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

if T«B^/ilL^. The negative sign before the time-increase of this quantity points to exhaustion of energy, aa before. If so, we should still represent the flux of energy by Veh. But, of course, T is an almost vanishing quantity when fi is small enough, or v big enough. Note that h is not a negligible quantity, though the product /uh 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 fbroe 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, thoimh only to a seiisiblu 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 traiisl;itional +

• These equations are analogous to (4) and (5), § 36. t This is analogous to (12), § 70, with the impressed forces made nro. X This does not exclude rotational motion, which is, in hct, a difierentisl e£bct in a special kind of translational motion.

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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, recjuires us. to turn the circuital laws (17), (18) to

where q is the yelooity of the medium itself.^

It is needless to go into detail, because the matter may be regarded as a special and simplified case of my investigation of the forces in the electromagnetic field, with changed meanings of the symbols. It is sufiScient to point out that the stress in the field now becomes prominent as a working agent. It is of two sorts, one depending upon e and the other upon h, analo- gous to the electric and magnetic stresses. The one depending upon h is, of course, insignificant. The other consists of a pressure parallel to e combined with a lateral tension all round it, both of magnitude ^ee*. This was equtvalently suggested by Maxwell. Thus two bodies which appear to attract are pushed together. The case of two large parallel material planes exhibits this iu a marked manner, for e is very buiall between them, and relatively large on their furtlier sides.

But the above analogy, though interesting in its way, and serving to emphasise the non-necessity of the assumption of instantaneous or direct action of matter upon matter, does not enlighten us in the least about the ultimate nature of gravi- tational energy. It serves, in fact, to furtlier illustrate the mystery. For it must be confessed that the exhaustion of potential energy from a universal medium is a very unintelligi- ble and mysterious matter. When matter is infinitely widely separated, and the forces are least, the potential energy is at its greatest, and when the potential energy is most exhausted, the forces are most energetic 1

Now there is a magnetic problem in which we have a kind of similarity of behaviour, viz., when currents in material circuits are allowed to attract one another. Let, for completeness, the initial state be one of infinitely wide separation of infinitely

• The additional terms are analogous to the uiotional electric and mag- netic forces of § 44 and § 66, &c See alio ((£), § 132.

curl (6 + fiVah) = fih, .

curl (h + cVeq) = /ou - ce,

(19)

(20)

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BLBCTROMAONBTIO THIBOBT.

AFP. B.

small filamentary currents in closed circuits. Then, ou con- centration to any other state, the work done by the attractive forces is represented by 2)J/aH-, where fx is the inductivity and H the mc^netic force. This has its equivalent in the energy of motion of the circuits, or may be imagined to be so converted, or else wasted by friction, if we like. But, over and above this energy, the same amount, represents the

energy of the magnetic field, whicli can be got out of it in work. It was zero at the beginning. Now, as Lord Kelvin showed, this double work is aooounted for by astia work in the batteries -or other sources required to maintain the cumeDts constant. (I have omitted reference to the waste of energy due to elec- trical resistance^ to avoid complications.) In the gravitational case there is a partial analogy, but the matter is all along assamed to be incapable of variation, and not to require any supply of energy to keep it constant. If we asserted that Jee* was stored energy, then its double would be the work done per unit volume by letting bodies attract from infinity, with- out any apparent source. But it is merely the exhaustion of potential energy of unknown amount and distribution.*

Potential energy, when regarded merely as expressive of tba work that can be done by forces depending upon configuration, -does not admit of much argument. It is little more than a madiematioal idea, for there is scarcely any physics in it. It explains nothing. But in the consideration of physics in general, it is scaroely possible to avoid the idea that potential energy should be capable of localisation equally as well as kinetic. That the potential energy may be itself ultimately kinetic is a separate question. Perhaps the best definition of the former is contained in these words : — Potential energy is energy that is not known to be kinetic. But, however this be, there is a practical distinction between them which it is found useful to carry out. Now, when energy can be distinctly localised, its flux can also be traced (subject to circuital inde- terminateness, however). Also, this flux of energy forms a useful working idea when action at a distance is denied (eveo though the speed of transmission be infinitely great, or be

• It would aj^Msr that we must go to the ether to find the louree of all eueifiy.

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assumed to be so). Any distinct and practical localisation of energy is therefore a useful step, wholly apart from the debatable question of the identity of energy advocated by Prof. Lodge.

From this point ol view, then, we ought to localise gravita- tional energy as a preliniinary to a better understanding of that mysterious agency. It cannot be said that the theory of the potential enecgy of gravitation exhausts the subject. The flux of gravitational energy in the form above given is, per- haps, somewhat more distinoti since it considers the flux only and the changes in the amount loMdised, without any state- ment of the gross amount Perhaps the above analogy may be useful, and suggest something better.

Part n.

In the foregoing I partly assumed a knowledge on the part of the reader of ray theory of convective currents of electrifi- cation^ ("Electrical Papers," Vol. II., p. 495 and after), and only very briefly mentioned the modified law of the inverse squares which is involved, viz., with a lateral concentratioA of the lines of force. The remarks of the Editorf and of Prof. Lodget on gravitational aberration, lead me to point out now some of the consequences of the modified law which arises when we assume that the ether is the working agent in gravitational efibcts, and that it propagates disturbances at speed V in the manner sup- posed in Part L There is, so far as I can see at present, no aberrational effect^ but only a slight alteration in the intensi^ of force, in different directions round a moving body considered as an attractor.

Thus, take the case of a big Sun and small Earth, of masses S and E, at distauce r apart. Let / be the uumoditiod force of S on E, thus

/-S'

using rational units in order to harmonise with the electro- magnetic laws when rationally expressed. Also, let F be the

• See also 52 to 62, and §§ 163, 4.

t The Electrician, July 14, p. 277, and July 28, p. 340.

X The EUetrieian, July 28, p. 347.

464 ELECTROMAONBTIC THBOBT. AFP. B,

modified force when the Sua is in motion at speed u through the ether. Then*

where s is the small quantity and 6 is the angle between r and the line of motion. (" Electrical Papers," Vol. IL, pp. 495, 499).

Therefore, if the Sun ia at rest, there is no disturbance of the Newtonian law, because its " field of force " is stationary. Bat if it has a motion through space, there is a slight weakening of the force in the line of motion, and a slight strengthening equatoriallj. The direction is still radial.

To show the size of the effect, let

= 3 X 10'' centim. per sec.

ntim. per sec. )

\ . . . (3).

This value of u is not very different from the speed attri- buted to fast stars, and the value of v is the speed of light itself. So we have

iu^f one-millionth. All perturbing forces of the first order are, therefore, of the order of magnitude of only one-millionth of the full force, even when the speed of propagation is as small as that of light.

The simplest case is when the common motion of the Sun and Earth is perpendicular to the plane of the orbit. Than ^■•^ all round the orbit, and

F=/(l+i*), (5)

showing an increase in the force of attraction of S on £ of one two-millionth part, without alteration of direction or vEziation in the orbit.t

• This is the ca.se of steady motion. There is no simple formula '.vlicn the motion is un8tca<ly. Equation (2), above, is immediately derivable from equation (40), § 163.

t But Prof. Lodge tells me that our own particular Sun is considered to move only 10'9 miles per second, mils is stupendously slow. TbmnmU i is reduced to about 1/360 part of that in the tazt^ and the Hun* applies to the corrections depending upon it.

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465

Bat when the common motion of the Sun and Earth is in their plane, 0 varies from 0 to 27r in a revolution, so that the attraction on £, whilst towards the Sun's centre always, under- goes a periodic yariation from

F-/(l-f) (6)

when 0 = 0, to F=/(l+j8), (7)

when 0 = ^ir. The extreme variation is, therefore, J-^/j accord- ing to the data used. The result is a slight change in the shape of the orbit.

But, to be consistent, havingmadev finite by certain supposi- tions, we should carry out the consequences more fully, and allow not merely for the change in the Newtonian law, as above, but for the force brought in by the finiteness of v which is analogous to the '* electromagnetic force." This is very small truly, but so is the above change in the Newtonian law, and since they are of the same order of magnitude, we should also count the auxiliary force. Call it O. Then*

0=Fx5^xVa,VriHi, .... (8)

V*

where F is as before^ in (2) above, q is the actual speed of the Earth (not the same as ti), and in the third vectorial factor, Qp Up and are unit vectors drawn parallel to the direction of the Earth's motion, of the Sun's motion, and from the Sun to the Earth. We see at once that the order of magnitude cannot be greater than that of the departore of F'from^ before con- sidered, because u and q will be of the same order, at least when u is big. As for it is simply a numerical factor, which cannot exceed 1, and is probably

The simplest case is when the motion of the Sun is per- pendicular to the orbit of the Earth. Then

G = Fxa:a (9)

gives the tensor or size of the auxiliary force. It is radial, but outwards, so that the result is merely to reduce the size of

• See § 46, equation (9), nnd §§82 and 85. To apply to the present case, we have to note that G in the " electromagnetic force " of the Sun's " mag- neUo induction " on the Earth's ''electric current, " or rather, the analogue thereof ; and also that the direotaon must be reversed, jost sa the ordinary "deetrio force" is taken rerersed.

BB

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RLBCrrBOllAGNETIO TBBOBT.

APP. B.

the previoos correction, tie., the diflMBrenee of F from / in the

same motional ciroumstances.

But when the line of motion of the Sun is in the plane of the orbit, the case is much more complicated. The force O is neither constant (for the same distance) nor radial, except iu four positions, viz., two in the line of motion of the Sun, whea the auxiliary force vanishes, and two when 0 = ± |:r, when it is greatest. But this force is still in the plane of the orbit, which is an important thing, and is, moreover, periodic, so that the tangential component is as much one way as the other in a period.

All we need expect, then, so far as I can see from the above considerations, are small perturbations due to the yariation of the force of gravity in different directions, and to the auxiliary fotbe. Of course, there will be numerous minor perturbations.*

I! variations of the force of the size considered above are too small to lead to observable perturbations of motion, then the striking conclusion is that the speed of gravity may even be the same as that of light. If they are observable, then, if existent, they should turii up, but if non-existent then the speed of gravity should be greater. Furthermore, it is to be observed that there may be other ways of expressing the propagation of gravity.

But I am mindful of the good old adage about the shoemaker

and his last, and am, therefore, reluctant to make any more remarks about perturbations. The question of the ether in its gravitational aspect must be faced, however, and solved sooner or later, if it be possible. Perhaps, therefore, my suggestion* may not be wholly useless.

• The solution for steady rectilinear motion has been employed. The i4 justification thereof is the smallaess of i»/v and the large periodic time, f >( If ve allow tor the small oiirvature of path of an attraottng body, we shall introduce corrections of the second order of small quantities.

/

END OF YOLUHB I, 4

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