A History of the Theories of Aether and Electricity (1910) — part 19 of 29

It may seem strange that Maxwell, having successfully employed his electromagnetic theory to explain the propagation of light in isotropic media, in crystals, and in metals, should have omitted to apply it to the problem of reflexion and refrac- tion. This is all the more surprising, as the study of the optics of crystals had already revealed a close analogy between the electromagnetic theory and MacCullagh's elastic-solid theory; and in order to explain reflexion and refraction electro- magnetically, nothing more was necessary than to transcribe MacCullagh's investigation of the same problem, interpreting e (the time-flux of the displacement of MacCullagh's aether) as the magnetic force, and curl e as the electric displacement. As

• Ann. d. Phys. Ix (1897), p. 454.

t Rubens and Aschkinass, Ann. d. Phys. Ixiv (1898).

296 Maxwell.

in MacCullagh's theory the difference between the contiguous media is represented by a difference of their elastic constants, so in the electromagnetic theory it may be represented by a difference in their specific inductive capacities. From a letter which Maxwell wrote to Stokes in 1864, and which has been preserved,* it appears that the problem of reflexion and refrac- tion was engaging Maxwell's attention at the time when he was preparing his Eoyal Society memoir on the electromagnetic field; but he was not able to satisfy himself regarding the conditions which should be satisfied at the interface between the media. He seems to have been in doubt which of the rival elastic-solid theories to take as a pattern ; and it is not unlikely that he was led astray by relying too much on the analogy between the electric displacement and an elastic displacement. t For in the elastic-solid theory all three components of the dis- placement must be continuous across the interface between two contiguous media ; but Maxwell found that it was impossible to explain reflexion and refraction if all three components of the electric displacement were supposed to be continuous across the interface ; and, unwilling to give up the analogy which had hitherto guided him aright, yet unable to disprove^ the Greenian conditions at bounding surfaces, he seems to have laid aside the problem until some new light should dawn upon it.

This was not the only difficulty which beset the electro- magnetic theory. The theoretical conclusion, that the specific inductive capacity of a medium should be equal to the square of its refractive index with respect to waves of long period, was not as yet substantiated by experiment; and the theory of displacement-currents, on which everything else depended, was

• Stokes's Scientific Correspondence, ii, pp. 25, 26.

t It must be remembered tbat Maxwell pictured tbe electric displacement as a real displacement of a medium. "My theory of electrical forces," he \vrote, " is that they are called into play in insulating media by slight electric displacements, which put certain small portions of the medium into a state of distortion, which, being resisted by the elasticity of the medium, produces an electromotive force." Campbell and Garnett's Life of Maxwell, p. 244.

| The letter to Stokes already mentioned appears to indicate that Maxwell for a time doubted the correctness of Green's conditions.

Maxwell. 297

unfavourably received by the most distinguished of Maxwell's contemporaries. Helmholtz indeed ultimately accepted it, but only after many years ; and W. Thomson (Kelvin) seems never to have thoroughly believed it to the end of his long life. In 1888 he referred to it as a "curious and ingenious, but not wholly tenable hypothesis,"* and proposedf to replace it by an extension of the older potential theories. In 1896 he had some inclination? to speculate that alterations of electrostatic force due to rapidly-changing electrification are propagated by con- densational waves in the luminiferous aether. In 1904 he admittedg that a bar-magnet rotating about an axis at right angles to its length is equivalent to a lamp emitting light of period equal to the period of the rotation, but gave his final judgment in the sentence|| : — " The so-called electromagnetic theory of light has not helped us hitherto."

Thomson appears to have based his ideas of the propagation of electric disturbance on the case which had first become familiar to him — that of the transmission of signals along a wire. He clung to the older view that in such a disturbance the wire is the actual medium of transmission ; whereas in \ Maxwell's theory the function of the wire is merely to guide the disturbance, which is resident in the surrounding dielectric.

This opinion that conductors are the media of propagation of electric disturbance was entertained also by Ludwig Lorenz (&. 1829, d. 1891), of Copenhagen, who independently developed an electromagnetic theory of lightH a few years after the publication of Maxwell's memoirs. The procedure which Lorenz followed was that which Kiemann had suggested** in 1858 — namely, to modify the accepted formulae of electro- dynamics by introducing terms which, though too small to be

• Nature, xxxviii (1888) p. 571. t Brit. Assoc. Report, 1888, p. 567.

J Cf. Bottouiley, in Nature, liii (1896), p. 268 ; Kelvin, ib., p. 316 ; J. Willard Gibbs, ib., p. 509.

§ Baltimore Lectures (ed. 1904), p. 376. || Ibid., preface, p. 7.

H Oversiyt over det K. danske Vid. Selskaps Forhandliiiger, 1867, p. 26; Annul, der Phys. cxxxi (1867), p. 243 ; Phil. Mag., xxxiv (1867), p. 287.

** Cf. p. 268. Riemann's memoir was, however, published only in the same year (1867) as Lorenz's.

298 Maxwell.

appreciable in ordinary laboratory experiments, would be capable of accounting for the propagation of electrical effects through space with a finite velocity. We have seen that in Neumann's theory the electric force E was determined by the equation

-a, (1)

where <£ denotes the electrostatic potential defined by the equation

4>-{\(p'lr) dx'dy'dz',

p being the density of electric charge at the point (x, y, z'), and where a denotes the vector-potential, defined by the equation

a={[(i'lr)dx'dy'dz,

J J J

i' being the conduction-current at (x', y\ z'). We suppose the specific inductive capacity and the magnetic permeability to be everywhere unity.

Lorenz proposed to replace these by the equations

= \{p(t-r/c)/r\dx'dy'dz', {i'(t-r/c)/r}dx'dy'd3f'9

the change consists in replacing the values which p and i' have at the instant t by those which they have at the instant (t - r/c], which is the instant at which a disturbance travelling with velocity c must leave the place (x', y, z) in order to arrive at the place (x, y, z) at the instant t. Thus the values of the potentials at (x, y, z] at any instant t would, according to Lorenz's theory, depend on the electric state at the point (x', y', z') at the previous instant (t - r/c) : as if the potentials were propagated outwards from the charges and currents with velocity c. The functions <f> and a formed in this way are generally known as the retarded potentials.

Maxwell. 299

The equations by which (f> and a have been defined are equivalent to the equations

V2</> - $1* = - 4^, (2)

V2a - a/c2 = - 47ri, (3)

while the equation of conservation of electricity,

div i + p = 0 gives

div a + <f> = 0. (4)

From equations (1), (2), (4), we may readily derive the equation

divE = 47rcV; (I)

and from (1), (3), (4), we have

curl H = E/c2 + 47rt, (II)

where H or curl a denotes the magnetic force : while from (1) we have

curl E = - H. (Ill)

The equations (I), (II), (III) are, however, the fundamental equations of Maxwell's theory; and therefore the theory of L. Lorenz is practically equivalent to that of Maxwell, so far as concerns the propagation of electromagnetic disturbances through free aether. Lorenz himself, however, does not appear to have clearly perceived this ; for in his memoir he postulated the presence of conducting matter throughout space, and was consequently led to equations resembling those which Maxwell had given for the propagation of light in metals. Observing that his equations represented periodic electric currents at right angles to the direction of propagation of the disturbance, he suggested that all luminous vibrations might be constituted by electric currents, and hence that there was " no longer any reason for maintaining the hypothesis of an aether, since we can admit that space contains sufficient ponderable matter to enable the disturbance to be propagated."

Lorenz was unable to derive from his equations any explana- tion of the existence of refractive indices, and his theory lacks

300 Maxwell.

the rich physical suggestiveness of Maxwell's ; the value of his memoir lies chiefly in the introduction of the retarded potentials. It may be remarked in passing that Lorenz's retarded potentials are not identical with Maxwell's scalar and vector potentials ; for Lorenz's a is not a circuital vector, and Lorenz's <£ is not, like Maxwell's, the electrostatic potential, but depends on the positions occupied by the charges at certain previous instants.

For some years no progress was made either with Maxwell's theory or with Lorenz's. Meanwhile, Maxwell had in 1865 resigned his chair at King's College, and had retired to his estate in Dumfriesshire, where he occupied himself in writing a connected account of electrical theory. In 1871 he returned to Cambridge as Professor of Experimental Physics; and two years later published his Treatise on Electricity and Magnetism.

In this celebrated work is comprehended almost every branch of electric and magnetic theory; but the intention of the writer was to discuss the whole as far as possible from a single point of view, namely, that of Faraday; so that little or no account was given of the hypotheses which had been pro- pounded in the two preceding decades by the great German electricians. So far as Maxwell's purpose was to disseminate the ideas of Faraday, it was undoubtedly fulfilled ; but the Treatise was less successful when considered as the exposition of its author's own views. The doctrines peculiar to Maxwell — the existence of displacement-currents, and of electromagnetic vibrations identical with light — were not introduced in the first 'volume, or in the first half of the second volume ; and the account which was given of them was scarcely more complete, and was perhaps less attractive, than that which had been furnished in the original memoirs.

Some matters were, however, discussed more fully in the Treatise than in Maxwell's previous writings ; and among these was the question of stress in the electromagnetic field.

It will be remembered* that Faraday, when studying the

• Cf. p. 209.

Maxwell. 301

curvature of lines of force in electrostatic fields, had noticed an apparent tendency of adjacent lines to repel each other, as if each tube of force were inherently disposed to distend laterally ; and that in addition to this repellent or diverging force in the transverse direction, he supposed an attractive or contractile force to be exerted at right angles to it, that is to say, in the direction of the lines of force.

Of the existence of these pressures and tensions Maxwell was fully persuaded ; and he determined analytical expressions suitable to represent them. The tension along the lines of force must be supposed to maintain the ponderomotive force which acts on the conductor on which the lines of force terminate ; and it may therefore be measured by the force which is exerted on unit area of the conductor, i.e., *E2/87rc2 or iDE. The pressure at right angles to the lines of force must then be determined so as to satisfy the condition that the aether is to be in equilibrium.

For this purpose, consider a thin shell of aether included between two equipotential surfaces. The equilibrium of the, portion of this shell which is intercepted by a tube of force- requires (as in the theory of the equilibrium of liquid films), that the resultant force per unit area due to the above- mentioned normal tensions on its two faces shall have the- value T(l/pi + l//o2), where pi and pz denote the principal radii of curvature of the shell at the place, and where T denotes, the lateral stress across unit length of the surface of the shell,, T being analogous to the surface-tension of a liquid film.

Now, if t denote the thickness of the shell, the area inter- cepted on the second face by the tube of force bears to the area intercepted on the first face the ratio (pi + t) (pz + t)/p!p2 > and by the fundamental property of tubes of force, D and E vary inversely as the cross-section of the tube, so the total force on the second face will bear to that on the first face the ratio

piptKpi + 1} (pz + 1),

or approximately

302 Maxwell.

the resultant force per unit area along the outward normal is therefore

• IDE . t . (l//t>i + I//*), and so we have

T = - IDE . t ;

or the pressure at right angles to the lines of force is |DE per unit area — that is, it is numerically equal to the tension along the lines of force.

The principal stresses in the medium being thus determined, it readily follows that the stress across any plane, to which the unit vector N is normal, is

(D.N)E-i(D-E)N-

Maxwell obtained* a similar formula for the case of magnetic fields ; the ponderornotive forces on magnetized matter and on conductors carrying currents may be accounted for by assuming a stress in the medium, the stress across the plane N" being represented by the vector

1(B.K).H-1(B.H).N. ;j

This, like the corresponding electrostatic formula, represents a tension across planes perpendicular to the lines of force, and a pressure across planes parallel to them.

It may be remarked that Maxwell made no distinction between stress in the material dielectric and stress in the aether : indeed, so long as it was supposed that material bodies when displaced carry the contained aether along with them, no distinction was possible. In the modifications of Maxwell's theory which were developed many years afterwards by his followers, stresses corresponding to those introduced by Maxwell were assigned to the aether, as distinct from ponderable matter ; and it was assumed that the only stresses set up in material bodies by the electromagnetic field are produced indirectly: they may be calculated by the methods of the theory of elasticity, from a knowledge of the ponderomotive forces exerted on the electric charges connected with the bodies.

• Maxwell's Treatise on Electricity and Magnetism, § 643.

Maxwell. 303

Another remark suggested by Maxwell's theory of stress in the medium is that he considered the question from the purely statical point of view. He determined the stress so that it might produce the required forces on ponderable bodies, and be self-equilibrating in free aether. But* if the electric and magnetic phenomena are not really statical, but are kinetic in their nature, the stress or pressure need not be self-equilibrating. This may be illustrated by reference to the hydrodynamical models of the aether shortly to be described, in which perforated solids are immersed in a moving liquid : the ponderomotive forces exerted on the solids by the liquid correspond to those which act on conductors carrying currents in a magnetic field, and yet there is no stress in the medium beyond the pressure of the liquid.

Among the problems to which Maxwell applied his theory of stress in the medium was one which had engaged the attention of many generations of his predecessors. The ad- herents of the corpuscular theory of light in the eighteenth century believed that their hypothesis would be decisively con- firmed if it could be shown that rays of light possess momentum : to determine the matter, several investigators directed powerful beams of light on delicately-suspended bodies, and looked for evidences of a pressure due to the impulse of the corpuscles. Such an experiment was performed in 1708 by Homberg,f who imagined that he actually obtained the effect in question ; but Mairan and Du Fay in the middle of the century, having repeated his operations, failed to confirm his conclusion.*

The subject was afterwards taken up by Michell, who "some years ago," wrote Priestley § in 1772, " endeavoured to ascertain the momentum of light in a much more accurate manner than those in which M. Homberg and M. Mairan had attempted it." He exposed a very thin and delicately-suspended copper plate

• Cf. V. Bjeiknes, Phil. Mag. ix (1905), p. 491. t Histoire de 1'Acad., 1708, p. 21. % J. J. (ie Mairan, Traite de V A urore boreale, p. 370. § History of Vision, i, p. 387.

304 Maxwell.

to the rays of the sun concentrated by a mirror, and observed a deflexion. He was not satisfied that the effect of the heating of the air had been altogether excluded, but " there seems to be no doubt," in Priestley's opinion, " but that the motion above mentioned is to be ascribed to the impulse of the rays of light." A similar experiment was made by A. Bennet,* who directed the light from the focus of a large lens on writing-paper delicately suspended in an exhausted receiver, but " could not perceive any motion distinguishable from the effects of heat." " Perhaps," he concluded, " sensible heat and light may not be caused by the influx or rectilineal projections of fine particles, but by the vibrations made in the universally diffused caloric or matter of heat, or fluid of light." Thus Bennet, and after him Young, f regarded the non-appearance of light- repulsion in this experiment as an argument in favour of the undulatory system of light. " For," wrote Young, " granting the utmost imaginable subtility of the corpuscles of light, their effects might naturally be expected to bear some proportion to the effects of the much less rapid motions of the electrical fluid, which are so very easily perceptible, even in their weakest states."

This attitude is all the more remarkable, because Euler many years before had expressed the opinion that light-pressure might be expected just as reasonably on the undulatory 'as on the corpuscular hypothesis. "Just as," he wrote, J "a vehement sound excites not only a vibratory motion in the particles of the air, but there is also observed a real movement of the small particles of dust which are suspended therein, it is not to be doubted but that the vibratory motion set up by the light causes a similar effect." Euler not only inferred the existence of light-pressure, but even (adopting a suggestion of Kepler's) accounted for the tails of comets by supposing that the solar rays, impinging on the atmosphere of a comet, drive off from it the more subtle of its particles.

• Phil. Trans., 1792, p. 81. + Ibid., 1802, p. 46.

J Histoire de /' Acad. de Berlin, ii (1748), p. 117.

Maxwell. 305

The question was examined by Maxwell* from the point of view of the electromagnetic theory of light ; which readily furnishes reasons for the existence of light-pressure. For suppose that light falls on a metallic reflecting surface at perpendicular incidence. The light may be regarded as con- stituted of a rapidly-alternating magnetic field ; and this must induce electric currents in the surface layers of the metal. But a metal carrying currents in a magnetic field is acted on by a ponderomotive force, which is at right angles to both the magnetic force and the direction of the current, and is there- fore, in the present case, normal to the reflecting surface : this ponderomotive force is the light-pressure. Thus, according to Maxwell's theory, light-pressure is only an extended case of effects which may readily be produced in the laboratory.

The magnitude of the light-pressure was deduced by Maxwell from his theory of stresses in the medium. We have seen that the stress across a plane whose unit-normal is N is represented by the vector

(D . N) . E - J (D . E) . N + — (B . N) . H - ~ (B . H) . N.

47T O7T

Now, suppose that a plane wave is incident perpendicularly on a perfectly reflecting metallic sheet: this sheet must support the mechanical stress which exists at its boundary in the aether. Owing to the presence of the reflected wave, D is zero at the surface ; and B is perpendicular to N, so (B . N) vanishes. Thus the stress is a pressure of magnitude (l/8?r) (B . H) normal to the surface : that is, the light-pressure is equal to the density of the aethereal energy in the region immediately outside the metal. This was Maxwell's result.

This conclusion has been reached on the assumption that the light is incident normally to the reflecting surface. If, on the other hand, the surface is placed in an enclosure completely surrounded by a radiating shell, so that radiation falls on it from all directions, it may be shown that the light-pressure is measured by one-third of the density of aethereal energy.

• Maxwell's Treatise on Electricity and Magnetism, § 792. X

306 Maxwell.

A different way of inferring the necessity for light-pressure was indicated in 1876 by A. Bartoli,* who showed that, when radiant energy is transported from a cold body to a hot one by means of a moving mirror, the second law of thermodynamics would be violated unless a pressure were exerted on the mirror by the light.

The thermodynamical ideas introduced into the subject by Bartoli have proved very fruitful. If a hollow vessel be at a definite temperature, the aether within the vessel must be full of radiation crossing from one side to the other : and hence the aether, when in radiative equilibrium with matter at a given temperature, is the seat of a definite quantity of energy per unit volume.

If U denote this energy per unit volume, and P the light- pressure on unit area of a surface exposed to the radiation, we may applyf the equation of available energy!

U-TdF P ~ 1 dT

Since, as we have seen,

this equation gives .„ dU

dT'

and therefore U must be proportional to T*. From this it may be inferred that the intensity of emission of radiant energy by a body at temperature T is proportional to the fourth power of the absolute temperature — a law which was first discovered experimentally by Stefan§ in 1879.

In the year in which Maxwell's treatise was published, Sir William Crookes|| obtained experimental evidence of a pressure accompanying the incidence of light; but this was

• Bartoli, Sopra i movimenti prodotti dalla luce e dal calore e sopra il radiometro di Crookes. Firenze, 1876. Also Nuovo Cimento (3) xv (1884), p. 193 ; and Exner's Rep., xxi (1885), p. 198.

t Boltzmann, Ann. d. Phys. xxii (1884), p. 31. Cf. also B. Galitzine, Ann. d. Phys. xlvii (1892), p. 479.

| Cf. p. 240. § Wien. Ber. Ixxix (1879), p. 391.

|| Phil. Trans, clxiv (1874), p. 501. The radiometer was discovered in 1875.

Maxwell. 307

soon found to be due to thermal effects ; and the existence of a true light-pressure was not confirmed experimentally* until 1899. Since then the subject has been considerably developed, especially in regard to the part played by the pressure of radiation in cosmical physics.

Another matter which received attention in Maxwell's Treatise was the influence of a magnetic field on the propagation of light in material substances. We have already seenf that the theory of magnetic vortices had its origin in Thomson's speculations on this phenomenon ; and Maxwell in his memoir of 1861-2 had attempted by the help of that theory to arrive at some explanation of it. The more complete investigation which is given in the Treatise is based on the same general assumptions, namely, that in a medium subjected to a magnetic field there exist concealed vortical motions, the axes of the vortices being in the direction of the lines of magnetic force ; and that waves of light passing through the medium disturb the vortices, which thereupon react dynamically on the luminous motion, and so affect its velocity of propagation.

The manner of this dynamical interaction must now be more closely examined. Maxwell supposed that the magnetic vortices are affected by the light-waves in the same way as vortex-filaments in a liquid would be affected by any other coexisting motion in the liquid. The latter problem had been already discussed in Helrnholtz'js great memoir on vortex- motion ; adopting Helmholtz's results, Maxwell assumed for the additional term introduced into the magnetic force by the dis- placement of the vortices the value 9e/B0, where e denotes the displacement of the medium (i.e. the light vector), and the operator d/dO denotes H^/dx + Hy'dj^y + Hzd/dz, H denoting the imposed magnetic field. Thus the luminous motion, by dis- turbing the vortices, gives rise to an electric current in the medium, proportional to curl

*P. Lebedew, Archives des Sciences Phys. et Nat. (4) viii (1899), p. 184. Ann. d. Phys. vi (1901), p. 433. E. F. Nichols and G. F. Hull, Phys. Rev. xiii (1901), p. 293 ; Astrophys. Jour., xvii (1903), p. 315. t Cf. p. 274.

X 2

308 Maxwell.

Maxwell further assumed that the current thus produced interacts dynamically with the luminous motion in such a manner that the kinetic energy of the medium contains a term proportional to the scalar product of e and curl de/30. The total kinetic energy of the medium may therefore be written

\p& + Jcr (e . curl 9e/a0),

where p denotes the density of the medium, and cr denotes a constant which measures the capacity of the medium to rotate the plane of polarization of light in a magnetic field.

The equation of motion may now be derived as in the elastic- solid theories of light : it is

32

pe = %V2e - o- r— - curl e. ot cu

When the light is transmitted in the direction of the lines of force, and the axis of x is taken parallel to this direction, the equation reduces to

and these equations, as we have seen,* furnish an explanation of Faraday's phenomenon.

It may be remarked that the term

J(T (e . curl 9e/80)

in the kinetic energy may by partial integration be transformed into a term

Jo- (curie. 9e/90),t

together with surface-terms ; or, again, into

• Jo- (curl e . 8e/80),

together with surf ace- terms. These different forms all yield

• Cf. p. 215.

f This form was suggested by Fitz Gerald six years later, Phil. Trans., 1880, p. 691 : Fitz Gerald's Scientific Writings, p. 45.

Maxwell. 309

the same equation of motion for the medium; but, owing to the differences in the surface-terms, they yield different con- ditions at the boundary of the medium, and consequently give rise to different theories of reflexion.

The assumptions involved in Maxwell's treatment of the magnetic rotation of light were such as might scarcely be justified in themselves ; but since the discussion as a whole proceeded from sound dynamical principles, and its conclu- sions were in harmony with experimental results, it was fitted to lead to the more perfect explanations which were afterwards devised by his successors. At the time of Maxwell's death, which happened in 1879, before he had completed his forty- ninth year, much yet remained to be done both in this and in the other investigations with which his name is associated; and the energies of the next generation were largely spent in extending and refining that conception of electrical and optical phenomena whose origin is correctly indicated in its name of Maxwell's Theory.

( 310 )

CHAPTEK 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* of 1847 and Maxwell's memoirf 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 1855J and afterwards amplified by Helmholtz§ — regard mag- netic 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|| he showed tha

• Cf. p. 270. t Of. p. 276. % Cf. p. 271. § Cf. p. 274.

|| Kelvin's Math, and Phys. Papers, iii, p. 436.

Models of the Aether. 311

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 pro- duce 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 Mac Cullagh^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 permea- bility, the normal component of magnetic induction will be continuous across an interface, as it should be.*

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,f 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,

4-TrD = curl e, H = e,

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

• Thomson inclined to believe (Papers, iii, p. I&5) that light might he correctly represented by the vibratory motion of such a solid.

t Wilberforce, Trans. Camb. Phil. Soc. xiv (1887), p. 170 ; Lodge, Phil. Trans, clxxxix (1897), p. 149.

312 Models of the Aether.

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,* afterwards abandoned this conception in favour of that which represents magnetic phenomena as rotatory. "According to Ampere and all his followers," he wrote in 1870,f " 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 asso- ciated 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 itj by showing that if magnetic induction is com- pared to fluid velocity, then electric currents correspond to vortex-filaments in the fluid. Two years afterwards Kirchhoff § 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 incom- pressible 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

• Cf. p. 276.

t Proc. Lond. Math. Soc. iii (1870), p. 224 ; Maxwell's Sclent. Papers, ii, p. 263. J Cf. p. 274.

§ Journnl fur Math. Ixxi (1869) ; Kirchhoff's Ge*amm. AbhandL, p. 404. Cf. also C. Neumann, Leipzig Berichte, xliv (1892), p. 86.

Models of the Aether. 3 13

each other if they were traversed by electric currents pro- portional to the circulations.