A History of the Theories of Aether and Electricity/Chapter 9

by Edmund Taylor Whittaker

Chapter IX.

Models of the Aether.

The early attempts of Thomson and Maxwell to represent the electric medium by mechanical models opened up a new field of research, to which investigators were attracted as much by its intrinsic fascination as by the importance of the services which it promised to render to electric theory.

Of the models to which reference has already been made, some—such as those described in Thomson's memoir^[1] of 1847 and Maxwell's memoir^[2] of 1861-2—attribute a linear character to electric force and electric current, and a rotatory character to magnetism; others—such as that devised by Maxwell in 1855^[3] and afterwards amplified by Helmholtz^[4]—regard magnetic force as a linear and electric current as a rotatory phenomenon. This distinction furnishes a natural classification of models into two principal groups.

Even within the limits of the former group diversity has already become apparent; for in Maxwell's analogy of 1861-2, a continuous vortical motion is supposed to be in progress about the lines of magnetic induction; whereas in Thomson's analogy the vector-potential was likened to the displacement in an elastic solid, so that the magnetic induction at any point would be represented by the twist of an element of volume of the solid from its equilibrium position; or, in symbols,

a = e, E = -e, B = curl e

where a denotes the vector-potential, E the electric force, B the magnetic induction, and e the elastic displacement.

Thomson's original memoir concluded with a notice of his intention to resume the discussion in another communication His purpose was fulfilled only in 1890, when^[5] he showed tha in his model a linear current could be represented by a piece of endless cord, of the same quality as the solid and embedded in it, if a tangential force were applied to the cord uniformly all round the circuit. The forces so applied tangentially produce a tangential drag on the surrounding solid; and the rotatory displacement thus caused is everywhere proportional to the magnetic vector.

In order to represent the effect of varying permeability, Thomson abandoned the ordinary type of elastic solid, and replaced it by an aether of MacCullagh's type; that is to say, an ideal incompressible substance, having no rigidity of the ordinary kind (i.e. elastic resistance to change of shape), but capable of resisting absolute rotation-a property to which the name gyrostatic rigidity was given. The rotation of the solid representing the magnetic induction, and the coefficient of gyrostatic rigidity being inversely proportional to the permeability, the normal component of magnetic induction will be continuous across an interface, as it should be.^[6]

We have seen above that in models of this kind the electric force is represented by the translatory velocity of the medium. It might therefore be expected that a strong electric field would perceptibly affect the velocity of propagation of light; and that this does not appear to be the case,^[7] is an argument against the validity of the scheme.

We now turn to the alternative conception, in which electric phenomena are regarded as rotatory, and magnetic force is represented by the linear velocity of the medium; in symbols,

${\begin{cases}4\pi \mathbf {D} ={\text{curl }}\mathbf {e} ,\\\mathbf {H} ={\dot {\mathbf {e} }}\end{cases}}$

where D denotes the electric displacement, H the magnetic force, and e the displacement of the medium. In Maxwell's memoir of 1855, and in most of the succeeding writings for many years, attention was directed chiefly to magnetic fields of a steady, or at any rate non-oscillatory, character; in such fields, the motion of the particles of the medium is continuously progressive; and it was consequently natural to suppose the medium to be fluid.

Maxwell himself, as we have seen,^[8] afterwards abandoned this conception in favour of that which represents magnetic phenomena as rotatory. "According to Ampère and all his followers," he wrote in 1870,^[9] "electric currents are regarded as a species of translation, and magnetic force as depending on rotation. I am constrained to agree with this view, because the electric current is associated with electrolysis, and other undoubted instances of translation, while magnetism is associated with the rotation of the plane of polarization of light."

But the other analogy was felt to be too valuable to be altogether discarded, especially when in 1858 Helmholtz extended it^[10] by showing that if magnetic induction is compared to fluid velocity, then electric currents correspond to vortex-filaments in the fluid. Two years afterwards Kirchhoff^[11] developed it further. If the analogy has any dynamical (as distinguished from a merely kinematical) value, it is evident that the ponderomotive forces between metallic rings carrying electric currents should be similar to the ponderomotive forces between the same rings when they are immersed in an infinite incompressible fluid; the motion of the fluid being such that its circulation through the aperture of each ring is proportional to the strength of the electric current in the corresponding ring. In order to decide the question, Kirchhoff' attempted, and solved, the hydrodynamical problem of the motion of two thin, rigid rings in an incompressible frictionless fluid, the fluid motion being irrotational; and found that the forces between the rings are numerically equal to those which the rings would exert on each other if they were traversed by electric currents proportional to the circulations.

There is, however, an important difference between the two cases, which was subsequently discussed by W. Thomson, who pursued the analogy in several memoirs.^[12] In order to represent the magnetic field by a conservative dynamical system, we shall suppose that it is produced by a number of rings of perfectly conducting material, in which electric currents are circulating; the surrounding medium being free aether. Now any perfectly conducting body acts as an impenetrable barrier to lines of magnetic force; for, as Maxwell showed,^[13] when a perfect conductor is placed in a magnetic field, electric currents are induced on its surface in such a way as to make the total magnetic force zero throughout the interior of the conductor.^[14] Lines of force are thus deflected by the body in the same way as the lines of flow of an incompressible fluid would be deflected by an obstacle of the same form, or as the lines of flow of electric current in a uniform conducting mass would be deflected by the introduction of a body of this form and of infinite resistance. If, then, for simplicity we consider two perfectly conducting rings carrying currents, those lines of force which are initially linked with a ring cannot escape from their entanglement, and new lines cannot become involved in it. This implies that the total number of lines of magnetic force which pass through the aperture of each ring is invariable. If the coefficients of self and mutual induction of the rings are denoted by L₁, L₂, L₁₂, the electrokinetic energy of the system may be represented by

$T={\tfrac {1}{2}}(L_{1}i_{1}^{2}+2L_{12}i_{1}i_{2}+L_{2}i_{2}^{2})$,

where i₁, i₂, in denote the strengths of the currents; and the condition that the number of lines of force linked with each circuit is to be invariable gives the equations

$L_{1}i_{1}+L_{12}i_{2}={\text{ constant}}$,

$L_{12}i_{1}+L_{2}i_{2}={\text{ constant}}$.

It is evident that, when the system is considered from the point of view of general dynamics, the electric currents must be regarded as generalized velocities, and the quantities

$(L_{1}i_{1}+L_{12}i_{2})$ and $(L_{12}i_{1}+L_{2}i_{2})$

as momenta. The electromagnetic ponderomotive force on the rings tending to increase any coordinate x is ∂T/∂x. In the analogous hydrodynamical system, the fluid velocity corresponds to the magnetic force: and therefore the circulation through each ring (which is defined to be the integral ∫vds, taken round a path linked once with the ring) corresponds kinematically to the electric current; and the flux of fluid through each ring corresponds to the number of lines of magnetic force which pass through the aperture of the ring. But in the hydrodynamical problem the circulations play the part of generalized momenta; while the fluxes of fluid through the rings play the part of generalized velocities. The kinetic energy may indeed be expressed in the form

$K={\tfrac {1}{2}}(N_{1}\kappa _{1}^{2}+2N_{12}\kappa _{1}\kappa _{2}+N_{2}\kappa _{2}^{2})$,

where κ₁, κ₂, denote the circulations (so that κ₁ and κ₂ are proportional respectively to i₁, and i₂), and N₁, N₁₂, N₂, depend on the positions of the rings; but this is the Hamiltonian (as opposed to the Lagrangian) form of the energy-function,^[15] and the ponderomotive force on the rings tending to increase any coordinate x is - ∂K/∂x. Since ∂K/∂x is equal to ∂T/∂x, we see that the ponderomotive forces on the rings in any position in the hydrodynamical system are equal, but opposite, to the ponderomotive forces on the rings in the electric system.

The reason for the difference between the two cases may readily be understood, The rings cannot cut through the lines of magnetic force in the one system, but they can cut through the stream-lines in the other: consequently the flux of fluid through the rings is not invariable when the rings are moved, the invariants in the hydrodynamical system being the circulations. If a thin ring, for which the circulation is zero, is introduced into the fluid, it will experience no ponderomotive forces; but if a ring initially carrying no current introduced into a magnetic field, it will experience ponderomotive forces, owing to the electric currents induced in it by its motion.

Imperfect though the analogy is, it is not without interest. A bar-magnet, being equivalent to a current circulating in a wire wound round it, may be compared (as W. Thomson remarked) to a straight tube immersed in a perfect fluid, the fluid entering at one end and flowing out by the other, so that the particles of fluid follow the lines of magnetic force. If two such tubes are presented with like ends to each other, they attract; with unlike ends, they repel. The forces are thus diametrically opposite in direction to those of magnets; but in other respects the laws of mutual action between these tubes and between magnets are precisely the same.^[16]

Thomson, moreover, investigated^[17] the ponderomotive forces which act between two solid bodies immersed in a fluid, when one of the bodies is constrained to perform small oscillations. If, for example, a small sphere immersed in an incompressible fluid is compelled to oscillate along the line which joins its centre to that of a much larger sphere, which is free, the free sphere will be attracted if it is denser than the fluid; while if it is less dense than the fluid, it will be repelled or attracted according as the ratio of its distance from the vibrator to its radius is greater or less than a certain quantity depending on the ratio of its density to the density of the fluid. Systems of this kind were afterwards extensively investigated by C. A. Bjerknes.^[18] Bjerknes showed that two spheres which are immersed in an incompressible fluid, and which pulsate (i.e., change in volume) regularly, exert on each other (by the mediation of the fluid) an attraction, determined by the inverse square law, if the pulsations are concordant; and exert on each other a repulsion, determined likewise by the inverse square law, if the phases of the pulsations differ by half a period. It is necessary to suppose that the medium is incompressible, so that all pulsations are propagated instantaneously: otherwise attractions would change to repulsions and vice versa at distances greater than a quarter wave-length.^[19] If the spheres, instead of pulsating, oscillate to and fro in straight lines about their mean positions, the forces between them are proportional in magnitude and the same in direction, but

opposite in sign, to those which act between two magnets oriented along the directions of oscillation.^[20]

The results obtained by Bjerknes were extended by A. H. Leahy^[21] to the case of two spheres pulsating in an clastic medium; the wave-length of the disturbance being supposed large in comparison with the distance between the spheres. For this system Bjerknes' results are reversed, the law being now that of attraction in the case of unlike phases, and of repulsion in the case of like phases: the intensity is as before proportional to the inverse square of the distance.

The same author afterwards discussed^[22] the oscillations which may be produced in an elastic medium by the displacement, in the direction of the tangent to the crosssection, of the surfaces of tubes of small sectional area: the tubes either forming closed curves, or extending indefinitely in both directions. The direction and circumstances of the motion are in general analogous to ordinary vortex-motions in an incompressible fluid; and it was shown by Leahy that, if the period of the oscillation be such that the waves produced are long compared with ordinary finite distances, the displacement due to the tangential disturbances is proportional to the velocity due to vortex-rings of the same form as the tubular surfaces. One of these "oscillatory twists," as the tubular surfaces may be called, produces a displacement which is analogous to the magnetic force due to a current flowing in a curve coincident with the tube; the strength of the current being proportional to b²ω sin pt, where b denotes the radius of the twist, and ω sin pt its angular displacement. If the field of vibration is explored by a rectilineal twist of the same period as that of the vibration, the twist will experience a force at right angles to the plane containing the twist and the direction of the displacement which would exist if the twist were removed; if the displacement of the medium be represented by F sin pt, and the angular displacement of the twist by ω sin pt, the magnitude of the force is proportional to the vector-product of F (in the direction of the displacement) and ω (in the direction of the axis of the twist).

A model of magnetic action may evidently be constructed on the basis of these results. A bar-magnet must be regarded as vibrating tangentially, the direction of vibration being parallel to the axis of the body. A cylindrical body carrying a current will have its surface also vibrating tangentially; but in this case the direction of vibration will be perpendicular to the axis of the cylinder. A statically electrified body, on the other hand, may, as follows from the same author's earlier work, be regarded as analogous to a body whose surface vibrates in the normal direction.

We have now discussed models in which the magnetic force is represented as the velocity in a liquid, and others in which it is represented as the displacement in an elastic solid. Some years before the date of Leahy's memoir, George Francis FitzGerald (b. 1851, d. 1901)^[23] had instituted a comparison between magnetic force and the velocity in a quasi-elastic solid of the type first devised by MacCullagh.^[24] An analogy is at once evident when it is noticed that the electromagnetic equation

$4\pi {\dot {\mathbf {D} }}={\text{curl }}\mathbf {H}$

is satisfied identically by the values

${\begin{cases}4\pi \mathbf {D} ={\text{curl }}\mathbf {e} ,\\\mathbf {H} ={\dot {\mathbf {e} }},\end{cases}}$

where e denotes, any vector; and that, on substituting these values in the other electromagnetic equation,

$-{\text{curl}}(4\pi c^{2}\mathbf {D} /\epsilon )={\dot {\mathbf {H} }}$,    ·

we obtain the equation

$\epsilon {\ddot {\mathbf {e} }}+c^{2}{\text{ curl curl }}\mathbf {e} =0$,

which is no other than the equation of motion of MacCullagh's aether,^[25] the specific inductive capacity e corresponding to the reciprocal of MacCullagh's constant of elasticity. In the analogy thus constituted, electric displacement corresponds to the twist of the elements of volume of the aether; and electric charge must evidently be represented as an intrinsic rotational strain. Mechanical models of the electromagnetic field, based on FitzGerald's analogy, were afterwards studied by A. Sommerfeld,^[26] by R. Reiff,^[27] and by Sir J. Larmor.^[28] The last-named author^[29] supposed the electric charge to exist in the form of discrete electrons, for the creation of which he suggested the following ideal process^[30]:—A filament of aether, terminating at two nuclei, is supposed to be removed, and circulatory motion is imparted to the walls of the channel so formed, at each point of its length, so as to produce throughout the medium a rotational strain. When this has been accomplished, the channel is to be filled up again with aether, which is to be made continuous with its walls. When the constraint is removed from the walls of the channel, the circulation imposed on them proceeds to undo itself, until this tendency is balanced by the elastic resistance of the aether with which the channel has been filled up; thus finally the system assumes a state of equilibrium in which the nuclei, which correspond to a positive and a negative electron, are surrounded by intrinsic rotational strain.

Models in which magnetic force is represented by the velocity of an aether are not, however, secure from objection, It is necessary to suppose that the aether is capable of lowing like a perfect fluid in irrotational motion (which would correspond to a steady magnetic field), and that it is at the same time endowed with the power (which is requisite for the explanation of electric phenomena) of resisting the rotation of any element of volume.^[31] But when the aether moves irrotationally in the fashion which corresponds to a steady magnetic field, each element of volume acquires after a finite time a rotatory displacement from its original orientation, in consequence of the motion, and it might therefore be expected that the quasi-elastic power of resisting rotation would be called into play—i.e., that a steady magnetic field would develop electric phenomena.^[32]

A further objection to all models in which magnetic force corresponds to velocity is that a strong magnetic field, being in such models represented by a steady drift of the aether, might be expected to influence the velocity of propagation of light, The existence of such an effect appears, however, to be disproved by the experiments of Sir Oliver Lodge;^[33] at any rate, unless it is assumed that the aether has an inertia at least of the same order of magnitude as that of ponderable matter, in which case the motion might be too slow to be measurable.

Again, the evidence in favour of the rotatory as opposed to the linear character of magnetic phenomena has perhaps, on the whole, been strengthened since Thomson originally based his conclusion on the magnetic rotation of light. This brings us to the consideration of an experimental discovery.

In 1879 E. H. Hall,^[34] at that time a student at Baltimore, repeating an experiment which had been previously suggested by H. A. Rowland, obtained a new action of a magnetic field on electric currents. A strip of gold leaf mounted on glass, forming part of an electric circuit through which a current was passing, was placed between the poles of an electromagnet, the plane of the strip being perpendicular to the lines of magnetic force. The two poles of a sensitive galvanometer were then placed in connexion with different parts of the strip, until two points at the same potential were found. When the magnetic field was created or destroyed, a deflection of the galvanometer needle was observed, indicating a change in the relative potential of the two poles. It was thus shown that the magnetic field produces in the strip of gold leaf a new electromotive force, at right angles to the primary electromotive force and to the magnetic force, and proportional to the product of these forces.

From the physical point of view we may therefore regard Hall's effect as an additional electromotive force generated by the action of the magnetic field on the current; or alternatively we may regard it as a modification of the ohmic resistance of the metal, such as would be produced if the molecules of the metal assumed a helicoidal structure about the lines of magnetic force. From the latter point of view, all that is needed is to modify Ohm's law

$\mathbf {S} =k\mathbf {E}$

(where S denotes electric current, k specific conductivity, and E electric force) so that it takes the form

$\mathbf {S} =k\mathbf {E} +h[\mathbf {E.H} ]$

where H denotes the imposed magnetic force, and h denotes a constant on which the magnitude of Hall's phenomenon depends. It is a curious circumstance that the occurrence, in the case of magnetized bodies, of an additional term in Ohm's law, formed from a vector-product of E, had been expressly suggested in Maxwell's Treatise^[35]: although Maxwell had not indicated the possibility of realizing it by Hall's experiment.

An interesting application of Hall's discovery was made in the same year by Boltzmann,^[36] who remarked that it offered a prospect of determining the absolute velocity of the electric charges which carry the current the strip. For if it is supposed that only one kind (vitreous or resinous) of electricity is in motion, the force on one of the charges tending to drive it to ono side of the strip will be proportional to the vector-product of its velocity and the magnetic intensity. Assuming that Hall's phenomenon is a consequence of this tendency of charges to move to one side of the strip, it is evident that the velocity in question must be proportional to the magnitude of the Hall electromotive force due to a unit magnetic field. On the basis of this reasoning, 1. von Ettingshausen^[37] found for the current sent by one or two Daniell's cells through a gold strip a velocity of the order of 0·1 cm. per second. It is clear, however, that, if the current consists of both vitreous and resinous charges in motion in opposite directions, Boltzmann's argument fails; for the two kinds of electricity would give opposite directions to the current in Hall's phenomenon.

In the year following his discovery, Hall^[38] extended his researches in another direction, by investigating whether a magnetic field disturbs the distribution of equipotential lines in a dielectric which is in an electric field; but no effect could be observed.^[39] Such an effect, indeed,^[40] was not to be expected on theoretical grounds; for when, in a material system, all the velocities are reversed, the motion is reversed, it being understood that, in the application of this theorem to electrical theory, an electrostatic state is to be regarded as one of rest, and a current as a phenomenon of motion, and if such a reversal be performed in the present system, the poles of the electromagnet are exchanged, while in the dielectric no change takes place.

We must now consider the bearing of Hall's effect on the question as to whether magnetism is a rotatory or a linear phenomenon.^[41] If magnetism be linear, electric currents must be rotatory, and if Hall's phenomenon be supposed to take place in a horizontal strip of metal, the magnetic force being directed vertically upwards, and the primary current flowing horizontally from north to south, the only geometrical entities involved are the vertical direction and a rotation in the east-and-west vertical plane; and these are indifferent with respect to a rotation in the north-and-south vertical plane, so that there is nothing in the physical circumstances of the system to determine in which direction the secondary current shall flow, The hypothesis that magnetism is linear appears therefore to be inconsistent with the existence of Hall's effect.^[42] There are, however, some considerations which may be urged on the other side. Hall's effect, like the magnetic rotation of light, takes place only in ponderable bodies, not in free aether; and its direction is sometimes in one sense, sometimes in the other, according to the nature of the substance. It may therefore be doubted whether these phenomena are not of a secondary character, and the argument based on them invalid. Moreover, as FitzGerald remarked,^[43] the magnetic lines of force associated with a system of currents are circuital and have no open ends, making it difficult to imagine how alteration of rotation inside them could be produced.

Of the various attempts to represent electric and magnetic phenomena by the motions and strains of a continuous medium, none of those hitherto considered has been found free from objection.^[44] Before proceeding to consider models which are not constituted by a continuous medium, mention must be made of a suggestion offered by Riemann in his lectures^[45] of 1861. Riemann remarked that the scalar-potential φ and vector-potential a, corresponding to his own law of force between electrons, satisfy the equation

${\dot {\phi }}+{\text{div }}\mathbf {a} =0$;

an equation which, as we have seen, is satisfied also by the potentials of L. Lorenz^[46] This appeared to Riemann to indicate that φ might represent the density of an aether, of which a represents the velocity. It will be observed that on this hypothesis the electric and magnetic forces correspond to second derivates of the displacement—a circumstance which makes it somewhat difficult to assimilate the energy possessed by the electromagnetic field to the energy of the model.

We must now proceed to consider those models in which the aether is represented as composed of more than one kind of constituent: of these Maxwell's model of 1861-2, formed of vortices and rolling particles, may be taken as the type. Another device of the same class was described in 1885 by FitzGerald^[47]; this was constituted of a number of wheels, free to rotate on axes fixed perpendicularly in a plane board; the axes were fixed at the intersections of two systems of perpendicular lines; and each wheel was geared to each of its four neighbours by an indiarubber band. Thus all the wheels could rotate without any straining of the system, provided they all had the same angular velocity; but if some of the wheels were revolving faster than others, the indiarubber bands would become strained, It is evident that the wheels in this model play the same part as the vortices in Maxwell's model of 1861-2: their rotation is the analogue of magnetic force; and a region in which the masses of the wheels are largo corresponds to a region of high magnetic permeability. The indiarubber bands of FitzGerald's model correspond to the medium in which Maxwell's vortices were embedded; and a strain on the bands represents dielectric polarization, the line joining the tight and slack sides of any band being the direction of displacement. A body whose specific inductive capacity is large would be represented by a region in which the elasticity of the bands is feeble. Lastly, conduction may be represented by a slipping of the bands on the wheels.

Such a model is capable of transmitting vibrations analogous to those of light. For if any group of wheels be suddenly set in rotation, those in the neighbourhood will be prevented by their inertia from immediately sharing in the motion; but presently the rotation will be communicated to the adjacent wheels, which will transmit it to their neighbours; and so a wave of motion will be propagated through the medium. The motion constituting the wave is readily seen to be directed in the place of the wave, i.e. the vibration is transverse. The axes of rotation of the wheels are at right angles to the direction of propagation of the wave, and the direction of polarization of the bands is at right angles to both these directions.

The elastic bands may be replaced by lines of governor balls:^[48] if this be done, the energy of the system is entirely of the kinetic type.^[49]

Models of types different from the foregoing have been suggested by the researches of Helmholtz and W. Thomson on vortex-motion. The earliest attempts in this direction, however, were intended to illustrate the properties of ponderable matter rather than of the luminiferous medium. A vortex existing in a perfect fluid preserves its individuality throughout all changes, and cannot be destroyed; so that if, as Thomson^[50] suggested in 1867, the atoms of matter are constituted of vortex-rings in a perfect fluid, the conservation of matter may be immediately explained. The mutual interactions of atoms may be illustrated by the behaviour of smoke-rings, which after approaching each other closely are observed to rebound: and the spectroscopic properties of matter may be referred to the possession by vortex-rings of free periods of vibration.^[51]

There are, however, objections to the hypothesis of vortex-atoms. It is not easy to understand how the large density of ponderable matter as compared with aether is to be explained; and further, the virtual inertia of a vortex-ring increases as its energy increases; whereas the inertia of a ponderable body is, so far as is known, unaffected by changes of temperature. It is, moreover, doubtful whether vortex-atoms would be stable. "It now seems to me certain," wrote W. Thomson^[52] (Kelvin) in 1905, "that if any motion be given within a finite portion of an infinite incompressible liquid, originally at rest, its fate is necessarily dissipation to infinite distances with infinitely small velocities everywhere; while the total kinetic energy remains constant. After many years of failure to prove that the motion in the ordinary Helmholtz circular ring is stable, I came to the conclusion that it is essentially unstable, and that its fate must be to become dissipated as now described."

The vortex-atom hypothesis is not the only way in which the theory of vortex-motion has been applied to the construction of models of the aether, It was shown in 1880 by W. Thomson^[53] that in certain circumstances a mass of fluid can exist in a state in which portions in rotational and irrotational motion are finely mixed together, so that on a large scale the mass is homogeneous, having within any sensible volume an equal amount of vortex-motion in all directions. To a fluid having such a type of motion he gave the name vortex-sponge.

Five years later, FitzGerald^[54] discussed the suitability of the vortex-sponge as a model of the aether. Since vorticity in a perfect fluid cannot be created or destroyed, the modification of the system which is to be analogous to an electric field must be a polarized state of the vortex motion, and light must be represented by a communication of this polarized motion from one part of the medium to another. Many distinct types of polarization may readily be imagined: for instance, if the turbulent motion were constituted of vortex-rings, these might be in motion parallel to definite lines or planes; or if it were. constituted of long vortex filaments, the filaments might be bent spirally about axes parallel to a given direction. The energy of any polarized state of vortex-motion would be greater than that of the unpolarized state; so that if the motion of matter had the effect of reducing the polarization, there would be forces tending to produce that motion. Since the forces due to a small vortex vary inversely as a high power of the distance from it, it seems probable that in the case of two infinite planes, separated by a region of polarized vortex-motion, the forces due to the polarization between the planes would depend on the polarization, but not on the mutual distance of the planes—a property which characteristic of plane distributions whose elements attract according to the Newtonian law.

It is possible to conceive polarized forms of vortex-motion which are steady so far as the interior of the medium is concerned, but which tend to yield up their energy in producing motion of its boundary—a property parallel to that of the aether, which, though itself in equilibrium, tends to move objects immersed in it.

In the same year Hicks^[55] discussed the possibility of transmitting waves through a medium consisting of an incompressible fluid in which small vortex-rings are closely packed together. The wave-length of the disturbance was supposed large in comparison with the dimensions and mutual distances of the rings; and the translatory motion of the latter was supposed to be 30 slow that very many waves can pass over any one before it has much changed its position. Such a medium would probably act as a fluid for larger motions. The vibration in the wavefront might be either swinging oscillations of a ring about a diameter, or transverse vibrations of the ring, or apertural vibrations; vibrations normal to the plane of the ring appear to be impossible. Hicks determined in each case the velocity of translation, in terms of the radius of the rings, the distance of their planes, and their cyclic constant.

The greatest advance in the vortex-sponge theory of the aether was made in 1887, when W. Thomson^[56] showed that the equation of propagation of laminar disturbances in a vortex-sponge is the same as the equation of propagation of luminous vibrations in the aether. The demonstration, which in the circumstances can scarcely be expected to be either very simple or very rigorous, is as follows:—

Let (u, v, w) denote the components of velocity, and p the pressure, at the point (x, y, z) in an incompressible fluid. Let the initial motion be supposed to consist of a laminar motion {f(y), 0, 0}, superposed on a homogeneous, isotropic, and finegrained distribution (u′₀, v₀, w₀): so that at the origin of time the velocity is {f(y) + u′₀, v₀, w₀}: it is desired to find a function f(y, t) such that at any time t the velocity shall be {f(y, t) + u′, v, w}, where u′, v, w, are quantities of which every average taken over a sufficiently large space is zero.

Substituting these values of the components of velocity in the equation of motion

${\frac {\partial u}{\partial t}}=-u{\frac {\partial u}{\partial x}}-v{\frac {\partial u}{\partial y}}-w{\frac {\partial u}{\partial z}}-{\frac {\partial p}{\partial x}}$,

there results

${\frac {\partial f(y,t)}{\partial t}}+{\frac {\partial u^{\prime }}{\partial t}}=-f(y,t){\frac {\partial u^{\prime }}{\partial x}}-v{\frac {\partial f(y,t)}{\partial y}}-u^{\prime }{\frac {\partial u^{\prime }}{\partial x}}-v{\frac {\partial u^{\prime }}{\partial y}}$

$-w{\frac {\partial u^{\prime }}{\partial z}}-{\frac {\partial p}{\partial x}}.$

Take now the xz-averages of both members. The quantities ∂u′/∂t, ∂u′/∂x, v, ∂p/∂x have zero averages; so the equation takes the form

${\frac {\partial f(y,t)}{\partial t}}=-A.\left(u^{\prime }{\frac {\partial u^{\prime }}{\partial x}}+v{\frac {\partial u^{\prime }}{\partial y}}+w{\frac {\partial u^{\prime }}{\partial z}}\right)$,

if the symbol A is used to indicate that the xz-average is to be taken of the quantity following. Moreover, the incompressibility of the fluid is expressed by the equation

${\frac {\partial u^{\prime }}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0$

whence

$0=-A.\left(u^{\prime }{\frac {\partial u^{\prime }}{\partial x}}+u^{\prime }{\frac {\partial v}{\partial y}}+u^{\prime }{\frac {\partial w}{\partial z}}\right)$.

When this is added to the preceding equation, the first and third pairs of terms of the second member vanish, since the x-average of any derivate ∂Q/∂x vanishes if Q is finite for infinitely great values of x; and the equation thus becomes

${\frac {\partial f(y,t)}{\partial t}}=-A.{\frac {\partial u^{\prime }v}{\partial y}}.\qquad \qquad$(1)

From this it is seen that if the turbulent motion were to remain continually isotropic as at the beginning, f (y, t) would constantly retain its critical value f(y). In order to examine the deviation from isotropy, we shall determine A∂(u′v)/∂t, which may be done in the following way:—Multiplying the u- and v-equations of motion by v, u′ respectively, and adding, we have

$v{\frac {\partial f(y,t)}{\partial t}}+{\frac {\partial u^{\prime }v}{\partial t}}=-f(y,t){\frac {\partial u^{\prime }v}{\partial x}}-v^{2}{\frac {\partial f(y,t)}{\partial y}}-u^{\prime }{\frac {\partial u^{\prime }v}{\partial x}}$

$-v{\frac {\partial u^{\prime }v}{\partial y}}-w{\frac {\partial u^{\prime }v}{\partial z}}-v{\frac {\partial p}{\partial x}}-u^{\prime }{\frac {\partial p}{\partial x}}$

Taking the xz-average of this, we observe that the first term of the first member disappears, since A . v is zero, and the first term of the second member disappears, since A . ∂(u′v)/∂x is zero. Denoting by 1/3R² the average value of u², v², or ω², so that R may be called the average velocity of the turbulent motion, the equation becomes

${\frac {\partial }{\partial t}}\{A.(u^{\prime }v)\}=-{\tfrac {1}{3}}R^{2}{\frac {\partial f(y,t)}{\partial t}}-Q$,

where

$Q=A.\left\{u^{\prime }{\frac {\partial (u^{\prime }v)}{\partial x}}+v{\frac {\partial (u^{\prime }v)}{\partial y}}+w{\frac {\partial (u^{\prime }v)}{\partial z}}+v{\frac {\partial p}{\partial x}}+u^{\prime }{\frac {\partial p}{\partial y}}\right\}$.

Let p be written ($r^{\prime }+{\bar {\omega }}$), where p′ denotes the value which p would have if f were zero. The equations of motion immediately give

$-\nabla ^{2}p=\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial v}{\partial y}}\right)^{2}+\left({\frac {\partial w}{\partial z}}\right)^{2}+2{\frac {\partial v}{\partial z}}{\frac {\partial w}{\partial y}}+2{\frac {\partial w}{\partial x}}{\frac {\partial u}{\partial z}}+2{\frac {\partial u}{\partial y}}{\frac {\partial v}{\partial x}}$;

and on subtracting the forms which this equation takes in the two cases, we have

$-\nabla ^{2}{\bar {w}}=2{\frac {\partial f(y,t)}{\partial y}}{\frac {\partial v}{\partial x}}$,

which, when the turbulent motion is fine-grained, so that f(y, t) is sensibly constant over ranges within which u′, v, 'ω pass through all their values, may be written

${\bar {w}}=-2{\frac {\partial f(y,t)}{\partial y}}\nabla ^{-2}{\frac {\partial v}{\partial x}}$.

Moreover, we have

$O=A\left\{u^{\prime }{\frac {\partial (u^{\prime }v)}{\partial x}}+v{\frac {\partial (u^{\prime }v)}{\partial y}}+w{\frac {\partial (u^{\prime }v)}{\partial z}}+v{\frac {\partial p^{\prime }}{\partial x}}+u^{\prime }{\frac {\partial p^{\prime }}{\partial y}}\right\}$;

for positive and negative values of u′, v, ω are equally probable; and therefore the value of the second member of this equation is doubled by adding to itself what it becomes when for u′, v, ω we substitute -u′, -v, -ω; which (as may be seen by inspection of the above equation in Δ²p) does not change the value of p′. Comparing this equation with that which determines the value of Q, we have

$Q=A.\left(v{\frac {\partial {\bar {w}}}{\partial x}}+u^{\prime }{\frac {\partial {\bar {w}}}{\partial y}}\right)$,

or substituting for ${\bar {w}}$,

$Q=-2{\frac {\partial f(y,t)}{\partial y}}.A.\left(v{\frac {\partial }{\partial x}}+u^{\prime }{\frac {\partial }{\partial y}}\right).\nabla ^{-2}{\frac {\partial v}{\partial x}}$.

The isotropy with respect to x and z gives the equation

$2A.\left(v_{0}{\frac {\partial }{\partial x}}+u_{0}^{\prime }{\frac {\partial }{\partial y}}\right).\nabla ^{-2}{\frac {\partial v_{0}}{\partial x}}=A.\left\{v_{0}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)\right.\qquad \qquad$

$\left.+\left(u_{0}^{\prime }{\frac {\partial }{\partial x}}+w_{0}{\frac {\partial }{\partial z}}\right){\frac {\partial }{\partial y}}\right\}\nabla ^{-2}v_{0}$.

But by integration by parts we obtain the equation

$\left(u_{0}^{\prime }{\frac {\partial }{\partial x}}+w_{0}{\frac {\partial }{\partial z}}\right){\frac {\partial }{\partial y}}.\nabla ^{-2}v_{0}=-A\left({\frac {\partial u_{0}^{\prime }}{\partial x}}+{\frac {\partial w_{0}}{\partial z}}\right){\frac {\partial }{\partial y}}\nabla ^{-2}v_{0}$;

and by the condition of incompressibility the second member may be written

$A.(\partial v_{0}/\partial y).(\partial /\partial y).\nabla ^{-2}v_{0},$    or    $-A.v_{0}.(\partial ^{2}/\partial y^{2}).\nabla ^{-2}v_{0}$^{[errata 1]};

So we have

$Q_{0}={\frac {\partial f(y,t)}{\partial y}}A.\left\{v_{0}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}\right)\right\}.\nabla ^{-2}v_{0}$.

On account of the isotropy, we may write 1/3 for

$\left({\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)\nabla ^{-2}$;

so $Q_{0}={\frac {1}{9}}R^{2}{\frac {\partial f(y,t)}{\partial y}}$; and, therefore,

${\frac {\partial }{\partial t}}_{(}t=0)\{A.(e^{\prime }v)\}=-{\tfrac {2}{9}}R^{2}\left\{{\frac {\partial f(y,t)}{\partial y}}\right\}_{t=0}$.

The deviation from isotropy shown by this equation is very small, because of the smallness of ∂f(y,t)/∂y. The equation is therefore not restricted to the initial values of the two members, for we may neglect an infinitesimal deviation from (2/9) R² in the first factor of the second member, in consideration of the smallness of the second factor. Hence for all values of t we have the equation

${\frac {\partial }{\partial t}}A.(u^{\prime }v)={\tfrac {2}{9}}R^{2}{\frac {\partial f(y,t)}{\partial y}}$,

which, in combination with (1), yields the result

${\frac {\partial ^{2}}{\partial t^{2}}}f(y,t)={\tfrac {2}{9}}R^{2}{\frac {\partial ^{2}}{\partial y^{2}}}f(y,t)$;

the form of this equation shows that laminar disturbances are propagated through the vortex-sponge in the same manner as waves of distortion in a homogeneous elastic solid.

The question of the stability of the turbulent motion remained undecided; and at the time Thomson seems to have thought it likely that the motion would suffer diffusion. But two years later^[57] he showed that stability was ensured at any rate when space is filled with a set of approximately straight hollow vortex filaments. FitzGerald^[58] subsequently determined the energy per unit-volume in a turbulent liquid which is transmitting laminar waves. Writing for brevity

$(2/9)R^{2}=V^{2}$, $f(y,t)=P$, and $A(u^{\prime }v)=\gamma$,

the equations are

${\frac {\partial P}{\partial t}}=-{\frac {\partial \gamma }{\partial y}}$, and ${\frac {\partial \gamma }{\partial t}}=-V^{2}{\frac {\partial P}{\partial y}}$

If the quantity

$P^{2}+\gamma ^{2}/V^{2}=2\Sigma$

is integrated throughout space, and the variations of the integral with respect to time are determined, it is found that

${\frac {\partial }{\partial t}}\iiint \textstyle \sum \ dx\ dy\ dz=\iiint \left(P{\frac {\partial P}{\partial t}}+{\frac {\gamma }{V^{2}}}{\frac {\partial \gamma }{\partial t}}\right)\ dx\ dy\ dz$

$\iiint \left(P{\frac {\partial P}{\partial t}}-\gamma {\frac {\partial P}{\partial y}}\right)\ dx\ dy\ dz$

Integrating the second term under the integral by parts, and omitting the superficial terms (which may be at infinity, or wherever energy enters the space under consideration), we have

${\frac {\partial }{\partial t}}\iiint \Sigma \ dx\ dy\ dz=\iiint P\left({\frac {\partial P}{\partial t}}+{\frac {\partial \gamma }{\partial y}}\right)\ dx\ dy\ dz=0$.

Hence it appears that the quantity Σ, which is of the dimensions of energy, must be proportional to the energy per unit-volume of the medium—a result which shows that there is a pronounced similarity between the dynamics of a vortex-sponge and of Maxwell's elastic aether.

A definite vortex-sponge model of the aether was described by Hicks in his Presidential Address to the mathematical section of the British Association in 1895.^[59] In this the small motions whose function is to confer the quasi-rigidity were not completely chaotic, but were disposed systematically. The medium was supposed to be constituted of cubical elements of fluid, each containing a rotational circulation complete in itself: in any element, the motion close to the central vertical diameter of the element is vertically upwards: the fluid which is thus carried to the upper part of the element flows outwards over the top, down the sides, and up the centre again. In each of the six adjoining elements the motion is similar to this, but in the reverse direction. The rotational motion in the elements confers on them the power of resisting distortion, so that waves may be propagated through the medium as through an elastic solid; but the rotations are without effect on irrotational motions of the fluid, provided the velocities in the irrotational motion are slow compared with the velocity of propagation of distortional vibrations.

A different model was described four years later by FitzGerald.^[60] Since the distribution of velocity of a fluid in the neighbourhood of a vortex filament is the same as the distribution of magnetic force around a wire of identical form carrying an electric current, it is evident that the fluid has more energy when the filament has the form of a helix than when it is straight; so if space were filled with vortices, whose axes were all parallel to a given direction, there would be an increase in the energy per unit volume when the vortices were bent into a spiral form; and this could be measured by the square of a vector—say, E—which may be supposed parallel to this direction.

If now a single spiral vortex is surrounded by parallel straight ones, the latter will not remain straight, but will be bent by the action of their spiral neighbour. The transference of spirality may be specified by a vector H, which will be distributed in circles round the spiral vortex; its magnitude will depend on the rate at which spirality is being lost by the original spiral, and can be taken such that its square is equal to the mean energy of this new motion. The vectors E and H will then represent the electric and magnetic vectors; the vortex spirals representing tubes of electric force.

FitzGerald's spirality is essentially similar to the laminar motion investigated by Lord Kelvin, since it involves a flow in the direction of the axis of the spiral, and such a flow cannot take place along the direction of a vortex filament without a spiral deformation of a filament.

Other vortex analogues have been devised for electrostatical systems. Ono such, which was described in 1888 by W. M. Hicks,^[61] depends on the circumstance that if two bodies in contact in an infinite fluid are separated from each other, and if there be a vortex filament which terminates on the bodies, there will be formed at the point where they separate a hollow vortex filament^[62] stretching from one to the other, with rotation equal and opposite to that of the original filament. As the bodies are moved apart, the hollow vortex may, through failure of stability, dissociate into a number of smaller ones; and if the resulting number be very large, they will ultimately take up a position of stable equilibrium. The two sets of filaments—the original filaments and their hollow companions—will be intermingled, and each will distribute itself according to the same law as the lines of force between the two bodies which are equally and oppositely electrified.

Since the pressure inside a hollow vortex is zero, the portion of the surface on which it abuts experiences a diminution of pressure, the two bodies are therefore attracted. Moreover, as the two bodies separate further, the distribution of the filaments being the same as that of lines of electric force, the diminution of pressure for each line is the same at all distances, and therefore the force between the two bodies follows the same law as the force between two bodies equally and oppositely electrified. It may be shown that the effect of the original filaments is similar, the diminution of pressure being half as large again as for the hollow vortices.

If another surface were brought into the presence of the others, those of the filaments which encounter it would break off and rearrange themselves so that each part of a broken filament terminates on the new body. This analogy thus gives a complete account of electrostatic actions both quantitatively and qualitatively: the electric charge on a body corresponds to the number of ends of filaments abutting on it, the sign being determined by the direction of rotation of the filament as viewed from the body.

A magnetic field may be supposed to be produced by the motion of the vortex filaments through the stationary aether, the magnetic force being at right angles to the filament and to its direction of motion. Electrostatic and magnetic fields thus correspond to states of motion in the medium, in which, however, there is no bodily flow; for the two kinds of filament produce circulation in opposite directions.

It is possible that hollow vortices are better adapted than ordinary vortex-filaments for the construction of models of the aether. Such, at any rate, was the opinion of Thomson (Kelvin) in his later years.^[63] The analytical difficulties of the subject are formidable, and progress is consequently slow; but among the many mechanical schemes which have been devised to represent, electrical and optical phenomena, none possesses greater interest than that which pictures the aether as a vortex-sponge.

Notes

1. Cf. p. 270.
2. Cf. p. 276.
3. Cf. p. 271.
4. Cf. p. 274.
5. Kelvin's Math. and Phys. Papers, iii, p. 436.
6. Thomson inclined to believe (Papers, iii, p. 465) that light might be correctly represented by the vibratory motion of such a solid.
7. Wilberforce, Trans. Camb. Phil. Soc. xiv (1887), p. 170; Lodge, Phil. Trans. clxxxix (1897), p. 149.
8. Cf. p. 276.
9. Proc. Lond. Math. Soc. iii (1870), p. 224; Maxwell's Scient. Papers, ii, p. 263.
10. Cf. p. 274.
11. Journal für Matls. Ixxi (1869); Kirchhoff's Geramm. Abhandl., p. 404. Cf. also C. Neumann, Leipzig Berichte, xliv (1892), p. 86.
12. Thomson's Reprint of Papers in Elect. and Mag., §§ 573, 733, 751 (1870-1872).
13. Maxwell's Treatise on Elect. and Mag., § 664.
14. For this reason W. Thomson called a perfect conductor an ideal extreme diamagnetic.
15. Cf. Whittaker, Analytical Dynamics, § 109,
16. The mathematical analysis in this case is very simple. I narrow table through which water is flowing may be regarded as equivalent to a source at one end of the tube and a sink at the other; and the problem may therefore be reduced to the consideration of sinks in an unlimited fluid. If there are two sinks in such a fluid, of strengths m and m′, the velocity-potential is

    m/r + m′/r′,

    where r and r′ denote distance from the sinks. The kinetic energy per unit volume of the fluid is

    ${\tfrac {1}{2}}\rho \left[\left\{{\frac {\partial }{\partial x}}\left({\frac {m}{r}}+{\frac {m^{\prime }}{r^{\prime }}}\right)\right\}^{2}+\left\{{\frac {\partial }{\partial y}}\left({\frac {m}{r}}+{\frac {m^{\prime }}{r^{\prime }}}\right)\right\}^{2}+\left\{{\frac {\partial }{\partial z}}\left({\frac {m}{r}}+{\frac {m^{\prime }}{r^{\prime }}}\right)\right\}^{2}\right]$,

    where ρ denotes the density of the fluid; whence it is easily seen that she total energy of the fluid, when the two sinks are at a distance l apart, exceeds the total energy when they are at an infinite distance apart by an amount

    $\rho mm^{\prime }\iiint \left\{{\frac {\partial }{\partial x}}\left({\frac {1}{r}}\right){\frac {\partial }{\partial x}}\left({\frac {1}{r^{\prime }}}\right)+{\frac {\partial }{\partial y}}\left({\frac {1}{r}}\right){\frac {\partial }{\partial y}}\left({\frac {1}{r^{\prime }}}\right)+{\frac {\partial }{\partial z}}\left({\frac {1}{r}}\right){\frac {\partial }{\partial z}}\left({\frac {1}{r^{\prime }}}\right)\right\}\ dx\ dy\ dz$,

    the integration being taken throughout the whole volume of the fluid, except two small spheres s, s′, surrounding the sinks. By Green's theorem, this expression reduces at once to

    $\rho mm^{\prime }\iint {\frac {1}{r^{\prime }}}{\frac {\partial }{\partial n}}\left({\frac {1}{r}}\right)dS$,

    where the integration is taken over s and s′, and n devotes the interior portal to s or s′. The integral taken over s′ vanishes; evaluating the remaining integral, we have

    $-{\frac {\rho mm^{\prime }}{l}}\iint {\frac {\partial }{\partial r}}\left({\frac {1}{r}}\right)dS$,   or   4πρmm′/l.

    The energy of the fluid is therefore greater ben sinks of strengths m, m′ are at a mutual distance l than when sinks of the same strengths are at infinite distance apart by an amount 4πρmm′/l. Since, in the case of the tubes, the quantities m correspond to the fluxes of fluid, this expression corresponds to the Lagrangian form of the kinetic energy: and therefore the force tending to increase the coordinate x of one of the sinks is (∂/∂x) (4πρmm′/l). Whence it is seen that the like ends of two tubes attract, and the unlike ends repel, according to the inverse square law.

17. Phil. Mag. xli (1870), p. 427.
18. Repertorium d. Mathematik von Konisberger und Zeuner (1876), p. 268. Göttinger Nachrichten, 1876, p. 245. Comptes Rendus, lxxxiv (1877), p. 1377. Cf. Nature, xxiv (1881), p. 360.
19. On the mathematical theory of the force between two pulsating spheres in a fluid, cf. W. M. Hicks, Proc. Camb. Phil. Soc. iii (1879), p. 276: iv (1880), p. 29.
20. A theory of gravitation bas been based by Korn on the assumption that gravitating particles resemble slightly compressible spheres immersed in an incompressible perfect fluid: the spheres execute pulsations, whose intensity corresponds to the mass of the gravitating particles, and thus forces of the Newtonian kind are produced between them. Cf. Korn, Eine Theorie der Gravitation und der elect. Erscheinungen, Berlin, 1898.
21. Trans, Camb. Phil. Soc. xiv (1884), p. 45.
22. Trans. Camb. Phil. Soc. xiv (1885), p. 288.
23. Phil. Trans., 1880, p. 691 (presented October, 1878). FitzGerald's Scientific Writings, p. 45.
24. Cf. p. 155.
25. Cf. p. 155.
26. Ann. d. Phys, xlvi (1892), p. 139.
27. Reiff, Elasticität und Elektricität, Freiburg, 1893.
28. Phil. Trans. clxxxv (1893), p. 719.
29. In a supplement, of date August, 1894, to his above-cited memoir of 1893.
30. Phil. Trans. clxxxv (1894), p. 810; cxc (1897), p. 210; Larmor, Aether and Matter (1900), p. 326.
31. Larmor (loc. cit.) suggested the analogy of a liquid filled with magnetic molecules under the action of an external magnetic field.
    It has often been objected to the mathematical conception of a perfect fluid that it contains no safeguard against slipping between adjacent layers, so that there is no justification for the usual assumption that the motion of a perfect fluid is continuous. Larmor remarked that a rotational elasticity, such as is attributed to the medium above considered, furnishes precisely such a safeguard; and that without some property of this kind a continuous frictionless fluid cannot be imagined.
32. Larmor proposed to avoid this by assuming that the rotation which is resisted by an element of volume of the aether is the vector sum of the series of differential rotations which it has experienced.
33. Phil. Trans. clxxxix (1897), p. 149.
34. Am. Jour. Math. ii, p. 287; Am. J. Sci. xix, p. 200, and xx, p. 161; Phil. Mag. ix, p. 225, and x, p. 301.
35. Elect, and Mag., § 303. Cf. Hopkinson, Phil. Mag. x (1880), p. 430.
36. Wien Anz., 1880, p. 12. Phil. Mag. ix (1880), p. 307.
37. Ann, d. Phys. xi (1880), pp. 432, 1044.
38. Am. Jour, Sci, xx (1880), p. 164.
39. In 1885-6 E. van Aubel, Bull, de l'Acad. Roy, de Belgique (3) x, p. 609; xii, p. 280, repeated the investigation in an improved form, and confirmed the result that a magnetic field has no influence on the electrostatic polarization of dielectrics.
40. H. A. Lorentz, Arch. Neêrl. xix (1884), p. 123.
41. Cr. F. Kolácĕk, Ann. d. Phys. lv (1895), p. 503.
42. Further evidence in favour of the hypothesis that it is the electric phenomena which are linear is furnished by the fact that pyro-electric effects (the production of electric polarization by warming) occur in acentric crystals, and only in such. Cf. M. Abraham, Encyklopädie der math. Wiss. iv (2), p. 43.
43. Cf. Larmor, Phil. Trans. clxxxv, p.780.
44. Cf. H. Witte, Ueber den gegenwärtigen Stand der Frage nach einer mechanischen Erklärung der elektrischen Erscheinungen; Berlin, 1906.
45. Edited after his death by K. Hattendorff, under the title Schwere, Elektricitüt, und Magnetismus, 1875, p. 330.
46. Cf. p. 299.
47. Scient. Proc. Roy. Dublin Soc., 1885; Phil. Mag, June, 1885; FitzGerald's Scient. Writings, pp. 142, 157.
48. FitzGerald's Scient. Writings, p. 271.
49. It is of course possible to devise models of this clues in which the rotation nay he interpreted as having the electric instead of the magnetic character. Such a model was proposed by Boltzmann, Vorlesungen über Maxwell's Theorie, ii.
50. Phil. Mag. xxxiv (1867), p. 10; Proc. R.S. Edinb. vi, p. 94.
51. An attempt was made in 1883 by J. J. Thomson, Phil. Mag. xv (1883), p. 427, to explain the phenomena of the electric discharge through gases in terms of the theory of vortex-atoms. The electric field was supposed to consist in's distribution of velocity in the medium whose vortex-motion constituted the atoms of the gas; and Thomson considered the effect of this field on the dissociation and recoupling of vortex-rings.
52. Proc. Roy. Soc. Edinb., xxv (1905), p. 565.
53. Brit. Assoc. Rep., 1880, p. 473.
54. Scient. Proc, Roy. Dubin Soc., 1886; Scientific Writings of FitzGerald, p. 154.
55. Brit. Assoc. Rep., 1985, p. 930.
56. Phil. Mag. xxiv (1887), p. 342: Kelvin's Math.and Phys. Papers, iv, p. 308.
57. Proc. Roy. Irish Acad. (3) i (1889), p. 340; Kelvin's Math. and Phys. Papers, iv, p. 202,
58. Brit. Assoc. Rep., 1899. FitzGerald's Scientific Writings, p. 484.
59. Brit. Assoc. Rep., 1895, p. 595.
60. Proc. Roy. Dublin Soc., December 12, 1899; FitzGerald's Scientific Writings, p. 472.
61. Brit. Assoc. Rep., 1888, p. 577.
62. A hollow vortex is a cyclic motion existing in a fluid without the presence of any actual rotational filaments. On the general theory of. Hicks, Phil. Trans. clxxv (1883), p. 161; clxxvi (1885), p. 125, excii (1898), p. 33.
63. Proc. Roy. Iris. Acad., November 30, 1889: Kelvin's Math. and Phys. Papers, iv, p. 202. "Rotational vortex-cores," he wrote, "must be absolutely discarded; and we must have nothing but irrotational revolution and vacuous cores."

Errata

1. Original: $-A.v_{0}.(\partial /^{2}\partial y^{2}).\nabla ^{-2}v_{0}$ was amended to $-A.v_{0}.(\partial ^{2}/\partial y^{2}).\nabla ^{-2}v_{0}$