(A), we see that the effect of increasing u is to concentrate the displacement about the equatorial plane . Self-induction does it. In the limit, when u=v, the numerator vanishes, making E=0, H=0 everywhere except at the plane mentioned, where, by reason of the denominator becoming infinitely small in comparison with the numerator, the displacement is all concentrated in a sheet, and with it the induction, forming a plane electromagnetic wave, as described (and realized) in my previous communication.
If we terminate the field described in (A) and (B) on a spherical surface of radius a, instead of continuing it up to the charge q at the origin, we have the case of a perfectly conducting sphere of radius a possessing a total charge q, moving steadily at speed u through the dielectric ether. As the speed is increased to v, the charge all accumulates at the equator of the sphere. [See footnote on p. 514, later.]
But after that? This brings us to the third case of u>v, and here I have so-far failed to find any solution which will satisfy all the necessary conditions without unreality. The description at the close of Part II. must therefore be received as a suggestion, at present unconfirmed. I hope to consider the matter in a future communication.
P.S.—In a recent number Mr. W. P. Granville raised the question of action through a medium being only action at a short distance instead of a long one, and asked for instruction. His inquiry has elicited no response. This is not, however, because there is nothing to be said about it. The matter did not escape the notice of the "anti-distance-action sage." My own opinion is that the question involved is, if not metaphysical, dangerously near to being so; consequently, whole books might be devoted to it. At present, however, I think it is more useful to try to find out what happens, and to construct a medium to make it happen; after that, perhaps, the matter referred to may be more advantageously discussed. The well of truth is bottomless.
Part IV.
In previous communications [above] I have discussed this matter. Referring to the case of steady rectilinear motion, I gave a description of the result when the speed of the charge exceeds that of light, obtained mainly by general reasoning, and stated my inability to find a solution to represent it. The displacement cannot be outside a certain cone of semi-vertical angle whose sine equals the ratio v/u of the speed of light to that of the charge, which is at the apex.
In the Phil. Mag. for July, 1889, Prof. J. J. Thomson has examined this question. Like myself, he fails to find a solution within the cone; but concludes that the displacement is confined to its surface. If so, it must form, along with the magnetic induction, an electromagnetic wave. But it may be readily seen that such a wave is impossible, having no stability.
For as the charge moves from A to B, a given surface-element, C, would move to D. In doing so its area would vary directly as its distance from the apex, and the energy in the element would therefore
