This reproduction of Heaviside's article is an unedited copy of
the original, except that I have converted some formulas and all vector
equations appearing in the article to modern mathematical notation.

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
Maxwell'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. Furthermore, it
is assumed to have personal identity, so that the position and motion of
a definite particle of matter are definite, at any rate relative to an
assumed fixed space. Matter is recognised 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 be
the density of matter, and **e** the intensity of force, or the force
per unit matter, then

Now the force

Now when matter is allowed to fall together from any configuration to a closer one, the work done by the gravitational forcive is expressed by the increase made in the quantity . This is identically the same as the quantity

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 analogous 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 gravitational analogy. But this is not true.

Thus, if

Now if we multiply (5) by

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 instantaneous action, which involves

This, of course, might be inferred from the electromagnetic case.

In order that the speed

The gravitational-electromagnetic analogy may be further extended if we allow that the ether which supports and propagates the gravitational influence can have a translational motion of its own, thus carrying about and distorting the lines of force. Making allowance for this convection of

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

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 gravitational 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 unintelligible 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 small filamentary currents in closed circuits. Then, on concentration to any other state, the work done by the attractive forces is represented by , where is the inductivity and

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 indeterminateness, 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 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 gravitational 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, perhaps, somewhat more distinct, since it considers the flux only and the changes in the amount localised, without any statement of the gross amount. Perhaps the above analogy may be useful, and suggest something better.

In my first article on this subject (*The Electrician,
*July 14, 1893, p.281), I partly assumed a knowledge on the part of
the reader of my theory of convective currents of electrification ("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^{(1)} and of Prof. Lodge^{(2)}
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 supposed in my former article. 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 *f*
be the unmodified force of *S* on *E*, thus

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

This value of *u* is not very different from the speed attributed
to fast stars, and the value of *v *is the speed of light itself.

So we have

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

But when the common motion of the Sun and Earth is in their plane, varies from 0 to in a revolution, so that the attraction on

But, to be consistent, having made

The simplest case is when the motion of the Sun is perpendicular to the orbit of the Earth. Then

But when the line of motion of Sun is in the plane of the orbit, the case is much more complicated. The force

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

If 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 turn 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 suggestions may not be wholly useless.

1. The Electrician, July 14, p. 277, and July 23, p. 340.

2. The Electrician, July 28, p. 347.

3. This is the case of steady motion. There is no simple formula when the motion is unsteady.

4. But Prof. Lodge tells me that our own particular
Sun is considered to move only 109 miles per second. This is stupendously
slow. The size of *s* is reduced to about 1/360 part of that in the
text, and the same applies to the corrections depending upon it.

5. Heaviside uses the word "tensor" for the
magnitude of a force vector (O. D. J.).