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 magnetic 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- ticularly the case with telephonic currents, when imperfect penetration will still further reduce the effective inductance. So the increased L due to the sheath must be far less than large quantity of iron might suggest. Again, there is the the increased 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 cable 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

THEORY OF PLANE ELECTROMAGNETIC WAVES. 445

carried out in a direct and natural manner, there is, so far as I am aware, no experimental evidence yet published of the equi- valent 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 coils 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 uniform 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 inductance, and therefore each of half the inductance ; or mn coils each of one 7ith 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. But it would seem to be sufficient if the effect of the extra resistance be of minor importance compared with the effect of the increased inductance. This means using coils of low resistance and the largest possible time-constants. For suppose 4 ohms per kilom. is the natural resistance, and there be one coil per kilom. having 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 brought in in this way.

446 ELECTROMAGNETIC THEORY. CH. IV.

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 shunts involve 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 any electromagnetic combination the effects of electric and magnetic energy are antagonistic in some respects. Or we may say that magnetic inductances tend to neutralise electric 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 simply periodic voltage, the current lags, owing to the self-induction. 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 current 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 + L^, where R and L are its resistance and inductance, and p is the differentiator djdt ; and that of the condenser being (Sp)"1, where S is its permittance, when the two are in sequence the resistance operator of the combination is

Z = R + Lp + (Sp)-i, (1)

which, in a simply periodic state of frequency n/'2irt making ni, becomes

R + /L-AV . .

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

THEORY OP PLANE ELECTROMAGNETIC WAVES. 447

This shows that the effective inductance of the condenser, which is - (S/&2)"1, is additive with real inductance when in sequence therewith, and being negative, reduces the effective inductance of the combination from L to L - (S^2)"1.

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

Y = (R + L^)-1 + Sp ..... (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

R

Y-

from which we see that the combination is equivalent to a condenser of conductance R (R2 + L2^2)"1 and of permittance S - L (R2 + L2^2)"1. The steady conductance being R"1, 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 sec " Electrical Papers," Vol. II., p. 355, and elsewhere. The present remarks are merely definitive and introductory to the theory of waves sent along a circuit, either naturally, or partly controlled by subsidiary arrangements. The comparison with a coil is, I believe, more generally useful. Then we imagine any combination to be replaced by a simple coil whose resistance and inductance are those (effectively) of the combination. But there are cases when the other way is preferable. Then we imagine the combination replaced by a condenser whose conductance and permittance are those (effectively) of the combination. And in the following it will be convenient to use both ways in the same investigation.

Inductive Leaks applied to Submarine Gables. § 220. Now imagine the permittance of a submarine cable to be concentrated in lumps at a number of points. Let S be the

448 ELECTROMAGNETIC THEORY. CH. IV.

permittance of one of the equivalent condensers. If it be shunted by a coil of such R and L that S = L (R2 + lAi2)"1, we know by the above that the state of the cable, when simply periodic, will be the same as if the condenser and coil were removed and replaced by a simple leak of conductance R (R2 + L2?!2)""1. 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 effective 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- tance, 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 the dis- tortionless circuit. With inductive shunts we produce a par- tial neutralisation of the permittance, the amount of which may vary very widely during the transmission of telephonic currents. Remember, too, that during the change from one frequency to another there are other phenomena than those concerned 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 ap d than a theoretically perfect system, and may possess practu d 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 Pnff, Silvanus Thompson*

has mentioned that he has made manv"experiments, yet he has

fl>

• " Ocean Telephony," Chicago Congress, 1893.

THEORY OF PLANE ELECTROMAGNETIC WAVES. 449

only described one case in which an inductive 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 very 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.

General Theory of Transmission of Waves along a Circuit with or without Auxiliary Devices.

§ 221. In the meantime the following theory and formulae 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 V and C be the transverse voltage and the current, at distance

at time t. 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 circuit. In the simplest p*-p Z reduces to a resistance, and Y to a conductance, per up length. But in general Z and Y may have various forms, in unlimited number, being then functions of electrical constants and of the time differentiator p. When there are no ai xiliary devices, and the natural resist- ance R, inductance L, leakance K, and permittance S (all per unit length) are constants, t .en we have

.. (6)

GO

450 ELECTROMAGNETIC THEORY. CH. IV.

as 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 = ni, or p2 = - n2. 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 = ^V,say, .... (7) dx2

provided Y and Z are independent of x. This restriction is not a necessary one, but, of course, it makes an important practical simplification to have uniformity along the circuit. In (7) q2 represents the operator YZ, and in passing we may notice an interesting matter connected with partial differential equations of the type (7), or of the more general type

V2V = 22V, (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 q2, which involves the time-differentiator. It is usually a very simple rational function of p, 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 + cp2)»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,

THEORY OF PLANE ELECTROMAGNETIC WAVES. 451

so 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 equations involving two variables, one of which is then eliminated to make the characteristic. If either of the component equations involves an irrational operator, the cha- racteristic resultant will be irrational. It must not be imagined that the solutions of such equations obtained by physical reasoning are impossible or beyond the range of mathematics, 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 involves two arbitrary functions of the time, as in the form

V = cosh<^.V0 + !^^ZC0 ..... (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 = €-«*V0 ...... (11)

is the solution representing V at x, t due to V0 given as a func- tion of the time at x = Q, the origin. And the current is given by

by using the first of (5).

To see the final state due to the continued action of constant V0, 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 arrived at, and, more generally, to interpret (11), (12) when V0 is any given function of the time, demands the conversion of these

GG2

452 ELECTROMAGNETIC THEORY. CH. IV.

equations to proper algebraical form by execution of the analy- tical operations implied in q, Y, and Z. This we are not con- cerned with here, except as regards the case arising when V0 is a simply periodic function. Then put p = ni, and reduce Z to R + L#>, and Y to K + Sp, as before stated, equations (6), making (11), (12) become

........ (13)

where 2 = (R + I#)»(K + Sp)» ..... (15)

These expressions have next to be reduced to the form (A + Bp)V0, which is a matter of common algebra. When done we shall find that if

P or Q = (J)» [(R2 + L2/2) (K2 + S2^2)* ± (RK - LSrc2)]*, . (16) then the V solution is

V = <J€-psm(7i-Qa>), ..... (17) due to V0 = e sin nt at the origin. From this, by the use of the first of (5), giving

R-L dV

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

C = j£^8 [(KP + SnQ) sin + (SnP - KQ) coB](n« - QOJ), (19) where the value of P2 + Q2 is given by

. . (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 e"Pa!, by (17). That of C may be got from (19), and is expressed by

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

(22)

v

where v = (LS)-*.

THEORY OF PLANE ELECTROMAGNETIC WAVES. 453

Application of above Theory to Inductive Leakance.

§ 222. The above simply periodic solution is discussed in 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 leakance and permittance per unit length, the latter pair being supposed to be the same at all frequencies. (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 conclusions.) But the same formulae are applicable when auxiliary devices are employed, if they are numerous enough, by a process analogous to the obvious one of representing large 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 very 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 /> 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

(23)

where (/> + A-p)"1 is the leakance operator, taking the place of p-1, whilst o- is the steady permittance per unit length. We have to use this special Y in the above formulae 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 p is the effective permittance per unit length.

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

• <25>

454 ELECTROMAGNETIC THEORY. OH. IV.

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 produce too great a loss in transit.

Notice that A and p are immensely big when the unit length is a centimetre. Remember that m equal coils in parallel behave like one coil of one with the resistance and inductance of the individual coils. Thus, when ten shunts at the rate of one per kilometre take the place of one shunt per ten kilo- metres, each of the ten should have ten times the resistance and inductance of the one. This leads 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 formulae, since Y 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, further, we assume that L = 0, then Z is also constant, viz., R, and the solutions are simply

(26)

V. .... (27)

(I)'

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 = 0, the value of P reduces to

)l}J> • (28)

GRAVITATIONAL ANALOGY. 455

and the current amplitude, according to (21), becomes

/ri\ _ P €~ *e

in which, of course, p and Xn are not independent, having a relation fixed between them by the vanishing of S in (25). By giving a suitable value to <r, for instance, J 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 different values of the insulation resistance or its equivalent.

APPENDIX B.

A GRAVITATIONAL AND ELECTROMAGNETIC

ANALOGY.

Part I.

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 space are supposed to be occupied by matter, and its amount is supposed to be invariable. Further- more, it is assumed to have personal identity, so that the posi- tion and motion of a definite particle of matter are definite, at any rate relative to an assumed fixed space. Matter is recog-

456 ELECTROMAGNETIC THEORY. APP. B.

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, or momentum.

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

r=e/> ....... (i)

expresses the moving force on p, which has its equivalent in increase of the momentum. There 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*

where c is a constant. 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 2JP/9. This is identically the same as the quantity 2Jce2 summed through all space. If, for example, the matter be given

• See § 133.

t The density is expressed in terms of the intensity of force by p = convce. This is also the case in the extended analogy later, when the lines of e are slightly shifted.

GRAVITATIONAL ANALOGY. 457

initially in a state of infinitely fine division, infinitely widely separated, then the work done by the gravitational forcive in passing to any other configuration is 2JP/> or 2Jce2, which therefore expresses the " exhaustion of potential energy."* We may therefore assume that Jce2 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 is 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 formula would not be sufficiently comprehensive to suit other cases. Now what is there analj- 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 electromagnetism was absent in the gravita- tional analogy. But this is not true.

Thus, if u is the velocity of p, then /ou is the density of a current (or flux) of matter. It is analogous to a convective current of electrification. Also, when the matter /> enters any region 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

C = /3U-ce, . (4)

then 0 is a circuital flux. It is the analogue of Maxwell's true current; 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

curl h = /ou - ce (5)

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

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

t Observe that it is -e that is the analogue of electric force, and -ce of displacement current.

458 ELECTROMAGNETIC THEORY. APP. B-

netic force, for it bears the same relation to flux of matter as magnetic force does to convective current. We have*

h = curl pot 0, (6)

= curl A,

if A = pot 0. But, since instantaneous action is here involved, we may equally well take

A = potpu, (7)

and its curl will be h. Thus, whilst the ordinary potential P is the potential of the matter, the new potential A is that of its flux.

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

e curl h = e/ou - ece, (8)

or, which is the same,f

conv Veh = Fu-U, (9)

•

if U = |ce2. But U represents the rate of exhaustion of potential energy, so - U represents its rate of increase, whilst Fu repre- sents the activity of the force on p, increasing its kinetic energy. Consequently, the vector Ven expresses the flux of gravitational energy. More strictly, any circuital flux what- ever may be added. This Ven is analogous to the electromag- netic VEH found by Poynting and myself, f But there is a reversal of direction. Thus, comparing a single moving particle of matter with a similarly-moving electric charge, describe a sphere 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 case, although h still follows the lines of latitude positively, yet since the radial e is directed to instead of from the centre, the flux of energy is along the lines of longitude from the positive pole to the

• Sec§ 134, page 208.

t By the transformation (178), § 132.

t See § 70, equation (13). Take e0 = 0 = h0.

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 instanta- neous action, which involves!

curle = 0, ...... (10)

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

v2V2e = e*, ....... (11)

for thisj is the general characteristic of undissipated propagation at finite speed. Now,||

V2 = V div - curl2, so in space free from matter we have

-v2curPe = e ...... (12)

But we also have, by (5),

-curlli = ce, ..... (13)

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

g= --curlh ..... . (14)

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

(15)

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

t This equation is equivalent to (2) above, and it implies that the gravitational force is exactly dependent on the configuration of the matter.

J 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 understand this property through plane waves.

See (193), § 132.

460 ELECTROMAGNETIC THEORY. APP. B.

Or, if p is a new constant, such that

,W = 1, ...... (16)

then (15) may be written in the form

curle = /*h (17)

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

curlh = /ou-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 value that may be settled upon as the least possible, we have merely to make /x be of the necessary smallness. The equation of activity becomes, instead of (9),f

convVeh-Fu-U-f, .... (18)

if T = J/xh2. The negative sign before the time-increase of this quantity points to exhaustion of energy, as before. If so, we should still represent the flux of energy by Veh. But, of course, T is an almost vanishing quantity when ft is small enough, or v big enough. Note that h is not a negligible quantity, though the product /xh is. Thus results will be sensibly as in the common theory of instantaneous action, although expressed in terms of wave-propagation. Results showing signs of wave-propagation would require an inordi- nately large velocity of matter through the ether. It may be worth while to point out that the lines of gravitational force connected with a particle of matter will no longer converge to it uniformly from all directions when the velocity v is finite, but will show a tendency to lateral concentration, though only to a sensible extent when the velocity of the matter is not an in- sensible fraction of v.

The gravitational-electromagnetic analogy may be further extended if we allow that the ether which supports and pro- pagates the gravitational influence can have a translationalj

• These equations are analogous to (4) and (5), § 35. t This is analogous to (12), § 70, with the impressed forces made zero. J This does not exclude rotational motion, which is, in fact, a differential effect in a special kind of translational motion.

GRAVITATIONAL ANALOGY. 461

motion of its own, thus carrying about and distorting the lines of force. Making allowance for this convection of e by the medium, with the concomitant convection of h, requires us to turn the circuital laws (17), (18) to

curl(e + /AVah) = /ih, . ... (19) curl (h + cVea) = /)U-ce, . . . (20)

where q. is the velocity of the medium itself.*