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

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 sufficient 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 ^ce2. This was equivalently suggested by Maxwell. Thus two bodies which appear to attract are pushed together. The case of two large parallel material planes exhibits this in a marked manner, for e is very small between them, and relatively large on their further 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 further 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 !

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 motional electric and mag- netic forces of § 44 and § 66, &c. See also (d), § 132.

462 ELECTROMAGNETIC THEORY. APP. B.

small filamentary currents in closed circuits. Then, on con- centration to any- other state, the work done by the attractive forces is represented by 2J/AH2, where p is the inductivity and H the magnetic 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, 2J/xH2, represents the energy of the magnetic field, which can be got out of it in work. It was zero at the beginning. Now, as Lord Kelvin showed, this double work is accounted for by extra work in the batteries or other sources required to maintain the currents 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 assumed to be incapable of variation, and not to require any supply of energy to keep it constant. If we asserted that |ce2 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 the work that can be done by forces depending upon configuration, does not admit of much argument. It is little more than a mathematical idea, for there is scarcely any physics in it. It explains nothing. But in the consideration of physics in general, it is scarcely 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 (even though the speed of transmission be infinitely great, or be

• It would appear that we must go to the ether to find the source of all energy.

GRAVITATIONAL ANALOGY. 463

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

Part II.

In the foregoing I partly assumed a knowledge on the part of the reader of my 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 concentration of the lines of force. The remarks of the Editor f and of Prof. Lodge J 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 effects, and that it propagates disturbances at speed v in the manner sup- posed in Part I. There is, so far as I can see at present, no aberrational effect, but only a slight alteration in the intensity 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 distance r apart. Let / be the unmodified force of S on E, thus

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.

t The Electrician, July 28, p. 347.

464 ELECTROMAGNETIC THEORY. APP. B.

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

where s is the small quantity ^2/v2, and 0 is the angle between r and the line of motion. (" Electrical Papers," Vol. II., pp. 495, 499).

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

To show the size of the effect, let

u = 3 x 107 centim. per sec. )

• = 3xlO™ „ „ „ /

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

i.e., 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. Then 0 = \TT all round the orbit, and

F-/(1 + M ....... (5)

showing an increase in the force of attraction of S on E of one two-millionth part, without alteration of direction or variation in the orbit, f

• Tliis is the case of steady motion. There is no simple formula when the motion is unsteady. Equation (2), above, is immediately derivable from equation (40), § 163.

f But Prof. Lodge tells me that our own particular Sun is considered to move only 10*9 miles per second. This is stupendously slow. The size of « is reduced to about 1/360 part of that in the text, and the same applies to the corrections depending upon it.

GRAVITATIONAL ANALOGY. 465

But when the common motion of the Sun and Earth is in their plane, 0 varies from 0 to 2r in a revolution, so that the attraction on E, whilst towards the Sun's centre always, under- goes a periodic variation from

F =/(!-«) (6)

when 0 = 0, to F=/(1 + J«), (7)

when 6 = \v. The extreme variation is, therefore, f-*/, accord- ing to the data used. The result is a slight change in the shape of the orbit.

But, to be consistent, having made v 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 Cr. Then*

, .... (8)

where F is as before, in (2) above, q is the actual speed of the Earth (not the same as u\ and in the third vectorial factor, QJ, Up and rt 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 departure of F from /, before con- sidered, because u and q will be of the same order, at least when u is big. As for x, it is simply a numerical factor, which cannot exceed 1, and is probably f .

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

G = Fx#s (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), and §§82 and 85. To apply to the present case, we have to note that G Is the " electromagnetic force " of the Sun's "mag- netic induction " on the Earth's " electric current," or rather, the analogue thereof ; and also that the direction must be reversed, just as the ordinary "electric force" is taken reversed.

HH

466 ELECTROMAGNETIC THEORY., A

the previous correction, viz., the difference of F from / i same motional circumstances.

But when the line of motion of the Sun is in the pla the orbit, the case is much more complicated. The fore neither constant (for the same distance) nor radial, exce four positions, viz., two in the line of motion of the Sun, the auxiliary force vanishes, and two when 0= ±^, whe greatest. But this force is still in the plane of the orbit, is an important thing, and is, moreover, periodic, so tha tangential component is as much one way as the othei period.

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

If variations of the force of the size considered above a small to lead to observable perturbations of motion, the striking conclusion is that the speed of gravity may even 1 same as that of light. If they are observable, then, if exi they should turn up, but if non-existent then the spc gravity should be greater. Furthermore, it is to be obs that there may be other ways of expressing the propa< of gravity.

Bat I am mindful of the good old adage about the shoei and his last, and am, therefore, reluctant to make any remarks about perturbations. The question of the ether gravitational aspect must be faced, however, and solved £ or later, if it be possible. Perhaps, therefore, my sugge may not be wholly useless.

• The solution for steady rectilinear motion has been employed justification thereof is the smallness of u/v and the large periodi If we allow for the small curvature of path of an attracting body, \ introduce corrections of the second order of small quantities.

END OF VOLUME I,

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