CHAPTER II.

What physical explanation is possible of this production of a real image?

First of all we may note that the production of a real image of any disturbance is one of the commonest phenomena.

If a piece of indiarubber lying on the table be pressed downwards with the finger it will move up when the finger is removed. The yielding and the resuming its original form are movement and image movement.

If the disturbance is simply a displacement in one line, then, if the medium in which this displacement is produced is not permanently displaced, but on the whole maintains its equilibrium, there invariably accompanies any displacement its image displacement.

Moreover, to take the simple example of a wave propagated through water—the particles of the water on the whole move about a mean position; they are not displaced permanently in any one direction; and, taking the distance from the crest to the hollow of a wave, then from the hollow to the next crest, is the real image of the first part. Thus in the complete movement in the wave measured from crest to crest, there is displacement and its real image.

Thus there seems some consistency about this supposition of an image, about the production of a real image in nature.

But there are two observations which we can make.

_Firstly_, if it is true in these complicated cases it ought to be true in simpler cases also. That is, if this supposition is in harmony with electrical actions, it ought to fit in with other actions of a simpler kind.

_Secondly_, a supposition of this kind has no permanent value; it is rather a feeler, by which we trace out our way in the darkness, than any actual vision itself. In default of an actual realization of what the electrical relations are we can treat them by means of a supposition. But we must be ready at any moment to give up the supposition if it does not harmonize with the facts.

And in the first case does the idea of a real image hold good about the simplest possible actions?

If we push our fist towards a glass the image is that of a fist moving in the opposite direction.

Now, suppose a pressure exerted on a wall, as, for instance, a hard stone hitting it. The wall undergoes a displacement, but not as a whole—only that part of it where the stone hits. And this displacement is followed by the image displacement, for the wall in the part where it has been hit and pressed back moves forward, and by its reaction throws the stone off.

Every case of action and reaction is a case of a motion and its image motion.

If a bullet strikes the wall and goes with such velocity that it lodges in it, then the motion of the ball and the image motion of the wall destroy one another, and the result is a shattering of the wall in the path of the bullet.

Now in the case of a simple displacement of this kind there is a rule by which we can form the image displacement. Take a point on the wall, and about this point as a centre turn the displacement half way round, so that it does not come to be itself again, but is opposite to itself.

By this turning, the displacement becomes the image of itself; a movement into the wall becomes a movement out from the wall; and these follow one another if the wall is not injured. It should be noticed that the displacement is moved round this point, using a direction which is _not_ in the displacement itself. The displacement goes straight into the wall. The turning motion, which we suppose, needs another direction than this.

Now suppose, instead of a simple displacement like this, we take a displacement involving two directions, as in the case of a wave disturbance—it will be found that the conditions are just the same. If a wave movement falls on a medium which it does not destroy or move as a whole, the displacement calls up its image displacement. And the image displacement can be found, as before, by twisting the displacement round so as to become opposite to itself—by twisting it half-way round. But in this case, too, a direction must be used which is not used in the displacement itself.

Let us look at the wave disturbance more closely.

The horizontal central line in Diagram II. will represent the positions which a number of particles occupy when at rest. That is, let us suppose there to be a number of particles lying in a series forming this line.

We can think of the portions of an elastic cord. An indiarubber tube may be taken as an illustration, and made to vibrate by a motion of the hand.

If now one of the particles be deflected from its natural position—suppose it is moved to the position M—then we should have one particle at M out of its place, and all the others in their places.

But this does not happen. If the particle is pulled to M, the particles near it follow after it, and are also disturbed from their places, though not so much as the particle at M.

We should have a set of particles forming a shape like L M N, only much longer; in fact, the particles all along the cord would be raised.

If the cord is struck suddenly we do have a set arranging themselves like L M N, but only for a limited distance along the cord.

And here we notice a curious thing.

If a set of particles is forced to go like L M N, removed from their position of repose, then at once a set of particles goes like N O N′.

A displacement is accompanied by another displacement which is the opposite of it. And this displacement and opposite displacement travels along the elastic cord.

But the point of view which is the most natural one to regard it from is a little different from this. Let us consider a single point, P. When this is disturbed it moves above its original position to M, and below to the other end of the dotted line. Its complete movement is from one of these extremes to the other. And if we take the complete disturbance as exhibited in all its phases by different points, we ought to look at the portion of the diagram M N O. For here at N we have a point not displaced at all; at M, one displaced to its full extent upwards; at O, one displaced to its full extent downwards. And intermediate particles have intermediate displacements.

Now when a complex displacement of this kind is put into a cord, its image at once springs up. The displacement represented by M N O at once calls up the displacement represented by O N′ M′, and this condition of displacement and image displacement continues repeating itself till the cord comes to rest.

If the diagram be closely looked at, it will be seen that it exhibits the image relationship twice over. For the movement of the particle P from P to M has its image in the motion of another particle from its place of repose to the position O. The disturbance itself, M N O, consists of displacements and image displacements; and this disturbance, with its image O N′ M′, makes the wave from crest to crest.

The “twist” which we consider in these pages is like the wave motion, but with a third component added, so that in the complete motion there is a displacement coming out from the plane of the paper, as well as the displacements in the plane of the paper itself.

And just as the wave displacement produces a real image of itself in a medium which it does not distort as a whole, so there is nothing arbitrary in our assuming that an electric twist calls up the real image of itself in an insulating medium—that is, a medium which it cannot twist as a whole.

If L M N O is a wave motion, then L′ M′ N′ O is its image, as produced by moving it round out of the plane of the paper—Diagram II. If the wave disturbance is moved round in the plane of the paper, the original wave L M N O becomes L′ M′ N′ O—Diagram III.—a shape which bears no resemblance to the transmitted wave.

Consider O N M L to be a bent piece of wire lying on the paper; if it is moved round O, keeping on the paper, it becomes O N′ M′ L′. To become like O N′ M′ L′ in Diagram II. it must move up from the paper and down again on the right.

Thus adopting this artificial aid to thought—that a displacement calls up an image displacement—we get the rule that this displacement, the image, can be got from the original displacement by moving the original displacement half-way round, using as the plane in which the turning is made that plane which is given us by taking these two directions—the direction in which the wave is moving, and a direction at right angles to the directions in which the displacements which form the wave take place.

Thus, with the wave motion shown, if we take the direction towards the top of the page to be the up direction, and that from left to right to be the sideways direction, then out of the paper towards us is the “near” direction. So, too, in this case we have to turn the wave disturbance out of the plane of the paper, and each point of it, to produce the image, must turn in a circle (going half-way round it) lying in a plane which has the two directions near and sideways. The motions of the particles themselves are in the plane of the paper. So to get the image by turning we use a direction—the “near” direction, which is not involved in the wave motion itself.

Hence we may state, as a tentative principle, that when a disturbance takes place in a medium which will not be disturbed as a whole, then such disturbance is accompanied by a real image of itself; and this real image of itself is the configuration which would be obtained by twisting the original disturbance round in a direction not contained in the original disturbance.

Thus the disturbance O N′ M′ L′ is obtained by twisting the disturbance L M N O round. The direction in which it is twisted is the direction coming out from the plane of the paper.

Now if this plane disturbance is in nature accompanied by its real image, why should not a twist such as takes place in the electric current also be accompanied by its image twist when it impinges on a medium which it cannot twist as a whole—that is, when it comes to an insulator in its path?

The reason, obviously, is that we cannot conceive such an image produced mechanically. And the reason of this can be exhibited thus.

When we had a plane disturbance like L M N O we only used up two dimensions of space, and we have a third coming up from the plane; and this direction enables us to imagine a turning which will alter A B into its image.

But when we have a twist proceeding along an axis, as in the case of electricity, we have no direction left over in space whereby we may conceive the twist turned round.

Now when the displacement itself involves all these directions how will our rule hold?

How shall we get the image displacement? We can find what it is by using a looking-glass; but the same rule which served in previous cases ought to work here also.

We want a direction which is neither up and down, right and left, towards and away.

Now let us adopt a mathematical device, and suppose there is such a direction, and let us call it the X direction, the unknown direction.

Then if we turn the twist round, using this X direction, we shall get the image if our rule is correct. And as a matter of fact, by twisting a figure round in this way, using a direction different from any of the three mentioned above, we do get its image.

Hence the rule we have formed works consistently.

It will be found that if there was another direction so that the spiral disturbance could be turned independently of the directions used up in it, that just as a plane disturbance can be turned into its image disturbance, so the spiral disturbance of electricity could be turned into its image spiral by a simple turning.

In this argument we have not looked at the matter directly, but from the outside. To see it immediately requires us to gain a familiarity with the properties of space with four independent directions, and that would take too long for the present paper. The same conclusion can be arrived at mathematically; but in these papers as far as possible we avoid symbolism. We want to gain hold of scientific facts in a warm and living way, to unwrap them from conventionalities and formulæ.

Thus if we suppose that in the minute motions which go on about us there is a possibility of moving in a four-dimensional way, then it is perfectly legitimate to assume that in a medium which cannot be twisted, but which is elastic, a twist calls up a real image twist.

And thus the assumptions which we have made as the basis of an electrical theory are justified on the assumption of a four-dimensional space, are untenable except on that supposition.

The matter is of course perfectly open. The only way is this, by adopting the assumption of a higher space to predict what the actions of the molecules will be, then if a number of predictions are verified the evidence will become strong. And I feel sure that there are some very curious things to be made out here. For my own part the evidence of the reality of four-dimensional space—in the sense in which we say that our space is real—does not rest on the consideration of the molecular movements about which it is not easy to get clear ideas, but on the study of the facts of space. I hardly think that any one who spent a few years in becoming familiar with the facts of space, not by the means of symbolism or reasoning but by pure observation, could doubt that there are really four dimensions.

In noticing the simpler actions and their image actions we find that the real image does not coexist with its original, but rather follows and succeeds it. If we push against a board the board yields, and springs back when we leave off pushing. If the original displacement is permanent as a point pressed against an elastic surface and making the surface yield, then the image of this displacement is potential; it is not actually there, but comes into play as soon as the original displacement is removed.

Now in the electrical actions we have assumed both the original twist and the image twist as concurrently existing.

In certain cases there is no doubt that they are coexistent as when a glass rod is rubbed by silk.

But if the case of the action of a charged poker on an uncharged one be examined it will be found that there is nothing to prove that the image twist comes into existence until the original one is removed.

When the charged poker is brought near the other, the remote end of the second is affected with the same kind of electricity as is on the charged poker.

The appearance is just the same as if a thin wall were exposed to a pressure on one side, and the other side were to bulge out. The displacement is transmitted through the conductor.

It is only when the original charged body is removed that the image charge is found to be in existence on the second conductor. There are some peculiarities, however, which make electrical displacements different in their appearances from ordinary displacements.

No body can be made to move in any direction without imparting an equal motion in an opposite direction to another body—_e.g._, the motion of a cannon ball is equalled by the recoil of the cannon.

And so no twist can be given to the particles of a body without an image twist being given to other particles.

Now the image displacement or rectilinear motion, in the case of a rectilinear motion, in straightforward movement seems to remain in the place where it was produced. The recoil of the gun carriage produces a strain on its bearings and friction, which produce heat, which gradually dissipates.

But the image displacement, in the case of electricity, seems to have a marvellous facility for running through the earth and meeting the original displacement. An indefinitely long line of action seems in electricity to take the place of a simple point. Our ordinary mechanical forces are located in centres, or points of action. In electricity the line seems to take the place of the point. Where the ordinary engineer deals with points the electrical engineer deals with lines.