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 relative motions will lead to equations which are not linear—as, for example, those of hydrodynamics—and the phenomena will be far more complexly involved. It is true that the theory of vortex rings in hydrodynamics is of a simpler type; but electric currents cannot be likened to permanent vortex rings, because their circuits can be broken and the element of cyclic steadiness on which the simplicity depends is thereby destroyed.

Dynamical Theories of the Aether.—The analytical equations which represent the propagation of light in free aether, and also in aether modified by the presence of matter, were originally developed on the analogy of the equations of propagation of elastic effects in solid media. Various types of elastic solid medium have thus been invented to represent the aether, without complete success in any case. In T. MacCullagh’s hands the correct equations were derived from a single energy formula by the principle of least action; and while the validity of this dynamical method was maintained, it was frankly admitted that no mechanical analogy was forthcoming. When Clerk Maxwell pointed out the way to the common origin of optical and electrical phenomena, these equations naturally came to repose on an electric basis, the connexion having been first definitely exhibited by FitzGerald in 1878; and according as the independent variable was one or other of the vectors which represent electric force, magnetic force or electric polarity, they took the form appropriate to one or other of the elastic theories above mentioned.

In this place it must suffice to indicate the gist of the more recent developments of the electro-optical theory, which involve the dynamical verification of Fresnel’s hypothesis regarding optical convection and the other relations above described. The aether is taken to be at rest; and the strain-forms belonging to the atoms are the electric fields of the intrinsic charges, or electrones, involved in their constitution. When the atoms are in motion these strain-forms produce straining and unstraining in the aether as they pass across it, which in its motional or kinetic aspect constitutes the resulting magnetic field; as the strains are slight the coefficient of ultimate inertia here involved must be great. True electric current arises solely from convection of the atomic charges or electrons; this current is therefore not restricted as to form in any way. But when the rate of change of aethereal strain—that is, of (f,g,h) specified as Maxwell’s electric displacement in free aether—is added to it, an analytically convenient vector (u,v,w) is obtained which possesses the characteristic property of being circuital like the flow of an incompressible fluid, and has therefore been made fundamental in the theory by Maxwell under the name of the total electric current.

As already mentioned, all efforts to assimilate optical propagation to transmission of waves in an ordinary solid medium have failed; and though the idea of regions of intrinsic strain, as for example in unannealed glass, is familiar in physics, yet on account of the absence of mobility of the strain no attempt had been made to employ them to illustrate the electric fields of atomic charges. The idea of MacCullagh’s aether, and its property of purely rotational elasticity which had been expounded objectively by W. J. M. Rankine, was therefore much vivified by Lord Kelvin’s specification (Comptes Rendus, 1889) of a material gyrostatically constituted medium which would possess this character. More recently a way has been pointed out in which a mobile permanent field of electric force could exist in such a medium so as to travel freely in company with its nucleus or intrinsic charge—the nature of the mobility of the latter, as well as its intimate constitution, remaining unknown.

A dielectric substance is electrically polarized by a field of electric force, the atomic poles being made up of the displaced positive and negative intrinsic charges in the atom: the polarization per unit volume (f&#8202;′,g′,h′) may be defined on the analogy of magnetism, and d/dt(f&#8202;′,g′,h′) thus constitutes true electric current of polarization, i.e. of electric separation in the molecules, specified per unit volume. The convection of a medium thus polarized involves electric disturbance, and therefore must contribute to the true electric current; the determination of this constituent of the current is the most delicate point in the investigation. The usual definition of the component current in any direction, as the net amount of electrons which crosses, towards the positive side, an element of surface fixed in space at right angles to that direction, per unit area per unit time, here gives no definite result. The establishment and convection of a single polar atom constitutes in fact a quasi-magnetization, in addition to the polarization current as above defined, the negative poles completing the current circuits of the positive ones. But in the transition from molecular theory to the electrodynamics of extended media, all magnetism has to be replaced by a distribution of current; the latter being now specified by volume as well as by flow so that (u,v,w) is the current in the element of volume. In the present case the total dielectric contribution to this current works out to be the change per unit time in the electric separation in the molecules of the element of volume, as it moves uniformly with the matter, all other effects being compensated molecularly without affecting the propagation. On subtracting from this total the current of establishment of polarization d/dt/(f&#8202;′,g′,h′) as formulated above, there remains vd/dx(f&#8202;′,g′,h′) as the current of convection of polarization when the convection is taken for simplicity to be in the direction of the axis of x with velocity v. The polarization itself is determined from the electric force (P,Q,R) by the usual statical formula of linear type which becomes for an isotropic medium

(f&#8202;′,g′,h′)＝$K−1⁄4c^{2}$(P,Q,R),

because any change of the dielectric constant K arising from the convection of the material through the aether must be independent of the sign of v and therefore be of the second order. Now the electric force (P,Q,R) is the force acting on the electrons of the medium moving with velocity v; consequently by Faraday’s electrodynamic law

(P,Q,R)＝(P′,Q′ − vc, R′+ vb)

where (P′, Q′, R′) is the force that would act on electrons at rest, and (a,b,c) is the magnetic induction. The latter force is, by Maxwell’s hypothesis or by the dynamical theory of an aether pervaded by electrons, the same as that which strains the aether, and may be called the aethereal force; it thereby produces an aethereal electric displacement, say (f,g,h), according to the relation

(f,g,h)＝(42) − (P′, Q′, R′),

in which is a constant belonging to the aether, which turns out to be the velocity of light. The current of aethereal displacement d/dt(f,g,h) is what adds on to the true electric current to produce the total circuital current of Maxwell.

We have now to substitute these data in the universally valid circuital relations—namely, (i) line integral of magnetic force round a circuit is equal to 4 times the current through its aperture, which may be regarded as a definition of the constitution of the aether and its relation to the electrons involved in it; and (ii) line integral of the electric force belonging to any material circuit (i.e. acting on the electrons situated on it which move with the velocity of the matter) is equal to minus the time-rate of change of the magnetic induction through that circuit as it moves with the matter, this being a dynamical consequence of the aethereal constitution assigned in (i).

We may now, as is somewhat the more natural course in the terrestrial application, take axes (x,y,z) which move with the matter; but the current must be invariably defined by the flux across surfaces fixed in space, so that we may say that relation (i) refers to a circuit fixed in space, while (ii) refers to one moving with the matter. These circuital relations, when expressed analytically, are then for a dielectric medium of types