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excited by simple translation, by means of the device of an outer shell imbedded in the æther and containing inside it masses with spring connexions; and such a system might also be adjusted so as to respond to simple rotation, and therefore be excited at the nodes of the wave-train instead of the antinodes. A theory based in this manner on difference of inertia must take the density of the æther to be very minute compared with that of matter; therefore if the molecule is to have free periods of the same order of magnitude as the periods of the incident light-waves, the elastic forces acting between the atoms and concerned in these periods must be very intense. But Lord Kelvin's well-known estimate of the rigidity of the æther on this hypothesis makes it very small compared with the ordinary rigidity of material bodies*. In fact on Pouillet's data, which imply a considerable underestimate, the energy of the solar radiation near the sun's surface is about 4 x 10-5 ergs per cubic centimetre; it easily follows that if the amplitude of the æthereal disturbance is, say, e times the wave-length, the density of the æther must be about 1/1022, and its rigidity, which is equal to the density multiplied by the square of the velocity of propagation, therefore 1/10e2. On an elastic solid theory it is desirable to have the density very small thus if we adopt 10-2 as the maximum likely value of €, the density of the æther comes out 10-18 of that of water, and its rigidity about 103, whereas the rigidity of steel or glass is of the order 1011.
Now at first sight it would appear that the elastic tractions exerted by an æther of such small rigidity on an imbedded molecule swaying backwards and forwards in it, would be vanishingly small compared with the elastic forces between its constituent atoms which are concerned with free vibrations of the kind of period under consideration; and that therefore they would be quite incompetent to produce violent disturbance in the molecule. But on a closer examination this difficulty may to a considerable extent be evaded.
Let us imagine an imbedded rigid nodule of linear dimension L, and let the force necessary to displace it in the æther in any manner through a distance x be La. Let us compare with it a similar nodule of linear dimensions L displaced through a distance xx. There is complete dynamical similarity between the two cases; the strains at corresponding points in the æther are equal, and therefore so are the tractions per unit area. Thus the forcive necessary in the latter case to produce the displacement κa is κ2La, and therefore to produce the same displacement x as in the previous case a *Cf. Maxwell, Encyc. Brit., article "Æther."
forcive L is required. If now instead of comparing the total forcives in the two cases we compare the forcives per unit volume, an increase of linear dimensions in the ratio of κ to one diminishes this forcive in the ratio of 2 to one. Thus, if only the atoms are taken small enough, an æther of very slight rigidity can exert a forcive on them which, estimated per unit volume, is of any order of magnitude we please. The features of the case are in fact analogous to those of the suspension of small bodies, such as motes, in a viscous fluid medium like the atmosphere: if only the particles are small enough they will float for an indefinitely great time against the force of gravity, even be they as dense as platinum,-the only limit being in that case the one imposed by the molecular discreteness of the air itself.
It would appear that the application of this principle does much to vivify the notion of an elastic solid æther. A medium of this kind, which is excessively rare and, as a consequence, of very feeble elasticity, would exert practically negligible tractions on the surfaces of a mass of matter in bulk, while it may exert relatively very powerful ones on the individual atoms of which the mass is composed if only they are sufficiently small, it being of course supposed that the structure of the medium itself is absolutely continuous. And it would even appear that a medium of very small density and rigidity may be competent to excite powerful vibrations in the molecules notwithstanding the strength of the forcives which hold them together.
We may thus imagine a working illustration of a ponderable transparent medium of elastic solid type as made up of very small spherical nodules of great density and rigidity dispersed through the æther and imbedded in it. We may even imagine these nodules to be collected into more or less independent groups, each of which will have free periods of relative vibrations of its own nearly independent of other groups, in the manner now well known in connexion with Prof. Ewing's model of a magnetic medium. A wave running across such a medium may excite these groups, and thus illustrate the theory of selective absorption by means of a system in which only the elasticity of the ambient medium is operative, but no other internal forcive.
The explanation of a very weak medium exciting such powerful tractions implies of course strains of enormous intensity, so that its limits of perfect elasticity must be taken enormously wide compared with anything we know in ordinary matter. The magnitude of the strains also requires that the displacement of an atom relative to the æther must
be a considerable fraction of its diameter: and this is sufficiently secured by the large value of e above that which is required to keep down the density of the æther, combined with the great relative density of the atom. It would thus seem to be possible to account for sufficiently large differential tractions between the component atoms of a molecule, especially if some of them lie well under the lee of others, to produce brisk internal vibration.
In this way we could imagine the construction of a sort of model illustrative of an elastic solid theory of refraction, including selective absorption and other such phenomena, in the form in which it is presented by von Helmholtz and others. In the simpler case, in which the atoms are not grouped into systems capable of synchronous free internal vibrations, let (, n, ) denote the mean displacement of the free æther, and (1, 1, 1) that of the atoms. Then the equations of vibration assume the forms
talline media could be included by assuming a vector-coefficient instead of the scalar a.
The conclusion, then, is that in this limited range an elasticsolid theory of a very rare æther is not so much at fault as would at first sight appear.
A theory based on difference of rigidity without difference of inertia, after MacCullagh's manner, would have to be realized by ascribing to the atom an atmosphere of intrinsic æthereal strain, instead of endowing it with great inertia ; and this could only be possible in a rotational æther, and would in fact form a mechanical representation of the electric theory. As such it must be expected to give an account of
* There are introduced by von Helmholtz (Wiss. Abh. ii. p. 216) in addition, a forcive proportional to the absolute displacement of the atom, and a frictional one proportional to its absolute velocity. The former is derived from the idea that the heavy central masses of the atoms are unmoved by the æther, and only outlying satellites are affected by its motion. On our present view this restriction might be dispensed with, except in so far as it renders possible an illustrative theory of absorption of an analytically simple character. The consequences of the above equations are set out by various writers, e. g. Carvallo (Comptes Rendus, cxii. p. 522).
the phenomena of electricity as well as those of light*, and in such an account is founded one of its chief claims.
A development of the electric theory has recently been essayed by von Helmholtz †, on the basis of the formal equations of Heaviside and Hertz, in which the free æther is still supposed to be an elastic medium of excessively small density in which the dense atoms are imbedded. If such a view should turn out to be the basis of a consistent body of theory, the considerations given above with respect to the intensities of molecular tractions would have a bearing on it also.
Let us now consider more particularly the explanation that would be offered by the electric theory of light. The difference between a material medium and a vacuum consists in an altered effective dielectric coefficient. This difference is simply and naturally explained by the hypothesis that the material molecules are polar owing to their associated atoms having atomic charges equal in amount but opposite in sign, and that they therefore possess electric moments just as the molecules of a magnet possess magnetic moments. An electric force thus tends to pull the two constituents of a molecule asunder ; and its full intensity is exerted in this manner, not merely its differential intensity over the range of the molecular volume. But a magnetic force has no such tendency even when we take the molecule to be magnetically polarized, because the two poles of a magnetic element cannot be dissociated from each other; the magnetic moment is thus directly associated with the atom, not with the molecule. In the case of the stationary light-waves the antinodes of the electric force are therefore places where alternating disturbances of a kind suitable to produce decomposition of the molecules are maintained, and may produce strong effects through sympathetic molecular vibration or otherwise; but at the intermediate antinodes of the magnetic force the individual ultimate atoms may be disturbed by the alternating magnetic force, but there is no tendency to separation of the constituents of the molecule. On the electric theory, therefore, there is abundant justification both for the magnitude of the effect produced, and for its localization as determined by Wiener's experimental investigation.
The theory, noticed first it seems by Weber, which ascribes molecular magnetism to the orbital rotation round each other of ionic charges, and which has very strong recommendations from the point of view of the dynamics of the æther, may form a partial exception to this statement. It leaves the Cf. "A Dynamical Theory . . .," Phil. Trans. 1884, §§ 122-124. † Wied. Ann. 1894.
question open as to whether the principal part of the magnetic moment is due to orbital motions in the atoms or to the motions of the constituent atoms in the molecules; though it suggests strongly the latter alternative. In that case there will usually be a differential magnetic action of the field as between these moving atoms; but the magnetic actions on positive and negative ions will be by no means equal and opposite, as is true of the electric actions. Thus, for example, in the limiting case of two equal and opposite ions revolving round each other, the elements of the equivalent ionic convection currents will be at each instant parallel, and there will be no differential magnetic forcive at all; there will also be no magnetic moment; but the electric differential action will retain its full force.
It is well understood, and in accordance with this explanation, that the energy of chemical combination of atoms into molecules is almost entirely that of electrostatic attraction of their atomic charges. In fact the electric attraction between them diminishes according to the law of inverse square with increasing distance, their magnetic attraction according to the law of the inverse fourth power: if these forces are of the same order of magnitude in the actual configuration of the atoms in the molecule, the work done by the former during their combination must be almost indefinitely greater than the work done by the latter.
If we contemplate the purely dynamical basis which must underlie the descriptive explanations of the electric theory of light, it is difficult to see how there can be any place for a theory of the æther loaded by the material molecules, which dynamical views usually associated with Fresnel's theory demand. There could be no polarity in the inertia of a mere load, such as the present considerations require. On the other hand, the presence of electrically polarized molecules is effectively a diminution of the elasticity of the luminiferous medium; and I have tried to show elsewhere* that the principles of MacCullagh's theory of optics are in substantial agreement with all the general features of our electrical and optical knowledge.
It is definitely implied in the electromotive, as distinguished from an electrodynamic, character of the electric theory of light, that the atomic charges vibrate in unison with the light-waves, quite unimpeded by any material inertia of their atoms. This hypothesis is conceivable and natural, independently of any particular explanation, on the theory that the
*Loc. cit. Phil. Trans. 1894.