Dynamical Theory of the Electric and Luminiferous Medium III

Introductory
1. In two previous memoires it has been explained, that the various hypotheses involved in the theory of electric and optical phenomena, which has been developed by and, can be systematized by assuming the æther to be a continuous, homogeneous, and incompressible medium, endowed with inertia and with elasticity purely rotational. In this medium unitary electric charges, or electrons, exist as point-singularities, or centres of intrinsic strain, which can move about under their mutual actions; while atoms of matter are in whole or in part aggregations of electrons in stable orbital motion. In particular, this scheme provides an consistent foundation for the electrodynamic laws, end agrees with the actual relations between radiation and moving matter.

An adequate theory of material phenomena is necessarily ultimately atomic. The older mathematical type of atomic theory which regards the atoms of matter as acting on each other from a distance by means of forces whose laws and relations are gradually evolved by observation and experiment, is in the present method expanded end elucidated by the introduction of a medium through whose intervention these actions between the material atoms take place. It is interesting to recall the circumstance that in his electrodynamic speculations, which remained unpublished during his lifetime, arrives substantially at this point of view ; after examining a law of attraction, of the ian type, between the "electric particles," he finally discards it and expresses his conviction, in a most remarkable letter to, , Werke, V., p. 629, letter to  of date 1845 ; quoted by , "Treatise" II., § 861. After the present memoir had been practically completed, my attention was again directed, through a reference by, to 's Memoir "Le Théorie Electromagnetique de et son application aux corps mouvants," Archives Néerlandaises 1892, in which (pp. 70 seqg.) ideas similar to the above are developed. The electrodynamic scheme at which he arrives is formulated differently from that given in § 13 infra, the chief difference being that in the expression for the electric force (P, Q, R) the term $$-d/dt$$ (F, G, H) is eliminated by introducing the æthereal displacement (f, g, h). This applies also to the later "Versuch einer Theoric . . . in bewegten Körpern," 1895. The author remarks on the indirect manner in which dynamical equations had to be obtained, mainly on account of the absence of any notion as to the nature of the connexion between the stagnant æther and the molecules that are moving through it. "Dans le chemin qui nous a conduit à ces équations nous avons rencontré plus d’une difficulté sérieuse, et on sera probablement peu satisfait d’une théorie qui, loin de dévoiler le mécanisme des phénomenes, nous laisse tout au plus l’espoir de le découvrir un jour" (§ 91). In the following year (1893) similar general ideas were introduced by, in his now well-known memoir on the electrical theory of optical dispersion, in which currents of conduction are included: but his argument is very difficult, and the results are in discrepancy with those of and the present writer in various respects in which the latter agree; moreover they are not consistent with the optical properties of moving material media. Both these discussions, of and of, are in the main confined to electromotive phenomena: the treatment of the mechanical forces acting on matter in bulk would require for basis a theory of the mechanical relations of molecular media such as is developed in this paper. The results in the paper by, above referred to, "On the Influence of Magnetism on the Light emitted by a substance," Verslagen Akad. Amsterdam, Nov. 28, 1896, have an important bearing on the view of the dynamical constitution of a molecule that has been advanced in these papers, and illustrated by calculation in an ideal simple case in Part I., §§ 114-8; cf. 'Roy. Soc. Proc.' 60, 1897, p. 514. [See 'Phil. Mag.,' Dec. 1897: where the loss of energy by radiation from the moving ions is also examined.] that the key-stone of electrodynamics will be found in an action propagated in time from one "electric particle" to another. The abstract philosophical distinction between actions at a distance and contact actions, which dates for modern science from 's adoption of the scholastic axiom Nulla actio fieri potest nisi per contactum, can have on an atomic theory of matter no meaning other than in the present sense. The question is simply whether a wider and more consistent view of the actions between the molecules of matter is obtained when we picture them as transmitted by the elasticity and inertia of a medium by which the molecules are environed, or when we merely describe them as forces obeying definite laws. But this medium itself, as being entirely supersensual, we must refrain from attempting to analyse further. It would be possible (cf. § 6) even to ignore the existence of an æther altogether, and simply hold that actions are propagated in time and space from one molecule of matter to the surrounding ones in accordance with the system of mathematical equations which are usually associated with that medium ; in strictness nothing could be urged against such a procedure, though, in the light of our familiarity with the transmission of stress and motion by elastic continuous material media such as the atmosphere, the idea of an æthereal medium supplies so overwhelmingly natural and powerful an analogy as for purposes of practical reason to demonstrate the existence of the æther. The aim of a theory of the æther is not the impossible one of setting down a system of properties in which everything that may hereafter be discovered in physics shall be virtually included, but rather the practical one of simplifying and grouping relations and of reconciling apparent discrepancies in existing knowledge.

2. It would be an unwarranted restriction to assume that the properties of the æther must be the same as belong to material media. The modes of transmission of stress by media sensibly continuous were however originally formulated in connexion with the observed properties of elastic matter ; and the growth of general theories of stress-action was throughout checked and vivified by comparison with those properties. It was thus natural in the first instance to examine whether a restriction to the material type of elastic medium forms an obstacle in framing a theory of the æther; but when that restriction has been found to other insuperable difficulties it seems to be equally natural to discard it. Especially is this the case when the scheme of properties which specifies an available medium turns out to be intrinsically simpler than the one which specifies ordinary isotropic elastic matter treated as continuous.

A medium, in order to be available at all, must transmit actions across it in time ; therefore there must be postulated for it the property of inertia, — of the same kind as ordinary matter possesses, for there can hardly be a more general kind, — and also the property of elasticity or statical resistance to change either of position or of form. In ordinary matter the elasticity has reference solely to deformation; while in the constitution here assumed for the æther there is perfect fluidity as regards form, but elastic resistance to rotational displacement. This latter is in various ways formally the simpler scheme; elasticity depending on rotation is geometrically simpler and more absolute than elasticity depending on change of shape ; and moreover no phenomenon has been discovered which would allow us to assume that the property of elasticity of volume, which necessarily exists in any molecularly constituted medium such as matter, is present in the æther at all. The objection that rotational elasticity postulates absolute directions in space need hardly have weight when it is considered that a definite space, or spacial framework fixed or moving, to which motion is referred, is a necessary part of any dynamical theory. The other fundamental query, whether such a scheme as the one here sketched could be consistent with itself, has perhaps been most convincingly removed by Lord 's actual specification of a gyrostatic cellular structure constituted of ordinary matter, which has to a large extent these very properties; although the deduction of the whole scheme of relations from the single formula of Least Action, in its ordinary form in which the number of independent variables is not unnaturally increased, includes its ultimate logical justification in this respect.

On Material Models and Illustrations of the Æther and its associated Electrons.
3. Although the ian aspect of the subject, which would simply assert that the primary atoms of matter exert actions on each other which are transmitted in time across space in accordance with 's equations, is a formally sufficient basis on which to construct physical theory, yet the question whether we can form a valid conception of a medium which is the seat of this transmission is of fundamental philosophical interest, quite independently of the fact that in default of the analogy at any rate of such a medium this theory would be too difficult for development. With a view to further assisting a judgment on this question, it is here proposed to describe a process by which a dynamical model of this medium can be theoretically built up out of ordinary matter, — not indeed a permanent model, but one which can be made to continue to represent the æther for any assignable finite time, though it must ultimately decay. The æther is a perfect fluid endowed with rotational elasticity; so in the first place we have — and this is the most difficult part of our undertaking — to construct a material model of a perfect fluid, which is a type of medium nowhere existing in the material world. Its characteristics are continuity of motion and absence of viscosity: on the other hand in an ordinary fluid, continuity of motion is secured by diffusion of momentum by the moving molecules, which is itself viscosity, so that it is only in motions such as vibrations and slight undulations where the other finite effects of viscosity are negligible, that we can treat an ordinary fluid as a perfect one. If we imagine an aggregation of frictionless solid spheres, each studded over symmetrically with a small number of frictionless spikes (say four) of length considerably less than the radius, so that there are a very large number of spheres in the differential element of volume, we shall have a possible though very crude means of representation of an ideal perfect fluid. There is next to be imparted to each of these spheres the elastic property of resisting absolute rotation; and in this we follow the lines of Lord 's gyrostatic vibratory æther. Consider a gyrostat consisting of a flywheel spinning with angular momentum $$\mu$$, with its axis AB pivoted as a diameter on a ring whose perpendicular diameter CD is itself pivoted on the sphere, which may for example be a hollow shell with the flywheel pivoted in its interior; and examine the effect of imparting a small rotational displacement to the sphere. The direction of the axis of the gyrostat will be displaced only by that component of the rotation which is in the plane of the ring; an angular velocity $$d\theta/dt$$ in this plane will produce a torque measured by the rate of change of the angular momentum, and therefore by the parallelogram law equal to $$\mu d\theta/dt$$ turning the ring round the perpendicular axis CD, thus involving a rotation of the ring round that axis with angular acceleration $$\mu/i\cdot d\theta/dt$$, that is with velocity $$\mu/i\cdot \theta$$, where $$i$$ is the aggregate moment of inertia of the ring and the flywheel about a diameter of the wheel. Thus when the sphere has turned through a small angle $$\theta$$, the axis of the gyrostat will be turning out of the plane of $$\theta$$ with an angular velocity $$\mu/i\cdot \theta$$, which will persist uniform so long as the displacement of the sphere is maintained. This angular velocity again involves, by the law of vector composition, a decrease of gyrostatic angular momentum round the axis of the ring at the rate $$\mu^{2}/i\cdot \theta$$; accordingly the displacement $$\theta$$ imparted to the sphere originates a gyrostatic opposing torque, equal to $$\mu^{2}/i\cdot \theta$$ so long as $$\mu/i\cdot\int\theta dt$$ remains small, and therefore of purely elastic type. If then there are mounted on the sphere three such rings in mutually perpendicular planes, having equal free angular momenta associated with them, the sphere will resist absolute rotation in all directions with isotropic elasticity. But this result holds only so long as the total displacement of the axes of the flywheels is small: it suffices however to confer rotatory elasticity, as far as is required for the purpose of the transmission of vibrations of small displacement through a medium constituted of a flexible framework with such gyrostatic spheres attached to its links, which is Lord Kelvin's gyrostatic model of the luminiferous working of the æther. For the present purpose we require this quality of perfect rotational elasticity to be permanently maintained, whether the disturbance is vibratory or continuous. Now observe that if the above associated free angular momentum $$\mu$$ is taken to be very great, it will require a proportionately long time for a given torque to produce an assigned small angular displacement, and this time we can thus suppose prolonged as much as we please: observe further that the motion of our rotational æther in the previous papers is irrotational except where electric force exists which produces rotation proportional to its intensity, and that we have been compelled to assume a high coefficient of inertia of the medium, and therefore an extremely high elasticity in order to conserve the ascertained velocity of radiation, so that the very strongest electric forces correspond to only very slight rotational displacements of the medium: and it follows that the arrangement here described, though it cannot serve as a model of a field of steady electric force lasting for ever, can yet theoretically represent such a field lasting without sensible decay for any length of time that may be assigned.

4. It remains to attempt a model (cf. Part I., § 116) of the constitution of an electron, that is of one of the point-singularities in the uniform æther which are taken to be the basis of matter, and at any rate are the basis of its electrical phenomena. Consider the medium composed of studded gyrostatic spheres as above: although the motions of the æther, as distinct from the matter which flits across it, are so excessively slow on account of its great inertia that viscosity might possibly in any case be neglected, yet it will not do to omit the studs and thus make the model like a model of a gas, for we require rotation of an individual sphere to be associated with rotation of the whole element of volume of the medium in which it occurs. Let then in the rotationally elastic medium a narrow tubular channel be formed, say for simplicity a straight channel AB of uniform section: suppose the walls of this channel to be grasped, and rotated round the axis of the tube, the rotation at each point being proportional for the straight tube to $$AP^{-2}+PB^{-2}$$: this rotation will be distributed through the medium, and as the result there will be lines of rotational displacement all starting from A and terminating at B: and so long as the walls of the channel are held in this position by extraneous force, A will be a positive electron in the medium, and B will be the complementary negative one. They will both disappear together when the walls of the channel are released. But now suppose that before this release the channel is filled up (except small vacuous nuclei at A and B which will assume the spherical form) with studded gyrostatic spheres so as to be continuous with the surrounding medium; the effort of release in this surrounding medium will rotate these spheres slightly until they attain the state of equilibrium in which the rotational elasticity of the new part of the medium formed by their aggregate provides a balancing torque, and the conditions all round A or B will finally be symmetrical. We shall thus have created two permanent conjugate electrons A and B; each of them can be moved about through the medium, but they will both persist until they are destroyed by an extraneous process the reverse of that by which they are formed. Such constraints as may be necessary to prevent division of their vacuous nuclei are outside our present scope; and mutual destruction of two complementary electrons by direct impact is an occurrence of infinitely small probability. The model of an electron thus formed will persist for any finite assignable time if the distribution of gyrostatic momentum in the medium is sufficiently intense: but the constitution of our model of the medium itself of course prevents, in this respect also, absolute permanence. It is not by any means here suggested that this circumstance forms any basis for speculation as to whether matter is permanent, or will gradually fade away. The position that we are concerned in supporting is that the cosmical theory which is used in the present memoirs as a descriptive basis for ultimate physical discussions is a consistent and thinkable scheme; one of the most convincing ways of testing the possibility of the existence of any hypothetical type of mechanism being the scrutiny of a specification for the actual construction of a model of it.

5. An idea of the nature and possibility of a self-locked intrinsic strain, such as that here described, may he facilitated by reference to the cognate example of a material wire welded into a ring after twist has been put into it. We can also have a closer parallel, as well as a contrast; if breach of continuity is produced across an element of interface in the midst of an incompressible medium endowed with ordinary material rigidity, for example by the creation of a lens-shaped cavity, and the material on one side of the breach is twisted round in its plane, and continuity is then restored by cementing the two sides together, a model of an electric doublet or polar molecule will be produced, the twist in the medium representing the electric displacement and being at a distance expressible as due to two conjugate poles in the ordinary manner. Such a doublet is permanent, as above; it can be displaced into a different position, at any distance, as a strain-form, without the medium moving along with it; such displacement is accompanied by an additional strain at each point in the medium, namely, that due to the doublet in its new position together with a negative doublet in the old one. A series of such doublets arranged transversely round a linear circuit will represent the integrated effect of an electric polarization-current in that circuit; they will imply irrotational linear displacement of the medium round the circuit after the manner of vortex motion, but this will now involve elastic stress on account of the rigidity. Thus with an ordinary elastic solid medium, the phenomena of dielectrics, including wave-propagation, may be kinematically illustrated; but we can thereby obtain no representation of a single isolated electric charge or of a current of conduction, and the laws of optical reflexion would be different from the actual ones. This material illustration will clearly extend to the dynamical laws of induction and electromagnetic attraction between alternating currents, but only in so far as they are derived from the kinetic energy; the law of static attraction between doublets of this kind would be different from the actual electric law.

6. According to the present scheme the ponderomotive forces acting on matter arise from the forces acting on the electrons which it involves; the application of the principle of virtual work to the expression for the strain-energy shows that, for each electron at rest, this force is equal to its charge multiplied by the intensity of the electric field where it is situated. It has been urged that a model of the æthereal electric field cannot be complete, and so must be rejected, unless it exhibits a direct mechanism by which the ponderomotive normal traction $$F^{2}/8\pi$$ is transmitted across the æther from the surface of one conducting region to that of another: but the position can be maintained that such a representation would transcend the limitations belonging to a mechanical model of a process which is in part mechanical and in part ultra-mechanical. Indeed if this force were transmitted in the ordinary elastic senee, the transmitting stress would have to be of the nature of a self-balancing stress involving the square of the æther-strain instead of its first power, and thus not directly related to elastic propagation. The model above described is so to speak made of æther, and ought to represent all the tractions that exist in æther, vanishing as they do over the surface of a conducting region: but the model does not in the ordinary sense represent matter at all, except in so far as the æthereal strain-form which constitutes the electron is associated with matter. It therefore cannot represent directly, after the manner of a stress across a medium, a force acting on matter, for that would from this ultimate standpoint be a force acting on a strain-form spreading from its nucleus all through the medium, not a traction on a definite surface bounding the matter.

The fact is that transmission of force by a medium, or by contact action so-called, remains merely a vague phrase until the strain-properties of that medium are described; the scientific method of describing them is to assign the mathematical function which represents its energy of strain, and thence derive its relations of stress by the principle of virtual work; a real explanation of the transmission of a force by contact action must be taken to mean this process. Now in an elastic medium permeated by centres of permanent intrinsic strain, whether it be the rotational æther with its contained electrons, or an ordinary elastic solid permeated by polar strain-nuclei as described above, the specification of the strain-energy of the medium involves a mathematical function, not only of the displacement at each material point of the medium, but also of the positions of these intrinsic strain-centres which can move independently through it. To derive the play of internal force, this energy function must be varied with respect to all these independent quantities; the result is elastic tractional stress in the medium across every ideal interface, together with forcive tending to displace each strain-centre, which we can consider either as resisted by extraneous constraint preventing displacement of the strain-centre, or as compensated by the reaction of the inertia of the strain-form against acceleration. Consider, for example, the analogy of the elastic solid medium, and suppose a portion of it to be slowly strained by extraneous force; two strains are thereby set up in it, namely that strain which would be thus originated if the solid were initially devoid of intrinsic strain, and that strain which has to be superposed in order to attain the new configuration of the intrinsic strain arising from the displacement of its nuclei. The latter part is conditioned by the displacement of these strain-centres, and in its production forces acting on them must be considered to assist, whose intensities may be determined as has been already done in the æthereal problem.

The attractions between material bodies are therefore not transmitted by the æther in the way that mechanical tractions are transmitted by an ordinary solid, for it is electric force that is so transmitted: but neither are they direct actions at a distance. The point of view has been enlarged: the ordinary notion of the transmission of force, as framed mathematically by and  for a simple elastic medium without singularities, is not wide enough to cover the phenomena of a medium containing intrinsic strain-centres which can move about independently of the substance of the medium. But the same mathematical principles lead to the necessary extension of the theory, when the energy function thus involves the positions of the strain-centres as well as the elastic displacement in the medium; and the theory which in the simpler case answers fairly to the description of transmission by contact action has features in the wider case to which that name does not so suitably apply. The strain-centres (that is, the matter) have, in the strict sense of the term, energy of position, or potential energy, due to their mutual configuration in the æther, which can come out as work done by mutual forces between them when that configuration is altered, which work may be used up either in accumulating other potential energy elsewhere, or in increasing the kinetic energy of the matter, which is itself, in whole or in part, energy in the æther arising from the movement of the strain-forms across it. Discussions as to transmission by contact are not the fundamental ones, as the above actual material illustration shows: the single comprehensive basis of dynamics into which all such partial modes of explanation and representation must fit and be coordinated is the formula of Stationary Action, including, as the particular case which covers all the domain of steady systems, the law that the mutual forces of such a system are derived from a single analytical function which is its available potential energy.

The circumstance that no mode of transmission of the mechanical forces, of the type of ordinary stress across the æther, can be put in evidence, thus does not derogate from the sufficiency of the present standpoint. The transmission of material traction by an ordinary solid, which is now often taken as the type to which all physical action must conform, is merely an undeveloped notion arising from experience, which must itself be analysed before it becomes of scientific value: the explanation thereof is the quantitative development of the notion from the energy function by the method of virtual work in the manner indicated in § 10 infra. This orderly development of the laws of action across a distance, from an analytical specification of a distribution of energy pervading the surrounding space, is the essence of the so-called principle of contact action. It is precisely what the present procedure carries out, with such generalization as the scope of the problem demands; besides attaining a correlation of the whole range of the phenomena, it avoids the antinomies of partial theories which accumulate on the æther contradictory and unrelated properties, and sometimes even save appearances by passing on to the simple fundamental medium those complex properties of viscous matter whose real origin is to be found in its molecular discreteness.

Æther contrasted with Matter.
7. The order of development here followed is thus avowedly based on the hypothesis that the æther is a very simple uniform medium, about which it may be possible to know all that concerns us; and the present state of the theories of optics and electricity does much to encourage that idea. This procedure is of course at variance with the extreme application of the inductive canon, which would not allow the introduction of any hypothesis not based on direct observation and. experiment. But though that philosophy has abundantly vindicated itself as regards the secondary properties of matter, which are amenable to direct examination, its rigid application would debar us from any theory of the æther at all, as we can only learn about it from circumstantial evidence. We could then merely go on heaping up properties on the æther, on the analogy of what is known of matter, as circumstances necessitated; and this medium would be a sort of sink to dispose of relations that could not be otherwise explained. Whereas matter, with which we are familiar, is the really complicated thing on which all the maze of physical phenomena depends, so that it is doubtful whether much can ever be known definitely as to its ultimate dynamical constitution; our best chance is to try to approach it through the presumably simple and homogeneous æther in which it subsists.

For example, it is found that the transmission of electrostatic force is affected by the constitution of the material dielectric through which it passes, and this is explained by a perfectly valid theory of polarization of the molecules of the matter: to press the analogy and ascribe the possibility of transmission through a vacuum to polarization of the rather may be convenient for some purposes of description, but in the majority of cases the impression is left that the so-called polarization of the rather is thereby explained. Whereas the processes being, almost certainly, of totally different character in the two cases, it will conduce to accurate thought to altogether avoid using the same term in the two senses, and to speak of the displacement of the æther which transmits electric force across a vacuum as producing polarization in the molecules of a material dielectric which exists in its path, which latter in turn affects the transmission of the electric force by reaction. In trying to pass beyond this stage, we may accumulate descriptive schemes of equations, which express, it may be with continually increasing accuracy, the empirical relations between these two phenomena; but we can never reach very far below the surface without the aid of simple dynamical working hypotheses, more or less a priori, as to how this interaction between continuous æther and molecular matter takes place.

8. On the present view, physical theory divides itself into two regions, but with a wide borderland common to both: the theory of radiation or the kinetic relations of this ultimate medium; and the theory of the forces of matter which deals for the most part with molecular movements so slow that the surrounding rather is at each instant practically in an equilibrium condition, so that the material atoms practically act on each other from a distance with forcives obeying definite laws, derivable from the formula for the energy. It is only in electromagnetic phenomena and molecular theory that non-vibrational movements of the æther are involved. The æther not being matter, it need not obey the laws of the dynamics of matter, provided it obey another scheme of dynamical laws consistent among themselves; these laws must however be such that we can construct in the æther an atomic system of matter which itself obeys the actual material laws. The sole spacial relations of the æther itself, on which its dynamics depend, those namely of incompressibility and rotational elasticity, are thus to be classed along with the existing Euclidean relations of measurements in space (which also might a priori be different from what they are) as part of the ultimate scheme of mental representation of the actual physical world. The elastic and other characteristics of ordinary matter, including its viscous relations, are on the other hand a direct consequence of its molecular constitution, in combination with the law of material energy which is itself a consequence of the fact that the energies of the atoms are wholly located in the surrounding simple continuous æther and are thus functions of their mutual configurations. In this way we come round again to an order of procedure similar to that by which and  originally based the elastic relations of material bodies on the mutual actions of their constituent molecules.

Consider any two portions of matter which have a potential-energy function depending, as above explained, on their mutual configuration alone, the material movements being thus comparatively slow compared with the velocity of radiation; any displacement of them as a single rigid system, whether translational or rotational, can involve no expenditure of work; hence the resultant forcive exerted by the first system on the second must statically equilibrate that exerted by the second system on the first, these forcives must in fact be equal and opposite wrenches on a common axis; and the energy principle thus involves the principle of the balance of action and reaction, in its most general form. This stress, between two molecules, is usually sensible only at molecular range; hence the action of the surrounding parts on a portion of a solid body is practically made up of tractions exerted over the interface between them. Further, since rotation of the body without deformation cannot alter the potential energy of mutual configuration of the molecules, it follows that for a rectangular element of ordinary solid matter the tangential components of these tractions must be self-conjugate, as they are taken to be in the ordinary theory of elasticity. On the other hand, for a medium not molecularly constituted we can hardly treat at all of mutual configuration of parts, and the self-conjugate stress-relation will not be a necessary one.

A certain similarity may be traced with the view of, who was disinclined to allow that ray-vibrations are transmitted by any medium of the molecular character of ordinary matter, but considered them rather as affections of the lines which represent electric force, the propagation being influenced by the material nuclei which in ponderable media disturb these lines. This propagation in time requires inertia and elasticity for its mathematical expression, and the problem of the free æther is to find what kind of each is requisite.

9. A theory which, like the present one, explains atoms of matter as made up of singularities of strain and motion in the æther, is bound to look for an explanation of gravitation by means of the properties of that medium; it cannot avail itself of 's dogma that gravitation at a distance is itself as fundamental and intelligible as any explanation thereof could be. In further development of the illustrative possibilities of the pulsatory theory of gravitation, mentioned in the previous papers, we can (ideally) imagine the pulsation to have been applied initially over the outside boundary of the æthereal universe, and thence instantaneously communicated throughout the incompressible medium to the only places that can respond to it, the vacuous nuclei of the electrons; and we can even imagine the pulsations thus established as spontaneously keeping time and phase ever after, when the exciting cause which established this harmony has been discontinued.

It has been noticed in Part I, § 103, that gravitation cannot be transmitted by any action of the nature of statical stress; for then the approach of two atoms would increase the strain, and therefore also the stress, and therefore also in a higher ratio the energy of strain which depends on their product, and hence the mutual forces of the atoms would resist approach. As gravitation must belong to the ultimate constituents of matter, that is on this theory to the electrons, and must be isotropic all round each of them, it would appear that no mediate æthereal representation of it is possible except the one here considered. The radially vibrating field might be described formally as the magnetic field of the electron considered as a unipolar magnet, necessarily of very rapidly alternating type because otherwise a field of gravitation would be an ordinary magnetic field. The bare groundwork of this hypothesis may thus be formally expressed in 's language and developed along his lines, by postulating that the electron is not only a centre of steady intrinsic electric force, but also a centre of alternating intrinsic magnetic force, instantaneously transmitted because it would otherwise involve condensation, each force being necessarily radial. The unsatisfactory feature is that this radial quasi-magnetic field is introduced for the sake of gravitation alone, which does not present itself as in any direct correlation with other physical agencies.

The following sections are occupied chiefly with an attempt to logically systematize, and in various respects extend, the electric aspect of molecular theory. The preceding paper dealt mainly with the molecular side of directly æthereal phenomena, such as electric and radiative fields; of the present one the earlier part follows up the same subject, and the remainder relates to the actions of the molecules of polarized material bodies on one another, and the material stresses and physical changes thereby produced. As in the preceding papers, the quantitative results are to a large extent independent of any special theory of the constitution of matter, such as is here employed to bind together and harmonize the separate groups of phenomena, and to form a mental picture of their mutual relations; so far as they are electric they may be based directly on 's equations of the electric field in free space, which form a sufficient description of the free æther, and have been verified by experiment. In the theory, however, as usually expounded, an explanation of these equations is found, explicitly or tacitly, in an assumption that the æther is itself polarizable in the same manner as a material medium, and æther is in fact virtually considered to be matter; on the present theory the equations for free space are an analytical statement of the ultimate dynamical definition of the continuous æthereal medium, and the polarization of material bodies with the resulting forcive are deduced from the relation of their molecules to this medium in which they have their being.

10. In the modern treatment of material dynamics, as based on the principle of energy, the notion of configuration is, as above explained, fundamental. The potential energy, from which the forces are derived, is a function of the mutual configurations of the parts of the material system. In the case of forces of elasticity the internal energy is primarily a function of the mutual configurations of the individual molecules, from which a regular or organised part (§ 49 infra) is separated which is expressible in terms of the change of configuration of the differential element of volume containing a great number of molecules, and from which alone is derived the stress that is mechanically transmitted. In connexion with the discussion of contact action in § 6 above, the mode of this derivation and transmission becomes a subject of interest. In the first place the primary notion of a force as acting from one point to another in a straight line, has to be generalized into a forcive in 's manner on the basis of the principle of virtual work: then the forcive arising from the internal strain-energy of the element of volume of the material is derived by variation of this organized energy, and appears primarily as made up of definite complex bodily forcives resisting the various types of strain that occur in the element: then these forcives are rearranged, by the process of integration by parts, into a uniform translatory force acting throughout the element of volume of the material (which must compensate the extraneous applied bodily forcive) together with tractions acting over its surface. When this is done also for adjacent elements of volume, other tractions arise which must compensate the previous ones over the part of the surface that is common to the two elements; and thus the uncompensated traction is passed on from element to element until finally the boundary of the material system is reached where it remains uncompensated and must be balanced extraneously. The outstanding irregular part of the aggregate mutual potential energy of the individual molecules, which cannot be included in a function of strain of the element of volume, cannot on that account take part in the transmission of mechanical forces, and is evidenced only in local changes of the physical properties and temperature of the material. Cf. § 48 infra.

The other main division of the energy is the kinetic part, which is specified in terms of the rate of change of configuration of the material system with respect to an extraneous spacial framework to which its position is referred. Whatever notions may commend themselves a priori as to the impossibility of absolute space and absolute time, the fact remains that it has not been found possible to construct a system of dynamics which has respect only to the relative positions of moving bodies; and the reason suggests itself, that there is an underlying part of the phenomena, which does not usually explicitly appear in abstract material dynamics, namely, the æthereal medium, and that the spacial framework in absolute rest, which was introduced by and was probably a main source of the great advance in abstract dynamics originated by the Principia, is in fact the quiescent underlying æther. In this way the purely a priori standpoint is pushed away a stage, and we may find justification against the reproach that a philosophical formulation of dynamics should be concerned only with relative motions.

Relation to Gas-Theory: Internal Molecular Energy.
11. The kinetic theory of gases is considerably affected by the view here taken of the constitution of a molecule. In those simple and satisfactory features which are concerned only with the translatory motion of the molecules, it stands intact; but it is different with problems, like that of the ratio of the specific heats, which involve the internal energy. According to the usual hypothesis of the theory of gases, all the internal kinetic energy of the molecule is taken to be thermal and in statistical equilibrium, through encounters, with the translatory energy. But on the present view, the energy of the steady orbital motions in the molecule (including therein slow free precessions) makes up both the energy of chemical constitution and the internal thermal energy; while it is only when these steady motions are disturbed that the resulting vibration gives rise to radiation by which some of the internal energy is lost. The amount of internal energy can however never fall below the minimum that corresponds to the actual conserved rotational momenta of the molecule; this minimum is the energy of chemical combination of its ultimate constituents, while the excess above it actually existing is the internal thermal energy. The present view requires that the energy of chemical constitution shall be very great compared with the thermal energy; but for this very reason our means of chemical decomposition are limited, so that only a part of that energy is experimentally realizable. This being the case, the alteration produced by external disturbance in the state of steady internal motions of the molecule consists in the superposition on it of very slow free precessional motions, which have practically no influence on its higher free periods: and this explains why change of temperature has no influence on the positions of the lines in a spectrum. As a gas at high temperature must contain molecules with all amounts of internal thermal energy from nothing upwards, we should on the other hand, on the ordinary gas-theory, expect both a shift of the brightest part of a spectral line when the temperature is raised, and also a widening of its diffuse margin.

The ordinary encounters between the molecules will influence this thermal energy or energy of slow precessional oscillation, without disturbing the state of steady constitutive motion on which it is superposed, therefore without exciting radiation, which depends on more violent disturbances involving dissociative action.

On this view the postulates of the theorem on the distribution of the internal energy in gases would not obtain, for the thermal energy of the molecule would not be expressible as a sum of squares. The ratio of the specific heats in a gas must still lie between 1 and $$\tfrac{5}{3}$$; but the nature of the similarity of molecular constitution in the more permanent gases, which makes the ratio of the total thermal energy to the translatory energy either $$\tfrac{5}{3}$$ or unity for most of them, would remain to be discovered. In those gases for which the latter value obtains, the energy of precessional motion in the molecule would be negligibly small, involving small resultant angular momentum and possibly small paramagnetic moment.

The necessity of a distinction such as that here drawn between the internal thermal energy and the energy of the vibratory disturbances of internal structure which maintain radiation, is well illustrated by the recent recognition (foreshadowed by and 's researches on the law of cooling) and application by  of the remarkable insulating power of a vacuum jacket as regards heat. If this distinction did not exist, both conduction and convection must ultimately depend on transfer by ordinary radiation at small distances, as Fourier imagined; and it would not appear why convection by a gas, even when highly rarefied, is so much more efficient in the transfer of heat than radiation.

12. The result obtained by and, and others, that all over the gas-liquid range the characteristic equations of the substances on which they experimented proved to be very approximately of the form $$p = aT + b$$, where $$a$$ and $$b$$ are functions of the density alone, also supplies corroboration to this view. Expressing the increment of energy per unit mass $$dE = Mdv + \kappa dt$$, we have for the increment of heat supplied $$dH = dE + pdv$$; and the fact that $$dE$$ and $$dH/T$$ are perfect differentials shows immediately that M is equal to $$-b$$ and $$\kappa$$ is independent of $$v$$, so that the total (non-constitutive) energy per unit mass consists of two independent parts, an energy of expansion and an energy of heating. The latter part is the thermal energy of the individual molecules; it is a function of their mean states and velocities alone, and constitutes almost all the energy in the gaseous state. The former part is the energy of mutual actions between the molecules; it is negative and bears a considerable ratio to the whole thermal energy in the liquid state, in the case of substances with high latent heats of evaporation; for all gases except hydrogen, inasmuch as they are cooled by transpiration through a porous plug, $$b$$ is negative at ordinary densities. Cf. § 62, infra.

There would be no warrant for a view that the forces of chemical affinity fall off and finally vanish as the ultimate zero of temperature is approached. The translatory motions of the molecules would diminish without limit, and therefore also the opportunities for reaction between them, so that many chemical changes would cease to take place for the same reason that a fire ceases to burn when the supply of air is insufficient, or coal gas ceases to explode when too much diluted with air: but the energies of affinity exist all the time in probably undiminished strength, while the forces of cohesion are modified by the fall of temperature but not necessarily in the direction of extinction.

The Equations of the Æthereal Field, with Moving Matter: various applications: influence of Motion through the Æther on the Dimensions of Bodies.
13. Let (u, v, w) represent the total circuital current, and (u, v', w') the conducted part of it, which will be taken to include the current (u$0$, v$0$, w$0$) of migration of the free electric charge as this is in all cases very small in comparison; let (f', g', h') denote the electric polarization of the material, and (f, g, h) the æthereal elastic displacement, so that the total circuital displacement of 's theory is their sum (f", g", h"); let the space of reference be fixed with respect to the stagnant æther, and (p, q, r) be the velocity with which the matter situated at the point (x, y, z) is moving, and let δ/dt represent $$d/dt+pd/dx+qd/dy+rd/dz$$; let (P, Q, R) denote the electric force, namely that which acts on the electrons, and (P', Q', R') the æthereal force, that which produces the æthereal electric displacement (f, g, h); let ρ denote density of free electric charge. Then the electromotive equations are

$\mathrm{P}=qc-rb-\frac{d\mathrm{F}}{dt}-\frac{d\Psi}{dx},\quad\mathrm{P}'=-\frac{d\mathrm{F}}{dt}-\frac{d\Psi}{dx},\quad f=\frac{1}{4\pi\mathrm{c}^{2}}\mathrm{P}',$|undefined

where

$\mathrm{F}=\int\frac{u}{r}d\tau+\int\left(\mathrm{B}\frac{d}{dz}-\mathrm{C}\frac{d}{dy}\right)\frac{1}{r}d\tau,\quad\alpha=\frac{d\mathrm{H}}{dy}-\frac{d\mathrm{G}}{dz};$|undefined

and

$u=u'+\frac{\delta f'}{dt}+\frac{df}{dt}+p\rho,$ [Added Sept. 14.—The term δf'/dt in u arises as follows. In addition to the change of the polarization in the element of volume, df/dt, there is the electrodynamic effect of the motion of the positive and negative electrons of the polar molecule. Now the movement of two connected positive and negative electrons is equivalent to that of a single positive electron round the circuit formed by joining together the ends of their paths: and a similar statement holds when there are more than two electrons in the molecule. Hence the motion of a polarized medium with velocity (p, q, r), which need not be constant from point to point, produces the electrodynamic effect of a magnetization ($rg'-qh',\ ph'-rf,\ qf'-pg'$) distributed throughout the volume: cf. Part I, § 125. And it has been shown in Part II, § 31 that any distribution of magnetism (A, B, C) may be represented as a volume distribution of electric current equal to curl (A, B, C), which is necessarily circuital, together with a surface current sheet equal to ($Bn-Cm,\ Cl-An,\ Am-Bl$). Thus, when (p, q, r) is uniform and (f, g', h') is circuital, the above magnetic distribution is equivalent to a current system ($pd/dx+qd/dy+rd/dz)\ (f,\ g',\ h')$ together with current sheets on interfaces of discontinuity: this system is to be added on to d/dt (f, g', h') in order to give the full electrodynamic effect. Thus in these special circumstances the formulation in the test is correct in so far as it leads to the correct differential equations for the element of the medium: the integral expression there given for F is however only correct either when it is reduced to the differential form $-\nabla^{2}\mathrm{F}/4\pi=u+d\mathrm{C}/dy-d\mathrm{B}/dz$, which is derivable on integration of its second term by parts, or else when, the velocity of the matter still being uniform, discontinuous interfaces are replaced in the analysis by gradual though rapid transitions. These conditions are satisfied in all the applications that follow: but they would not be satisfied for example in the problem of the reflexion of radiation from the surface of moving matter. But a formulation which is preferable to the above, in that it is absolutely general, is simply to implicitly include the above virtual magnetization directly in (A, B, C) and consequently change from δf'/dt to df'/dt in the expression for u: this will also involve that the relation $\mathrm{A}=\kappa\alpha$. which occurs lower down shall be replaced by $\mathrm{A}=\kappa\alpha+rg'-qh'$, but there will be no further alteration in the argument of the text. The boundary conditions of the text are unaltered.]

where

$\rho=\frac{d(f'+f)}{dx}+\frac{d(g'+g)}{dy}+\frac{d(h'+h)}{dz}.$

From the formula for (P, Q, R) Faraday's law follows that the line integral of electric force round a circuit in uniform motion with the matter is equal to the time-rate of diminution of the magnetic flux through its aperture. The line-integral of the æthereal force (P', Q', R') round a circuit fixed in the æther has the same value. Again if (F', G', H') be defined so that $$F'=\int u/r.d\tau$$, we have

$a=\frac{d\mathrm{H}'}{dy}-\frac{d\mathrm{G}'}{dz}+4\pi\mathrm{A}-\frac{d}{dx}\int\left(\mathrm{A}\frac{d}{dx}+\mathrm{B}\frac{d}{dy}+\mathrm{C}\frac{d}{dz}\right)\frac{1}{r}d\tau,$

so that

$\alpha+\frac{d\mathrm{V}'}{dx}=\frac{d\mathrm{H}'}{dy}-\frac{d\mathrm{G}'}{dz}$

where (α, β, γ) is magnetic force and V' is the potential of the magnetism: hence Ampère's law follows that the line integral of the magnetic force round any circuit is equal to 4π times the total current that flows through its aperture. These two circuital relations are coextensive with the previous equations involving the vector potential, and can thus replace them, when the difference between (P, Q, R) and (P', Q', R') is inessential, that is (i) when the displacement currents are negligible, or (ii) when the matter is at rest; the quantity Ψ then enters as an arbitrary function in the integration of the equations.

The mechanical force acting on the matter, or ponderomotive force, is (X, Y, Z) per unit volume, where (§ 38 infra)

$\mathrm{X}=\left(v-\frac{dg}{dt}\right)\gamma-\left(w-\frac{dh}{dt}\right)\beta+\mathrm{A}\frac{d\alpha}{dx}+\mathrm{B}\frac{d\alpha}{dy}+\mathrm{C}\frac{d\alpha}{dz}+f'\frac{d\mathrm{P}}{dx}+g'\frac{d\mathrm{P}}{dy}+h'\frac{d\mathrm{P}}{dz}+\rho\mathrm{P}.$|undefined

The mechanical traction on an interface will be considered later (§ 39). In a magnetic medium the magnetic force (α, β, γ) differs from the magnetic flux (a, b, c) simply by not including the influence of the local Amperean currents; thus $$\alpha=a-4\pi\mathrm{A}$$.

When there is no conductivity, the free charge must move along with the matter, so that

$\frac{d\rho}{dt}+\frac{d\rho p}{dx}+\frac{d\rho q}{dy}+\frac{d\rho r}{dz}=0;$

therefore, from the circuitality of the total current, we must have, identically,

$\frac{d\rho}{dt}=\frac{d}{dx}\left(\frac{\delta f'}{dt}+\frac{df}{dt}\right)+\frac{d}{dy}\left(\frac{\delta g'}{dt}+\frac{dg}{dt}\right)+\frac{d}{dx}\left(\frac{\delta h'}{dt}+\frac{dh}{dt}\right).$

The latter is the same as the convergence of $$(\delta/dt-d/dt)\ (f',\ g',\ h')$$, which asserts (for the case of uniform motion that is contemplated) that mere convection of the polarized medium does not produce separation of free electricity. The relation between (f, g', h') and (P, Q, R) must be such as to strictly satisfy this equation. The quantity Ψ occurs in the equations of the field as an undetermined potential which is sufficient in order to conserve the condition of bodily circuitality $$du/dx+dv/dy+dw/dz=0$$.

In order to express the conditions that must hold at an interface of transition, we notice that by definition F, G, H are continuous everywhere; but it is only when the media are non-magnetic that their rates of change along the normal (and therefore all their first differential coefficients) are also completely continuous. Across an interface the traction in the tether must be continuous, so that the tangential component of the æthereal force (P', Q', R') must be continuous, which is satisfied by continuity of Ψ. The continuity of the total electric current secures itself without further condition by a compensating distribution of electric charge on the interface, that is by a discontinuity in dΨ/dn. The tangential continuity of the elastic æther requires that the tangential component of the magnetic force (α, β, γ) must be continuous; the normal continuity of the magnetic flux is assured by the continuity of (F, G, H). It might be argued that if the electric force (P, Q, R) were not continuous tangentially, a perpetual motion could arise by moving an electron along one side of the interface and back again along the other side. But this reasoning requires that (p, q, r) shall be continuous across the interface, as otherwise the circuit returning on the other side could not be complete; and it also requires that there shall be no magnetization, as otherwise the mechanical force on the electrons in an element of volume, which is what we are really concerned with in the perpetual motion axiom, is different from the sum of the electric forces on the individual electrons, by involving (α, β, γ) instead of (a, b, c). We can thus assert continuity of the tangential electric force only in the cases in which it is already involved in that of the tangential æthereal force; and consistency is verified. The aggregate of all these interfacial electromotive conditions is thus continuity of the vector potential (F, G, H), and of Ψ, and of the tangential components of the magnetic force; they formally involve continuity of the tangential components of the æthereal force (P', Q', R'), and of the electric and magnetic fluxes. But further, in the equations from which 's circuital relation is derived above, it is only the normal space-variation of V' that is discontinuous; hence continuity of the tangential magnetic force is involved in that of F, G, H, Ψ by virtue of the mode of expression of (F, G, H) in terms of the currents and the magnetism. Thus there are in all cases only four independent interfacial conditions to be satisfied.

The scheme is thus far absolute, in the sense that the relations between the variables are independent of the special molecular constitution of the matter that is present. The system of equations must now be completed for material media by joining to it the relations which connect the conduction current in the matter with the electric force, and the electric polarization of the matter with the electric force, and the magnetic polarization of the matter with the magnetic force, in the cases in which these relations are definite and can be experimentally ascertained. In the simplest case of isotropic matter, polarizable according to a linear law, they are of types

$u'=\sigma\mathrm{P},\quad f'=(\mathrm{K}-1)/4\pi\mathrm{c}^{2}\mathrm{P},\quad\mathrm{A}=\kappa\alpha.$

The expression for ρ leads in homogeneous isotropic media to

$\mathrm{K}\nabla^{2}\Psi=-4\pi\mathrm{c}^{2}\rho+(\mathrm{K}-1)\left\{ d/dx(cq-br)+d/dy(ar-cp)+d/dz(bp-aq)\right\} $

so that Ψ is only in part an electrostatic potential. Inside a uniform isotropic conductor at rest, the condition of circuitality becomes $$\sigma\nabla^{2}\Psi=d\rho/dt$$; substituting this, we have $$d\rho/dt+4\pi c^{2}\sigma\mathrm{K}^{-1}\rho=0$$, so that $$\rho=\rho_{0}\exp(-4\pi\mathrm{c}^{2}\sigma\mathrm{K}^{-1}t)$$, showing that an initial volume density of free electricity would in that case be instantly driven to the boundary owing to the dielectric action. This proposition may be extended to aeolotropic media.

14. The nature of the foregoing electric scheme may be elucidated by aid of some simple applications.

(i.) When a conducting system is in steady motion so that there is no conduction current flowing into it, the electric force (P, Q, R) must be null throughout its substance. Thus for the case of a solid conductor rotating round an axis of symmetry in a uniform magnetic field parallel to that axis, with steady angular velocity ω, the electric force in it, namely $$(\omega cx-d\Psi_{1}/dx,\ \omega cy-d\Psi_{1}/dy,\ -d\Psi_{1}/dz)$$, must be null, so that $$\Psi_{1}=\tfrac{1}{2}\omega c(x^{2}+y^{2})+\mathrm{A}$$; the polarization in it is therefore null, but there is in it an æthereal displacement $$-(4\pi\mathrm{c}^{2})^{-1}(d/dx,\ d/dy,\ d/dz)\Psi_{1}$$. In outside space, the electric force and æthereal force are each $$-(d/dx,\ d/dy,\ d/dz)\Psi_{2}$$, where Ψ$0$ is that free electrostatic potential which is continuous with the surface value $$\tfrac{1}{2}\omega\mathrm{c}(x^{2}+y^{2})+\mathrm{A}$$ at the conductor. Inside the conductor this purely æthereal displacement involves an electrification of volume density $$\rho=-\omega c/2\pi\mathrm{c}^{2}$$, which will be a density of free electrons or ions as all true electrifications are; while there is a compensating surface density σ equal to the difference of the total normal electric displacements on the two sides, that is to $$(4\pi\mathrm{c}^{2})^{-1}(d\Psi_{2}/dn_{2}+d\Psi_{1}/dn_{1})$$, where dn$0$, dn$0$ are both measured towards the surface, the outside medium being air for which K is unity. The value of the constant A is determined by the circumstance that the aggregate of this volume and surface charge shall be null when the conductor is insulated and unelectrified, or equal to the given total charge when it is insulated and charged: when it is uninsulated, the constant is determined by the position of the point on it that is connected to Earth, and therefore at zero potential. The procedure of Part II., § 25 is thus justified, because there is in fact no dielectric polarization in the conductor, but only æthereal displacement.

It remains to consider whether the parts of this volume density ρ and surface density σ of electrification are carried round with the conductor in its motion, or slip back through its volume and over its surface so as to maintain fixed positions in space. It is clear (as in Part II., § 27) that the same cause, namely, viscous diffusion of momentum among moving ions and molecules, which produces Ohmic resistance to a steady current, will lead to the electrons constituting electric densities being wholly carried on by the matter whenever a steady state is attained. This necessary consequence of the theory is in keeping with Rowland's classical experiments on convection currents. The excessively minute magnetic field due to these convection currents themselves has been neglected in the above analysis, which has enabled us to specify the slight redistribution of free charge on the rotating conductor when under the influence of a powerful extraneous magnetic field: when the magnetic field is due solely to its own motion the redistribution is of course absolutely negligible.

(ii.) In the case of a dielectric (as also in the above) the restriction to a steady state and permanent configuration may be dispensed with; for the magnetic field arising from induced displacement currents can always be neglected in comparison with the inducing field. Thus, (a, b, c) being the extraneous inducing field, the electric forces inside and outside a rotating mass are

$(\omega cx-d\Psi_{1}/dx,\ \omega cy-d\Psi_{1}/dy,\ -\omega ax-\omega by-d\Psi_{1}/dz)$ and $-(d/dx,\ d/dy,\ d/dz)\Psi_{2}.$

As there can be no free electrification,

$\nabla^{2}\Psi_{1}=(1-\mathrm{K}^{-1})\omega\left\{ 2c+x(dc/dx-da/dz)-y(dc/dy-db/dz)\right\}$ and $\nabla^{2}\Psi_{2}=0;$

while at the surface

$\Psi_{1}=\Psi_{2}$, and $\mathrm{K}\ d\Psi_{1}/dn-(\mathrm{K}-1)\omega\left\{ cxl+cym-(ax+by)n\right\} =d\Psi_{2}/dn,$

the outside medium being air. If the dielectric body is a sphere rotating in a uniform field (0, 0, c) parallel to the axis, this gives by the usual harmonic analysis $$\Psi_{1}=\tfrac{1}{3}(1-\mathrm{K}^{-1})\omega cr^{2}+\mathrm{A}r\ \cos\theta+\mathrm{A}'$$ and $$\Psi_{2}+\mathrm{B}r^{-2}\cos\theta+\mathrm{B}'r^{-1}\ $$, where, r$2$ being the radius $$\mathrm{A}=\mathrm{B}/r_{1}^{3}=-3\mathrm{K}/(2\mathrm{K}+1)r_{1}.\mathrm{A}'=-\tfrac{3}{2}/(\mathrm{K}+2)r_{1}^{2}.\mathrm{B}'=(\mathrm{K}-1)/(\mathrm{K}+2).\omega cr_{1}$$; thus determining the electric potential Ψ$2$ in the space surrounding the rotating sphere.

15. More generally, let us consider steady distributions of electric charges on a system of conductors and dielectric bodies in motion through the æther. That there may be a steady state, without conduction currents, it is necessary that the configuration of the matter shall be permanent, and that its motion shall be the same at all times relative to this configuration and to the æther, and also to the extraneous magnetic field if there is one: this confines it to uniform spiral motion on a definite axis fixed in the æther. Referring to axes fixed in the material system, the vector potential has in the steady motion no time-variation: hence

$(\mathrm{P,Q,R})=-(d/dx,\ d/dy,\ d/dz)\mathrm{V},\quad(\mathrm{P',Q',R'})=(\mathrm{P}-qc+rb,\ \mathrm{Q}-ra+pc,\ \mathrm{R}-pb+qa).$

The magnetic induction through any circuit moving with the matter being constant, (P, Q, R) is derived (§12) from an electric potential function V. Inside a conductor the electric force must vanish, otherwise electric separation would be going on, therefore V must there be constant.

When the surrounding dielectric is free space, the total current in it, referred to these axes moving with the matter, is $$-(pd/dx+qd/dy+rd/dz)\ (f,\ g,\ h)$$. When the velocity (p, q, r) of the matter is uniform, it then follows from Ampère's circuital relation that $$(a,\ b,\ c)=4\pi(qh-rg, rf-ph, pg-qf)$$. Hence (f, g, h), given by $$4\pi\mathrm{c}^{2}f=\mathrm{P}-qc+rb$$, is expressed in terms of (P, Q, R) by equations of type $$\left(c^{2}-p^{2}-q^{2}-r^{2}\right)f=\mathrm{P}/4\pi-p/4\pi\mathrm{c}^{2}(p\mathrm{P}+q\mathrm{Q}+r\mathrm{R})$$. The circuital quality of (f, g, h) thus gives the characteristic equation of the single independent variable V of the problem, in the form $$\nabla^{2}\mathrm{V}=c^{-2}(pd/dx+qd/dy+rd/dz)^{2}\mathrm{V}$$, the boundary condition being that V is constant over each conductor.

Thus in the case of a system of conductors moving steadily through space with uniform velocity v in the direction of the axis of x, ε denoting $$\left(1-v^{2}/c^{2}\right){}^{-1}$$ we have $$(f,\ g,\ h)=(4\pi\mathrm{c}^{2})^{-1}(\mathrm{P},\ \epsilon\mathrm{Q},\ \epsilon\mathrm{R})$$, and therefore $$(d^{2}/dx^{2}+\epsilon\ d^{2}/dy^{2}+\epsilon\ d^{2}dz^{2})\mathrm{V}=0$$. The distribution of electric force is therefore precisely the same as if the system were at rest, and the isotropic dielectric constant unity of the surrounding space changed into an aeolotropic one (1, ε ε), cf. Part I. §115; and so would the surface density of true charge, which is the superficial discontinuity of total displacement, be the same, were it not that there is æthereal displacement inside the conductors which must be subtracted. The internal displacement current thence arising is $$-(4\pi\mathrm{c}^{2})^{-1}vd/dx(0,-vc,vb)$$; hence (a, b, c) is of the form $$\left\{ d/dx,\ (1+v^{2}/c^{2})^{-1}d/dy,\ (1+v^{2}/c^{2})^{-1}d/dz\right\} \phi$$, by Ampère's circuital relation: the circuitality of (a, b, c) then leads to a characteristic equation for &Phi;, which must be solved so as to give at the surface of the conductor a value for the normal component of (a, b, c) continuous with the already known outside value, and the internal displacement is thereby determined. There is no bodily electrification inside the conductors, since this displacement is circuital.

We can restore the above characteristic equation of V, the potential of the electric force, to an isotropic form by a geometrical strain of the system and the surrounding space, represented by $$(x',\ y',\ z')=\left(\epsilon^{\tfrac{1}{2}}x,\ y,\ z\right)$$: the actual distribution of potential around the original system in motion corresponds then to that isotropic distribution of potential round the new system at rest which has the same values over the conductors. The æthereal displacements through related elements of area δS and δS', of direction cosines (l, m, n) and (l', m', n') in the two spaces, multiplied by 4πc², will be

$-(ld/dx+\epsilon\ md/dy+\epsilon\ nd/dz)V\delta S$ and $-(l'd/dx'+m'd/dy'+n'd/dz')V'\delta S';$

of these the second is always $$\epsilon^{-\tfrac{1}{2}}$$ times the first; thus the elements of surface for which the total displacement is null correspond in the two systems, and therefore the lines and tubes of total displacement also correspond, the flux of displacement in these tubes being $$\epsilon^{-\tfrac{1}{2}}$$ times greater in the second system than in the first. But on account of the æthereal displacement in the interior, the outside tubes do not mark out the distribution of the charge on each conductor. If then a system of charged conductors has a velocity of uniform translation v through the æther: and an auxiliary system at rest is imagined consisting of the original system and its space each uniformly expanded in the ratio $$\epsilon^{\tfrac{1}{2}}$$ or $$\left(1-v^{2}/c^{2}\right){}^{\tfrac{1}{2}}$$ in the direction of the motion, and the charges on this new system are $$\epsilon^{\tfrac{1}{2}}$$ times those on the actual system: then the fields of æthereal displacement of the two systems agree in the surrounding spaces so as to be the same across corresponding areas, but the distributions of the charges on the conductors do not thus exactly correspond. [These results are obtained on the supposition that the structure of the matter is not affected by its motion. The conductors on which these charges are situated will, however, if the results of the more fundamental analysis of §16 are admitted, change their actual forms to a slight extent depending on (v/c)² when they are put in motion, and this change will react so that the distribution of charges and displacements will be the simple one there given.]

16. The circumstances of propagation of radiation in a material medium moving with uniform velocity v parallel to the axis of x will form another example. We may here (§13) employ the circuital relations, of types

$4\pi u=\frac{d\gamma}{dy}-\frac{d\beta}{dz},\quad\frac{\delta\alpha}{dt}=\frac{d\mathrm{R}}{dy}-\frac{d\mathrm{Q}}{dz}$|undefined

where

$u=\frac{df}{dt}+\frac{\delta f'}{dt},\quad(f',\ g',\ h')=\frac{K-1}{4\pi c^{2}}(\mathrm{P,\ Q,\ R}),\quad(f,\ g,\ h)=\frac{1}{4\pi c^{2}}(\mathrm{P,\ Q}+vc,\ \mathrm{R}-vb).$|undefined

There readily results, on eliminating the electric force (P, Q, R),

$4\pi(u,\ v,\ w)=curl(\alpha,\ \beta,\ \gamma),\quad\mathrm{D}^{2}/dt^{2}(a,\ b,\ c)=4\pi c^{2}curl(u,\ v,\ w),$

where

$\mathrm{D}^{2}/dt^{2}=d^{2}/dt^{2}+(\mathrm{K}-1)(d/dt+vd/dx)^{2};$

which agrees with the equation obtained in Part I. §124 and Part II. §13, leading to Fresnel's law of alteration of the velocity of propagation.

Now let us consider the free æther for which K and μ are unity, containing a definite system of electrons which are grouped into the molecules of a material body moving across the æther with uniform velocity v parallel to the axis of x; and let us remove the restriction to steadiness of § 15. We refer the equations of free æther, in which these electrons are situated, to axes moving with the body: the alteration thus produced in the fundamental æthereal equations

$4\pi d/dt.(f,\ g,\ h)=curl(a,\ b,\ c),\quad-d/dt.(a,\ b,\ c)=4\pi c^{2}curl(f,\ g,\ h)$

is change of d/dt into $$d/dt-v\ d/dx$$, leading to the forms

$4\pi d/dt.(f,\ g,\ h)=curl(a',\ b',\ c'),\quad-d/dt.(a,\ b,\ c)=4\pi c^{2}curl(f',\ g',\ h');$

where

$(a',\ b',\ c')=(a,\ b+4\pi vh,\ c-4\pi vg),\quad(f',\ g',\ h')=(f,\ g-vc/4\pi\mathrm{c}^{2},\ h+vb/4\pi\mathrm{c}^{2})$

from which eliminating the unaccented letters, neglecting (v/c)³, and writing as before ε for $$\left(1-v^{2}/c^{2}\right)^{-1}$$, we derive the system

$$\begin{array}{ccc} 4\pi\frac{df'}{dt}=\frac{dc'}{dy}-\frac{db'}{dz} & & -(4\pi\mathrm{c}^{2})^{-1}\frac{da'}{dt}=\frac{dh'}{dy}-\frac{dg'}{dz}\\ \\4\pi\epsilon\frac{dg'}{dt}=\frac{da'}{dz}-\left(\frac{d}{dx}+\frac{v}{c^{2}}\frac{d}{dt}\right)c' & & -(4\pi\mathrm{c}^{2})^{-1}\epsilon\frac{db'}{dt}=\frac{df'}{dz}-\left(\frac{d}{dx}+\frac{v}{c^{2}}\frac{d}{dt}\right)h'\\ \\4\pi\epsilon\frac{dh'}{dt}=\left(\frac{d}{dx}+\frac{v}{c^{2}}\frac{d}{dt}\right)b'-\frac{da'}{dy} & & -(4\pi\mathrm{c}^{2})^{-1}\epsilon\frac{dc'}{dt}=\left(\frac{d}{dx}+\frac{v}{c^{2}}\frac{d}{dt}\right)g'-\frac{df'}{dy}.\end{array}$$

Now change the time variable from t to t', equal to $$t-vx/c^{2}$$, so that $$d/dx+v/c^{2}-d/dt$$ becomes d/dx, and d/dt becomes d/dt', and these equations assume the form of an electric scheme for a crystalline medium at rest. Finally write x$1$ for $$x\epsilon^{\tfrac{1}{2}}$$, a$1$ for $$a'\epsilon^{-\tfrac{1}{2}}$$, f$2$ for $$f'\epsilon^{-\tfrac{1}{2}}$$, dt$1$ for $$dt'\epsilon^{-\tfrac{1}{2}}$$, keeping the other variables unchanged, and the system comes back to its original isotropic form for free æther. Thus the final variables (f$1$, g$1$, h$1$) and (a$1$, b$1$, c$1$) will represent the æthereal field for a correlative system of electrons forming the molecules of another material system at rest in the æther, of the form of the original one pulled out uniformly in the ratio $$\epsilon^{\tfrac{1}{2}}$$ along its direction of movement; the electric displacements through corresponding areas in the two systems are not equal, but their molecules are composed of equal electrons and are situated at corresponding points, and the individual electrons describe corresponding parts of their orbits in times shorter for the latter system in the ratio $$\epsilon^{-\tfrac{1}{2}}$$ or $$\left(1-\tfrac{1}{2}v^{2}/c^{2}\right)$$ while those less advanced in the direction of v are also relatively very slightly further on in their orbits on account of the difference of time-reckoning. Thus we have here two correlative systems each governed by the circuital relations, of the free æther: (i) a system in which the electric and magnetic displacements are (f, g, h) and (a, b, c), moving steadily with uniform velocity v parallel to the axis of x, (ii) the same system expanded in the direction of x in the ratio $$\epsilon^{\tfrac{1}{2}}$$ and at rest, the displacements at the corresponding points being $$\left(\epsilon^{-\tfrac{1}{2}}f,\ g-vc/4\pi c^{2},\ h+vb/4\pi c^{2}\right)$$ and $$\left(\epsilon^{-\tfrac{1}{2}}a,\ b-4\pi vh,\ c+4\pi vg\right)$$, and the molecules being situated in the corresponding positions with due regard to the varying time-origin. Inasmuch as the circuital relations form a differential scheme of the first order which determines step by step the subsequent stages of a system when its initial state is given, it follows that if these two æthereal systems are set free at any instant in corresponding states, they will be in corresponding states at each subsequent instant, their electrons or singularities being at corresponding points. If then the latter collocation represent a fixed solid body, the former will represent the same body in uniform motion; one consequence of the motion thus being that the body is shrunk in the direction of its velocity v in the ratio $$\epsilon^{-\tfrac{1}{2}}$$ or $$1-\tfrac{1}{2}v^{2}/c^{2}$$. It may be observed that there is here no question of verifying that the mechanical forces acting on the single electrons in the two cases are such as to maintain this correspondence; for in the present complete survey of the individual atoms there is no such entity as mechanical force, any more than there is on a free vortex ring in fluid; the notion of mechanical forces enters at a subsequent stage when we are treating of molecular aggregates considered as continuous bodies, and are examining the relations between the different groups into which our senses analyze their interactions (§ 48).

If this argument is valid, it will confirm the hypothesis of and, to which they were led as the ultimate resource for the explanation of the negative result of 's optical experiments; and conversely it will involve evidence that the constitution of a molecule is wholly electric, as here represented. The reasoning given in Part II., § 13, was insufficient, because the correlation between the two systems was not there pushed to their individual molecules.