? And if it have its analogue, what is there to correspond with electric current? At first glance it might seem that the whole of the magnetic side of electroniagnetism was absent in the gravita- tional analogy. But tliis is not true.

Thus, if u is the velocity of f}, then pu is the density of a current (or flux) of matter. It is analogous to a convective current of electrification. Also, when the matter p enters any r^on through its boundary, there is a simultaneous conver- gence of gravitational force into that region proportional to p. This is expressed by saying that iff

then 0 is a circuital flux. It is the analogue of Maxwell's true current j for although Maxwell did not include the convective term />u, yet it would be against his principles to ignore it. Being a circuital flux, it is the curl of a vector, say

This defines h except as regards its divergence, which is arbi- trary, and may be made Ecro. Then h is the analogue of mag-

• " Thomson and Tait," Vol. I., Part II., § 549.

t Obiierve that it id - e tliat ia the analogue of electric force, aiid — o6 of displacement current.

0 = pU- ce,

(4)

curl h = /au - ce

(5)

458 XLBOTROMAGNBnO IHIOBT. APP. B>

netic force, for it bean the same relation to flux of matter a» magnetic force does to convective current. We have**^

li»curlpotO, (6)

» curl A,

if A •■pot 0. But, since instantaneous action is here inTolved, we may equally well take

Aapotpu, (7)

and its curl will be h. Thus, whilst ^e ordinaiy potential P is the potential of the matter, the new potent A is that oT its flux.

Now if we multiply (5) by e, we obtain

e curl lL»e/)a-ee^ (8)

or, which is the 8ame,f

couv Veh^Pu-U, (9)

if U = M-. But U represents the rate of eshaustion of potential

energy, so - U represents its rate of increase, whilst Fa repre- sents the activity of the force on p, increasing its kinetic

energy. Consequently, the vector Yeh expresses the flux of gravitational energy. More strictly, any circuital flux what- ever may be added. This Veh is analogous to the electromag- netic VEH found by Poynting and myself. | But there is a reversal of direction. Thus, comparing a single moving particle of matter with a similarly-moving electric charge, describe a sjihere round each. Let the direction of motion be the axis, the positive pole being at the forward end. Then in the electrical case the magnetic force follows the lines of latitude with positive rotation about the axis, and the flux of energy coincides with the lines of longitude from the negative pole to the positive. But in the gravitational oase^ although h still follows the lines of latitude positively, yet since the radial a is directed to instead of from the centre, the flux of energy m- along the lines of longitude from the positive pole to the

• See §194, page 20a

t By the tiwiMforixuUon (178), § 132.

t Ste §70, equation (13). Tike e»=0=li«.

GRAVITATIONAL ANALOGY.

459

negative. This reversal arises from all matter being alike and attractive, whereas like electrifications repel one another.

The electromagnetic analogy may be pushed further. It is as incredible now as it was in Newton's time* that gravitative influence can be exerted without a medium ; and, granting a medium, we may as well consider that it propagates in time, although immensely fast. Suppose, then, instead of inatanta-

neons action, which involvesf

ourle«0 (10)

we assert that the gravitational force e in ether is propagated at a single finite speed v. This requires that

v«V«e«=8, (11)

for this^ is the general characceritttic of uudis&ipated propagation at finite speed. jNow,||

« V div - curl*, 80 in space free from matter we have

-«*ourl«o-». (12)

But we also have, by (5),

-curlh^ce, • • • • • (13)

away from matter. This gives a second value to 8, when we differentiate (13) to the time, say

8- -Icurlh. (14)

c

»

So, by (12) and (14), and remembering that we have already chosen h circuital, we derive

ev'curle-h. ...... (15)

• To Newton himself, as shown in his often -quoted letter to Bentley.

t This equation is equivalent to (2) above, and it Implies that the gravitational foroe is eteootfy dependent on the configuration of the matter.

X As shown by Poisson, the value of a quantity at a given place and

time, when controlled by this equation, depends solely on the state of things at distance vt and time t earlier. That is, disturbances travel at speed V. But it is much simpler to uuderstaud this property through plane waves.

460

ELECTROMAGNETIC THEORY.

APP. B.

Or, if /A is a new constant, such that

ftev*oil, (16) then (15) may be written in the form

curle=/A]L ...... (17)

To sum up, the first circuital law (5), or

curlh = pu-ce, .... (5) bis,

leads to a second one, namely (17), if we introduce the hypo- thesis of propagation at finite speed.* This, of course, might be inferred from the electromagnetic case.

In order that the speed v should be not less than any Talua that may be settled upon as the least possible, we hare merely to make /« be of the necessary smallness. The equation of activity becomes, instead of (9),t

conv Veh - Fu - U - f , . . . . (18)