A History of the Theories of Aether and Electricity (1910) — part 17 of 29

It will be seen that Kirchhoff's electrical researches were greatly influenced by those of Weber. The latter investiga- tions, however, did not enjoy unquestioned authority ; for there was still a question as to whether the expressions given by Weber for the mutual energy of two current elements, and for the mutual energy of two electrons, were to be preferred to the rival formulae of Neumann and Eiemann. The matter was examined in 1870 by Helmholtz, in a series of memoirs* to which reference has already been made.f Helmholtz remarked that, for two elements ds, ds', carrying currents i, i', the electro- dynamic energy is

n'(ds.ds') r ' according to Neumann, and

?V 5-(r.ds)(r.ds'),

according to Weber; and that these expressions differ from each other only by the quantity

• cos (ds . ds') + cos (r . ds) cos (r . ds') ] , dzr

or ^^

dsds

since this vanishes when integrated round either circuit, the two formulae give the same result when applied to entire currents. A general formula including both that of Neumann and that of Weber is evidently

n'(ds .ds') .., ffr

— + ki^ -j—-, dsds, r ds ds

where k denotes an arbitrary constant.^

Helmholtz's result suggested to Clausius§ a new form for the law of force between electrons ; namely, that which is

• Journal fur Math., Ixxii (1870), p. 57 : Ixxv (1873), p. 35: Ixxviii (1874), p. 273. t Cf. p. 229.

• Cf. H. Lamb, Proc. Lond. Math. Soc., xiv (1883), p. 301. § Journal fiir Math. Ixxxii (1877), p. 85 : Phil. Mag., x (1880), p. 255.

262 The Mathematical Electricians of the

obtained by supposing that two electrons of charges e, e', and velocities v, v', possess electrokinetic energy of amount

eef (v .v') 7 , d~r ,

— - - + kee -r— =-> w .

r dsds

Subtracting from this the mutual electrostatic potential energy, which is ee'c'/r, we may write the mutual kinetic potential of the two electrons in the form

(xx + ijy + zzf - c2) + kee' > vv',

where (x, y, z) denote the coordinates of e, and (X, y', z) those of ef.

The unknown constant k has clearly no influence so long as closed circuits only are considered: if k be replaced by zero, the expression for the kinetic potential becomes

ee'

— (xx + yy + zz - c2),

which, as will appear later, closely resembles the corresponding expression in the modern theory of electrons.

Clausius' formula has the great advantage over Weber's, that it does not compel us to assume equal and opposite velocities for the vitreous and resinous charges in an electric current; on the other hand, Clausius' expression involves the absolute velocities of the electrons, while Weber's depends only on their relative motion; and therefore Clausius' theory requires the assumption of a fixed aether in space, to which the velocities v and V may be referred.

When the behaviour of finite electrical systems is predicted from the formulae of Weber, Eiemann, and Clausius, the three laws do not always lead to concordant results. For instance, if a circular current be rotated with constant angular velocity round its axis, according to Weber's law there would be a development of free electricity on a stationary conductor in the neighbourhood ; whereas, according to Clausius' formula there would be no induction on a stationary body, but electrification

Middle of the Nineteenth Century. 263

would appear on a body turning with the circuit as if rigidly connected with it. Again,* let a magnet be suspended within a hollow metallic body, and let the hollow body be suddenly charged or discharged; then, according to Clausius' theory, the magnet is unaffected; but according to Weber's and Kiemann's theories it experiences an impulsive couple. And again, if an electrified disk be rotated in its own plane, under certain circumstances a steady current will be induced in a neighbouring circuit according to Weber's law, but not according to the other formulae.

An interesting objection to Clausius' theory was brought forward in 1879 by Frohlichf — namely, that when a charge of free electricity and a constant electric current are at rest relatively to each other, but partake together of the translatory motion of the earth in space, a force should act between them if Clausius' law were true. It was, however, shown by BuddeJ that the circuit itself acquires an electrostatic charge, partly as a result of the same action which causes the force on the external conductor, and partly as a result of electrostatic induction by the charge on the external conductor ; and that the total force between the circuit and external conductor is thus reduced to zero.§

We have seen that the discrimination between the different laws of electrodynamic force is closely connected with the question whether in an electric current there are two kinds of electricity moving in opposite directions, or only one kind moving in one direction. On the unitary hypothesis, that the

• The two following crudal experiments, with others, were suggested by E. Budde, Ann. d. Phys. xxx (1887), p. 100.

t Ann. d. Phys. ix (1880), p. 261.

• Ann. d. Phys. x (1880), p. 553.

§ This case of a charge and current moving side hy side was afterwards examined by Fitz Gerald (Trans. Boy. Dub. Soc. i, 1882 ; Scient. Writings of G. F. Fitz Gerald, p. Ill) without reference to Clausius' formula, from the standpoint of Maxwell's theory. The result obtained was the same — namely, that the electricity induced on the conductor carrying the current neutralizes the ponderomotive force between the current and the external charge.

264 The Mathematical Electricians of the

current consists in a transport of one kind of electricity with a definite velocity relative to the wire, it might be expected that a coil rotated rapidly about its own axis would generate a magnetic field different from that produced by the same coil at rest. Experiments to determine the matter were performed by A. Foppl* and by E. L. Nichols and W. S. Franklin,f but with negative results. The latter investigators found that the velocity of electricity must be such that the quantity conveyed past a specified point in a unit of time, when the direction of the current was that in which the coil was travelling, did not differ from that transferred when the current and coil were moving in opposite directions by as much as one part in ten million, even when the velocity of the wire was 9096 cm./sec. They considered that they would have been able to detect a change of deflexion due to the motion of the coil, even though the velocity of the current had been considerably greater than a thousand million metres per second.

During the decades in the middle of the century consider- able progress was made in the science of thermo-electricity, whose beginnings we have already described. J In Faraday's laboratory note-book, under the date July 28th, 1836, we read§ : — " Surely the converse of thermo-electricity ought to be obtained experimentally. Pass current through a circuit of antimony and bismuth."

Unknown to Faraday, the experiment here indicated had already been made, although its author had arrived at it by a different train of ideas. In 1834 Jean Charles Peltier|| (b. 1785, d. 1845) attempted the task, which was afterwards performed with success by Joule,1J of measuring the heat evolved by the passage of an electric current through a conductor. He found that a current produces in a homogeneous conductor an elevation

• Ann. d. Phys. xxvii (1886), p. 410.

t Amer. Jour. Sci., xxxvii (1889), p. 103.

J Cf. pp. 92, 93. § Bence Jones's Life of Faraday, ii, p. 76.

II Annales de Ciiimie, Ivi (1834), p. 371. If Cf. p. 239.

Middle of the Nineteenth Century. 265

of temperature, which is the same in all parts of the conductor where the cross-section is the same ; but he did not succeed in connecting the thermal phenomena quantitatively with the strength of .the current — a failure which was due chiefly to the circumstance that his attention was fixed on the rise of temperature rather than on the amount of the heat evolved. But incidentally the investigation led to an important discovery — namely, that when a current was passed in succession through two conductors made of dissimilar metals, there was an evolution of heat at the junction ; and that this depended on the direction of the current ; for if the junction was heated when the current flowed in one sense, it was cooled when the current flowed in the opposite sense. This Peltier effect, as it is called, is quite distinct from the ordinary Joulian liberation of heat, in which the amount of energy set free in the thermal form is unaffected by a reversal of the current ; the Joulian effect is, in fact, propor- tional to the square of the current-strength, while the Peltier effect is proportional to the current-strength directly. The Peltier heat which is absorbed from external sources when a current i flows for unit time through a junction from one metal B to another metal A may therefore be denoted by

where T denotes the absolute temperature of the junction. The function n^ (T) is found to be expressible as the difference of two parts, of which one depends on the metal A only, and the other on the metal B only ; thus we can write

In 1851 a general theory of thermo-electric phenomena was constructed on the foundation of Seebeck's* and Peltier's dis- coveries by W. Thomson.f Consider a circuit formed of two

• Cf. pp. 92, 93.

t Proc. R.S. Edinb. iii (1851), p. 91 ; Phil. Mag. iii (1852), p. 529 : Kelvin's Math, and Phys. Paper*, i, p. 316. Cf. also Trans. R. S. Edinb. xxi (1854), p. 123, reprinted in Papers, i, p. 232 : and Phil. Trans., 1856, reprinted in Papers, ii, p. 189.

266 The Mathematical Electricians of the

metals, A and B, and let one junction be maintained at a slightly higher temperature (T + $T) than the temperature T of the other junction. As Seebeck had shown, a thermo-electric current will be set up in the circuit. Thomson saw that such a system might be regarded as a heat-engine, which absorbs a certain quantity of heat at the hot junction, and converts part of this into electrical energy, liberating the rest in the form of heat at the cold junction. If the Joulian evolution of heat be neglected, the process is reversible, and must obey the second law of thermodynamics ; that is, the sum of the quantities of heat absorbed, each divided by the absolute temperature at which it is absorbed, must vanish. Thus we have

T+ST

so the Peltier effect H^(T) must be directly proportional to the absolute temperature T. This result, however, as Thomson well knew, was contradicted by the observations of Gumming, who had shown that when the temperature of the hot junction is gradually increased, the electromotive force rises to a maximum value and then decreases. The contradiction led Thomson to predict the existence of a hitherto unrecognized thermo-electric phenomenon — namely, a reversible absorption of heat at places in the circuit other than the junctions. Suppose that a current flows along a wire which is of the same metal throughout, but varies in temperature from point to point. Thomson showed that heat must be liberated at some points and absorbed at others, so as either to accentuate or to diminish the differences of temperature at the different points of the wire. Suppose that the heat absorbed from external sources when unit electric charge passes from the absolute temperature T to the temperature (T + $T) in a metal A is denoted by SA(T).ST. The thermodynamical equation now takes the corrected form

~ SA(T)}

Middle of the Nineteenth Century. 267

Since the metals A and B are quite independent, this gives

This equation connects Thomson's " specific heat of electricity" SA(T) with the Peltier effect.

In 1870 P. G. Tait* found experimentally that the specific heat of electricity in pure metals is proportional to the absolute temperature. We may therefore write SA(T) = aAT, where a A denotes a constant characteristic of the metal A. The thermodynamical equation then becomes

_d \UA(T)) dT ( T ~

or

where TTA denotes another constant characteristic of the metal. The chief part of the Peltier effect arises from the term irAT.

By the investigations which have been described in the present chapter, the theory of electric currents was considerably advanced in several directions. In all these researches, how- ever, attention was fixed on the conductor carrying the current as the seat of the phenomenon. In the following period, interest was centred not so much on the conductors which carry charges and currents, as on the processes which take place in the dielectric media .around them.

• Proc. R. S. Edinb. vii (1870), p. 308. Cf. also Batelli, Atti delia R. Ace. di Torino, xxii (1886), p. 48, translated Phil. Mug. xxiv (1887), p. 295.

( 268 ) CHAPTEE VIII.

MAXWELL.

SINCE the time of Descartes, natural philosophers have never ceased to speculate on the manner in which electric and magnetic influences are transmitted through space. About the middle of the nineteenth century, speculation assumed a definite form, and issued in a rational theory.

Among those who thought much on the matter was Karl Friedrich Gauss (b. 1777, d. 1855). In a letter* to Weber of date March 19, 1845, Gauss remarked that he had long ago proposed to himself to supplement the known forces which act between electric charges by other forces, such as would cause electric actions to be propagated between the charges with a finite velocity. But he expressed himself as determined not to publish his researches until he should have devised a mechanism by which the transmission could be conceived to be effected ; and this he had not succeeded in doing.

More than one attempt to realize Gauss's aspiration was made by his pupil Eiemann. In a fragmentary note,t which appears to have been written in 1853, but which was not published until after his death, Biemann proposed an aether whose elements should be endowed with the power of resisting compression, and also (like the elements of MacCullagh's aether) of resisting changes of orientation. The former pro- perty he conceived to be the cause of gravitational and electrostatic effects, and the latter to be the cause of optical and magnetic phenomena. The theory thus outlined was apparently not developed further by its author ; but in a short investigation^ which was published posthumously in 1867,§ he

• Gauss' Werke, v, p. 629. t Riemann's Werke, 2e Aufl., p. 526.

J Ann. d. Phys. cxxxi (1867), p. 237 ; Riemann's Werke, 2e Aufl., p. 288 ; Phil. Mag. xxxiv (1867), p. 368.

§ It had been presented to the Gottingen Academy in 1858, but afterwards withdrawn.

.

Maxwell. 269

returned to the question of the process by which electric action is propagated through space. In this memoir he proposed to replace Poisson's equation for the electrostatic potential, namely,

by the equation

according to which the changes of potential due to changing electrification would be propagated outwards from the charges with a velocity c. This, so far as it goes, is in agreement with the view which is now accepted as correct ; but Kiemann's hypothesis was too slight to serve as the basis of a complete theory. Success came only when the properties of the inter- vening medium were taken into account.

In that power to which Gauss attached so much importance, of devising dynamical models and analogies for obscure physical phenomena, perhaps no one has ever excelled W. Thomson*; and to him, jointly with Faraday, is due the credit of having initiated the theory of the electric medium. In one of his earliest papers, written at the age of seventeen,! Thomson compared the distribution of electrostatic force, in a region containing electrified conductors, with the distribution of the flow of heat in an infinite solid : the equipotential surfaces in the one case correspond to the isothermal surfaces in the other, and an electric charge corresponds to a source of heat.J

• As will appear from the present chapter, Maxwell had the same power in a very marked degree. It has always been cultivated hy the " Cambridge school " of natural philosophers.

t Camb. Math. Journal, iii (Feb. 1842), p. 71 ; reprinted in Thomson's Papers <JH Electrostatics and Magnetism, p. 1. Also Camb. and Dub. Math. Journal, Nov., 1845 ; reprinted in Papers, p. 15.

\ As regards this comparison, Thomson had been anticipated by Chasles, Journal de 1'Ec. Polyt. xv (1837), p. 266, who had shown that attraction accord- ing to Newton's law gives rise to the same fields as the steady conduction of heat, both depending on Laplace's equation v' V =• 0.

It will be remembered that Ohm had used an analogy between thermal conduction and galvanic phenomena.

270 Maxwell.

It may, perhaps, seem as if the value of such an analogy as this consisted merely in the prospect which it offered of comparing, and thereby extending, the mathematical theories of heat and electricity. But to the physicist its chief interest lay rather in the idea that formulae which relate to the electric field, and which had heen deduced from laws of action at a distance, were shown to be identical with formulae relating to the theory of heat, which had been deduced from hypotheses of action between contiguous particles.

In 1846 — the year after he had taken his degree as second wrangler at Cambridge — Thomson investigated* the analogies of electric phenomena with those of elasticity. For this purpose he examined the equations of equilibrium of an incompressible elastic solid which is in a state of strain ; and showed that the distribution of the vector which represents the elastic displacement might be assimilated to the distribution of the electric force in an electrostatic system. This, however, as he went on to show, is not the only analogy which may be perceived with the equations of elasticity ; for the elastic displacement may equally well be identified with a vector a, defined in terms of the magnetic induction B by the relation

curl a = B.

The vector a is equivalent to the vector-potential which had been used in the memoirs of Neumann, Weber, and Kirchhoff, on the induction of currents ; but Thomson arrived at it independently by a different process, and without being at the time aware of the identification.

The results of Thomson's memoir seemed to suggest a picture of the propagation of electric or magnetic force : might it not take place in somewhat the same way as changes in the elastic displacement are transmitted through an elastic solid ? These suggestions were not at the time pursued further by their author; but they helped to inspire another young

• Camb. and Dub. Math. Journ. ii (1847), p. 61 : Thomson's Math, and Phys. Papers, i, p. 76.

Maxwell. 271

Cambridge man to take up the matter a few years later. James Clerk Maxwell, by whom the problem was eventually solved, was born in 1831, the son of a landed proprietor in Dumfriesshire. He was educated at Edinburgh, and at Trinity College, Cambridge, of which society he became in 1855 a Fellow; and not long after his election to Fellowship, he communicated to the Cambridge Philosophical Society the first of his endeavours* to form a mechanical conception of the electro-magnetic field.

Maxwell had been reading Faraday's Experimental He- searches', and, gifted as he was with a physical imagination akin to Faraday's, he had been profoundly impressed by the theory of lines of force. At the same time, he was a trained mathematician ; and the distinguishing feature of almost all his researches was the union of the imaginative and the analytical faculties to produce results partaking of both natures. This first memoir may be regarded as an attempt to connect the ideas of Faraday with the mathematical analogies which had been devised by Thomson.

Maxwell considered first the illustration of Faraday's lines of force which is afforded by the lines of flow of a liquid. The lines of force represent the direction of a vector; and the magnitude of this vector is everywhere inversely proportional to the cross-section of a narrow tube formed by such lines. This relation between magnitude and direction is possessed by any circuital vector ; and in particular by the vector which represents the velocity at any point in a fluid, if the fluid be incompressible. It is therefore possible to represent the magnetic induction B, which is the vector represented by Faraday's lines of magnetic force, as the velocity of an incom- pressible fluid. Such an analogy had been indicated some years previously by Faraday himself,f who had suggested that along the lines of magnetic force there may be a " dynamic condition," analogous to that of the electric current, and

• Trans. Camb. Phil. Soc. x, p. 27; Maxwell's Scientific Papers, i, p. 155. t Exp. Res., § 3269 (1852).

272 Maxwell.

that, in fact, " the physical lines of magnetic force are currents."

The comparison with the lines of flow of a liquid is applicable to electric as well as to magnetic lines of force. In this case the vector which corresponds to the velocity of the fluid is, in free aether, the electric force E. But when different dielectrics are present in the field, the electric force is not a circuital vector, and, therefore cannot be represented by lines of force ; in fact, the equation

div E = 0 is now replaced by the equation

div(eE) = 0,

where g denotes the specific inductive capacity or dielectric constant at the place (x, y} z\ It is, however, evident from this equation that the vector cE is circuital ; this vector, which will be denoted by D, bears to E a relation similar to that which the magnetic induction B bears to the magnetic force H. It is the vector D which is represented by Faraday's lines of electric force, and which in the hydrodynamical analogy corresponds to the velocity of the incompressible fluid.

In comparing fluid motion with electric fields it is necessary to introduce sources and sinks into the fluid to correspond to the electric charges ; for D is not circuital at places where there, is free charge. The magnetic analogy is therefore somewhat the simpler.

In the latter half of his memoir Maxwell discussed how Faraday's "electrotonic state" might be represented in mathe- matical symbols. This problem he solved by borrowing from Thomson's investigation of 1847 the vector a, which is defined in terms of the magnetic induction by the equation

curl a = B ;

if, with Maxwell, we call a the electrotonic intensity, the. equation is equivalent to the statement that " the entire electrotonic intensity round the boundary of any surface measures the number of lines of magnetic force which pass,

Maxwell. 273

through that surface." The electromotive force of induction at the place (x, y, z) is - d&/dt : as Maxwell said, " the electromotive force on any element of a conductor is measured by the instantaneous rate of change of the electrotonic intensity on that element." From this it is evident that a is no other than the vector-potential which had been employed by Neumann, Weber, and Kirchhoff, in the calculation of induced currents ; and we may take* for the electrotonic intensity due to a current ir flowing in a circuit s' the value which results from Neumann's theory, namely,

., f *s' = t'

} r

It may, however, be remarked that the equation

curl a = B,

taken alone, is insufficient to determine a uniquely ; for we can choose a so as to satisfy this, and also to satisfy the equation

div a = ;//,

where i// denotes any arbitrary scalar. There are, therefore, an infinite number of possible functions a. With the particular value of a which has been adopted, we have

3 ., f dx' 8 f dy' 8 ., f dz div a = - i \ - + — ^' -2- + - i' \ — te I' r fy )8, r dz J, r

., *« ¥

= 0; so the vector-potential a which we have chosen is circuital.

In this memoir the physical importance of the operators curl and div first became evidentf ; for, in addition to those applications which have been mentioned, Maxwell showed that

• Cf . p. 224.

t These operators had, however, occurred frequently in the writings of Stokes especially in his memoir of 1849 on the Dynamical Theory of Diffraction.

T

274 Maxwell.

he connexion between the strength i of a current and the magnetic field H, to which it gives rise, may be represented by the equation

4?ri = curl H ;

this equation is equivalent to the statement that " the entire magnetic intensity round the boundary of any surface measures the quantity of electric current which passes through that surface."

In the same year (1856) in which Maxwell's investigation was published, Thomson* put forward an alternative inter- pretation of magnetism. He had now come to the conclusion, from a study of the magnetic rotation of the plane of polariza- tion of light, that magnetism possesses a rotatory character; and suggested that the resultant angular momentum of the thermal motions of a bodyf might be taken as the measure of the magnetic moment. " The explanation," he wrote, " of all phenomena of electromagnetic attraction or repulsion, or of electromagnetic induction, is to be looked for simply in the inertia or pressure of the matter of which the motions constitute heat. Whether this matter is or is not electricity, whether it is a continuous fluid interpermeating the spaces between molecular nuclei, or is itself molecularly grouped : or whether all matter is continuous, and molecular heterogeneous- ness consists in finite vortical or other relative motions of contiguous parts of a body: it is impossible to decide, and, perhaps, in vain to speculate, in the present state of science."

The two interpretations of magnetism, in which the linear and rotatory characters respectively are attributed to it, occur frequently in the subsequent history of the subject. The former was amplified in 1858, when Helmholtz published his researches^ on vortex motion ; for Helmholtz showed that if a

*Proc. Roy. Soc. viii (1856), p. 150 ; xi (1861), p. 327, foot-note: Phil. Mag. xiii (1857), p. 198; Baltimore Lectures, Appendix F.

t This was written shortly before the kinetic theory of gases was developed by Clausius and Maxwell.

• Journal fur Math. Iv (1858), p. 25; Helmholtz's Wiss. Abh. i, p. 101; translated Phil. Mag. xxxiii (1867), p. 485.

Harwell. 275

magnetic field produced by electric currents is compared to the flow of an incompressible fluid, so that the magnetic vector is represented by the fluid velocity, then the electric currents correspond to the vortex-filaments in the fluid. This analogy correlates many theorems in hydrodynamics and electricity ; for instance, the theorem that a re-entrant vortex-filament is equivalent to a uniform distribution of doublets over any surface bounded by it, corresponds to Ampere's theorem of the equivalence of electric currents and magnetic shells.

In his memoir of 1855, Maxwell had not attempted to construct a mechanical model of electrodynamic actions, but had expressed his intention of doing so. " By a careful study," he wrote,* " of the laws of elastic solids, and of the motions of viscous fluids, I hope to discover a method of forming a mechanical conception of this electrotonic state adapted to general reasoning " ; and in a foot-note he referred to the effort which Thomson had already made in this direction. Six years elapsed, however, before anything further on the subject was published. In the meantime, Maxwell became Professor of Natural Philosophy in King's College, London — a position in which he had opportunities of personal contact with Faraday, whom he had long reverenced. Faraday had now concluded the Experimental Researches, and was living in retirement at Hampton Court ; but his thoughts frequently recurred to the great problem which he had brought so near to solution. It appears from his note-book that in 1857f he was speculating whether the velocity of propagation of magnetic action is of the same order as that of light, and whether it is affected by the susceptibility to induction of the bodies through which the action is transmitted.

The answer to this question was furnished in 1861-2, when Maxwell fulfilled his promise of devising a mechanical conception of the electromagnetic field.*

• Maxwell's Scientific Papers, i, p. 188. t Bence Jones's Life of Faraday ii, p. 379.

I Phil. Mag. xxi (1861), pp. 161, 281, 338; xxiii (1862), pp. 12, 85; Maxwell's Scientific Papers, i, p. 451.

T 2

276 Maxwell.

In the interval since the publication of his previous memoir Maxwell had become convinced by Thomson's arguments that magnetism is in its nature rotatory. "The transference of electrolytes in fixed directions by the electric current, and the rotation of polarized light in fixed directions by magnetic force, are," he wrote, "the facts the consideration of which has induced me to regard magnetism as a phenomenon of rotation, and electric currents as phenomena of translation." This con- ception of magnetism he brought into connexion with Faraday's idea, that tubes of force tend to contract longitudinally and to expand laterally. Such a tendency may be attributed to centrifugal force, if it be assumed that each tube of force contains fluid which is in rotation about the axis of the tube. Accordingly Maxwell supposed that, in any magnetic field, the medium whose vibrations constitute light is in rotation about the lines of magnetic force; each unit tube of force may for the present be pictured as an isolated vortex.

The energy of the motion per unit volume is proportional to /jH2, where /j. denotes the density of the medium, and H denotes the linear velocity at the circumference of each vortex. But, as we have seen,* Thomson had already shown that the energy of any magnetic field, whether produced by magnets or by electric currents, is

where the integration is taken over all space, and where it denotes the magnetic permeability, and H the magnetic force. It was therefore natural to identify the density of the medium at any place with the magnetic permeability, and the circum- ferential velocity of the vortices with the magnetic force.

But an objection to the proposed analogy now presents itself. Since two neighbouring vortices rotate in the same direction, the particles in the circumference of one vortex must be moving in the opposite direction to the particles contiguous

• Cf. pp. 248, 250.

Maxwell. 277

to them in the circumference of the adjacent vortex ; and it seems, therefore, as if the motion would be discontinuous. Maxwell escaped from this difficulty by imitating a well-known mechanical arrangement. When it is desired that two wheels should revolve in the same sense, an " idle " wheel is inserted between them so as to be in gear with both. The model of the electromagnetic field to which Maxwell arrived by the intro- duction of this device greatly resembles that proposed by Bernoulli in 1736.* He supposed a layer of particles, acting as idle wheels, to be interposed between each vortex and the next, and to roll without sliding on the vortices ; so that each vortex tends to make the neighbouring vortices revolve in the same direction as itself. The particles were supposed to be not other- wise constrained, so that the velocity of the centre of any particle would be the mean of the circumferential velocities of the vortices between which it is placed. This condition yields (in suitable units) the analytical equation

47Ti = curl H,

where the vector i denotes the flux of the particles, so that its ^-component ix denotes the quantity of particles transferred in unit time across unit area perpendicular to the ^-direction. On comparing this equation with that which represents Oersted's discovery, it is seen that the flux i of the movable particles interposed between neighbouring vortices is the analogue of the electric current.

It will be noticed that in Maxwell's model the relation between electric current and magnetic force is secured by a connexion which is not of a dynamical, but of a purely kine- matical character. The above equation simply expresses the existence of certain non-holonomic constraints within the system.

If from any cause the rotatory velocity of some of the cellular vortices is altered, the disturbance will be propagated from that part of the model to all other parts, by the mutual

• Cf. p. 100.

278 Maxwell.

action of the particles and vortices. This action is determined, as Maxwell showed, hy the relation

fj$L = - curl E

which connects E, the force exerted on a unit quantity of particles at any place in consequence of the tangential action of the vortices, with H, the rate of change of velocity of the neighbouring vortices. It will be observed that this equation is not kinematical but dynamical. On comparing it with the electromagnetic equations

curl a = /*H,

Induced electromotive force = - a, it is seen that E must be interpreted electromagnetically as the induced electromotive force. Thus the motion of the particles constitutes an electric current, the tangential force with which they are pressed by the matter of the vortex-cells constitutes electromotive force, and the pressure of the particles on each other may be taken to correspond to the tension or potential of the electricity.