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.