A History of the Theories of Aether and Electricity (1910) — part 21 of 29

The greatest advance in the vortex-sponge theory of the aether was made in 1887, when W. Thomson* 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 {/(?/), 0, Oj, superposed on a homogeneous, isotropic, and fine- grained distribution (u'0t v0, w0) : so that at the origin of time the velocity is {/ (y) + u'0, v0, wn\ : it is desired to find a function / (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

du _ du du du dp

dt dx ~ dy~ dz ~ dx'

• Phil. Mug. xxiv (1887), p. 342 : Kelvin's Math, and Phys. Papers, iv, p. 308.

Models of the Aether. 329

there results

W dp

• w — - £.

dz dx

Take now the #2-averages of both members. The quantities du'/dt, du'/dx, v, dp/dx have zero averages; so the equation takes the form

df(y*t) ( ,W M

• = - A . [u -- + v — + w dt \ dx dy

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

whence

du' dv dw

• ~ + = '

f\ A I / *** / t/*1' ^ \JWJ

1 aaT"1 * ^+ l 9z

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

a)

From this it is seen that if the turbulent motion were to remain continually isotropic as at the beginning,/ (T/, t) would constantly retain its critical value /(y). In order to examine the deviation from isotropy, we shall determine Ad (u'v)/dt, which may be done in the following way : — Multiplying the u- and ^-equations of motion by v, u' respectively, and adding, we have

-.

' fa ty dx

d (u'v) d (u'v) dp , dp

-V- - ~ w ~V— -v^--uf^-

dy dz dx ty

330 Models of the Aether.

Taking the ^-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 . 3 (u'v]fix is zero. Denoting by %RZ the average value of uz, vz, or w1, so that R may be called the average velocity of the turbulent motion, the equation becomes

It ^ • (-

V * #2 y^

' y O

9)jm 3-"' ^

Vj

where

n. i la> «'>,,

9(^) d(u

y) 9p , 3p

i ^ _ f u dx dy

Let p be written (jp' + TO), where y denotes the value which p would have if / were zero. The equations of motion immediately give

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

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

Moreover, we have

. , , tyu'v) d(uv) 0

for positive and negative values of u, v, w 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, w we substitute - u', -v, -w, which (as may be seen by inspection of the above equation in V2^>) does not change the value of p'.

Models of the Aether. 331

Comparing this equation with that which determines the value of Q, we have

' d^

or substituting for CT,

The isotropy with respect to x and z gives the equation

, 8 a\ 8 _

-+ ^0- h" V ^«

But by integration by parts we obtain the equation

' U.v-^o=_

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

A . (tojty) . (d/ty) . V-2Vo, or - A . v0 . (d/zdf) . V-^o ; so we have

On account of the isotropy, we may write J for

and, therefore,

The deviation from isotropy shown by this equation is very small, because of the smallness of df(y, t)/dy. The equation is therefore not restricted to the initial values of the two members,

332 Models of the Aether.

for we may neglect an infinitesimal deviation from (2/9) IP 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

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

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* he showed that stability was ensured at any rate when space is filled with a set of approximately straight hollow vortex filaments. Fitz Geraldf subsequently determined the energy per unit-volume in a turbulent liquid which is transmitting laminar waves. Writing for brevity

(2/9) R* - V\ f(y, t) = P, and A (u'v) = 7, the equations are

s?.--h and h-.y*^ dt dy' ft " 8y

If the quantity

p-f jVP«2S

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

JIM-

• Proc. Roy. Irish Acad. (3) i (1889), p. 340 ; Kelvin's Math, and Phys. Papers, iv, p. 202.

t Brit. Assoc. Rep., 1899. Fitz Gerald's Scientific Writings, p. 484.

Models of the Aether. 333

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

0fa***.JJJp(*+g

Hence it appears that the quantity S, 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.* 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 Fitz Gerald, f Since the distribution of velocity of a fluid in the

• Brit. Assoc. Rep., 1895, p. 595.

t Proc. Roy. Dublin Soc., December 12, 1899; Fitz Gerald's Scientific Writings, p. 472.

334 Models of the Aether.

neighbourhood of a vortex filament is the same as the distribu- tion 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 dis- tributed 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.

Fitz Gerald'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 electro- statical systems. One such, which was described in 1888 by W. M. Hicks,* 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 filamentf stretching from one to the other, with rotation

• Brit. Assoc. Rep., 1888, p. 577.

} A hollow vortex is a cyclic motion existing in a fluid without the presence of any actual rotational filaments. On the general theory cf. Hicks, Phil. Trans, clxxv (1883), p. 161 ; clxxvi (1885), p. 725 ; cxcii (1898), p. 33.

Models of the Aether. 335

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 there- fore 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, how- ever, there is no bodily flow; for the two kinds of filament produce circulation in opposite directions.

336 Models of the Aether.

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.* 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.

• Proc. Roy. Irish Acad., November 30, 1889 ; Kelvin's Math, and Phys. Papers, iv, p. 202. "Rotational vortex-cores," he wrote, "must he absolutely discarded ; and we must have nothing hut irrotational revolution and vacuous cores."

( 337 )

CHAPTEE X.

THE FOLLOWERS OF MAXWELL.

THE most notable imperfection in the electromagnetic theory of light, as presented in Maxwell's original memoirs, was the absence of any explanation of reflexion and refraction. Before the publication of Maxwell's Treatise, however, a method of supplying the omission was indicated by Helmholtz.* The principles on which the explanation depends are that the normal component of the electric displacement D, the tangential components of the electric force E, and the magnetic vector B or H, are to be continuous across the interface at which the reflexion takes place; the optical difference between the con- tiguous bodies being represented by a difference in their dielectric constants, and the electric vector being assumed to be at right angles to the plane of polarization.-)- The analysis required is a mere transcription of MacCullagh's theory of reflexion,| if the derivate of MacCullagh's displacement e with respect to the time be interpreted as the magnetic force, fi curl e as the electric force, and curl e as the electric displace- ment. The mathematical details of the solution were not given by Helmholtz himself, but were supplied a few years later in the inaugural dissertation of H. A. Lorentz.§

In the years immediately following the publication of Maxwell's Treatise, a certain amount of evidence in favour of

• Journal fur Math. Ixxii (1870), p. 68, note.

t Helmholtz (loc. cit.) pointed out that if the optical difference between the media were assumed to be due to a difference in their magnetic permeabilities, it would be necessary to suppose the magnetic vector at right angles to the plane of polarization in order to obtain Fresnel's sine and tangent formulae of reflexion.

I Cf. pp. 148, 149, 154-156.

§ Zeitschrift fiir Math. u. Phys. xxii (1877), pp. 1, 205 : Over de theorie der terugkaatsing en breking van het licht, Arnhem, 1875. Lorentz's work was based on Helmholtz's equations, but remains substantially unchanged when Maxwell's formulae are substituted.

Z

338 The Followers of Maxwell.

his theory was furnished by experiment. That an electric field is closely concerned with the propagation of light was demon- strated in 1875, when John Kerr* showed that dielectrics subjected to powerful electrostatic force acquire the property of double refraction, their optical behaviour being similar to that of uniaxal crystals whose axes are directed along the lines of force.

Other researches undertaken at this time had a more direct bearing on the questions at issue between the hypothesis of Maxwell and the older potential theories. In 1875-6 Helmholtzf and his pupil Schiller^ attempted to discriminate between the various doctrines and formulae relative to unclosed circuits by performing a crucial experiment.

It was agreed in all theories that a ring-shaped magnet, which returns into itself so as to have no poles, can exert no ponderomotive force on other magnets or 011 closed electric currents. Helmholtz§ had, however, shown in 1873 that accord- ing to the potential-theories such a magnet would exert a ponderomotive force on an unclosed current. The matter was tested by suspending a magnetized steel ring by a long fibre in a closed metallic case, near which was placed a terminal of a Holtz machine. No ponderomotive force could be observed when the machine was put in action so as to produce a brush discharge from the terminal : from which it was inferred that the potential-theories do not correctly represent the phenomena, at least when displacement-currents and convection -currents (such as that of the electricity carried by the electrically repelled air from the terminal) are not taken into account.

The researches of Helmholtz and Schiller brought into prominence the question as to the effects produced by the

• Phil. Mag. (4) 1 (1875), pp. 337, 446 ; (5) viii (1879), pp. 85, 229 ; xiii (1882), pp. 153, 248.

t Monatsberichte d. Acad. d. Berlin/1875, p. 400. Ann. d. Phys., clviii (1876), p. 87. t Ann. d. Phys. clix (1876), pp. 456, 537 ; clx (1877), p. 333.

\ The valuable memoirs by Helmholtz in Journal fiir Math. Ixxii (1870), p. 57 ; Ixxv (1873), p. 35 ; Ixxviii (1874), p. 273, to which reference has already been made, contain a full discussion of the various possibilities of the potential- theories.

The Followers of Maxwell. 339

translatory motion of electric charges. That the convection of electricity is equivalent to a current had been suggested long before by Faraday.* "If," he wrote in 1838, "a baU be electrified positively in the middle of a room and be then moved in any direction, effects will be produced as if a current in the same direction had existed." To decide the matter a new experiment inspired by Helmholtz was performed by H. A. Kowlandf in 1876. The electrified body in Kowland's disposition was a disk of ebonite, coated with gold leaf and capable of turning rapidly round a vertical axis between two fixed plates of glass, each gilt on one side. The gilt faces of the plates could be earthed, while the ebonite disk received electricity from a point placed near its edge ; each coating of the disk thus formed a condenser with the plate nearest to it. An astatic needle was placed above the upper condenser-plate, nearly over the edge of the disk; and when the disk was rotated a magnetic field was found to be produced. This experiment, which has since been repeated under improved conditions by Kowland and Hutchinson,J H. Fender §, and Eichenwald,|| shows that the " convection-current " produced by the rotation of a charged disk, when the other ends of the lines of force are on an earthed stationary plate parallel to it, produces the same mag- netic field as an ordinary conduction-current flowing in a circuit which coincides with the path of the convection-current. When two disks forming a condenser are rotated together, the magnetic action is the sum of the magnetic actions of each of the disks separately. It appears, therefore, that electric charges cling to the matter of a conductor and move with it, so far as Rowland's phenomenon is concerned.

The first examination of the matter from the point of view of Maxwell's theory was undertaken by J. J. Thomson,1[ in 1881. If an electrostatically charged body is in motion, the change in

• Exper.Re*., § 1644.

t Monatsberichte d. Akad. d. Berlin, 1876, p. 211 : Ann. d. Phys. clviii (1876), p. 487 : Annales de Chim. et de Phys. xii (1877) p. 119.

i Phil. Mag. xxvii (1889), p. 445. \ Ibid, ii (1901), p. 179 : v (1903), p. 34. || Ann. d. Phys. xi (1901), p. 1. H Phil. Mag. xi (1881), p. 229.

Z 2

340 The Followers of Maxwell.

the location of the charge must produce a continuous alteration of the electric field at any point in the surrounding medium ; or, in the language of Maxwell's theory, there must be displacement- currents in the medium. It was to these displacement-currents that Thomson, in his original investigation, attributed the magnetic effects of moving charges. The particular system which he considered was that formed by a charged spherical conductor, moving uniformly in a straight line. It was assumed that the distribution of electricity remains uniform over the surface during the motion, and that the electric field in any position of the sphere is the same as if the sphere were at rest ; these assumptions are true so long as quantities of order (V/c)2 are neglected, where v denotes the velocity of the sphere and c the velocity of light.

Thomson's method was to determine the displacement- currents in the space outside the sphere from the known values of the electric field, and then to calculate the vector- potential due to these displacement-currents by means of the formula

where S' denotes the displacement-current at (x'y'zf). The magnetic field was then determined by the equation

H = curl A.

A defect in this investigation was pointed out by Fitz Gerald, who, in a short but most valuable note,* published a few months afterwards, observed that the displacement-currents of Thomson do not satisfy the circuital condition. This is most simply seen by considering the case in which the system consists of two parallel plates forming a condenser; if one of the plates is fixed, and the other plate is moved towards it, the electric field is annihilated in the space over which the moving plate travels : this destruction of electric displacement constitutes a displace- ment-current, which, considered alone, is evidently not a closed

• Proc. Roy. Dublin Soc., November, 1881 ; Fitz Gerald's Scientific Writings, p. 102.

The Followers #/ Maxwell. 341

current. The defect, as Fitz Gerald showed, may be immediately removed by assuming that a moving charge itself is to be counted as a current-element : the total current, thus composed of the displacement- currents and the convection-current, is circuital. Making this correction, Fitz Gerald found that the magnetic force due to a sphere of charge e moving with velocity v along the axis of z is curl (0, 0, ev/r) — a formula which shows that the displacement-currents have no resultant magnetic effect, since the term ev/r would be obtained from the convection-current alone.

The expressions obtained by Thomson and Fitz Gerald were correct only to the first order of the small quantity v/c. The effect of including terms of higher order was considered in 1889 by Oliver Heaviside,* whose solution may be derived in the following manner : —

Suppose that a charged system is in motion with uniform velocity v parallel to the axis of z ; the total current consists of the displacement- cur rent E/4?rc2 where E denotes the electric force, and the convection-current pv where p denotes the volume-density of electricity. So the equation which connects magnetic force with electric current may be written

E/c2 = curl H - 4:irpv. Eliminating E between this and the equation

curl E = - H, and remembering that H is here circuital, we have

H/c2 - V2H = 4?r curl pv. If, therefore, a vector-potential a be defined by the equation

a/c3 - V2a = 4?rpv,

the magnetic force will be the curl of a ; and from the equation for a it is evident that the components ax and ay are zero, and that az is to be determined from the equation az/c~ - V"az = 4npv.

• Phil. Mag. xxvii (1889), p. 324.

342 The Followers of Maxwell.

Now, let (x, y, £) denote coordinates relative to axes which are parallel to the axes (a;, y, z) , and which move with the charged bodies ; then az is a function of (x, y, £) only ; so we have

a a , a a

5 -IT and *' "Vr

and the preceding equation is readily seen to be equivalent to

where £1 denotes (1 - v'/c2)'^. But this is simply Poisson's equation, with & substituted for z; so the solution may be transcribed from the known solution of Poisson's equation : it is

/L»V dx' dy d%i'

the integrations being taken over all the space in which there are moving charges ; or

_rrr

jJJ

If the moving system consists of a single charge e at the point 5 = 0, this gives

ev

%(1 - tf sin8 0/c")* ' where sin2 0 = (a8 + y2)/r2.

It is readily seen that the lines of magnetic force due to the moving point-charge are circles whose centres are on the line of motion, the magnitude of the magnetic force being

ev (1 - v2/c2) sin 8

The electric force is radial, its magnitude being

r2(l - v2sin2 0/c2)f

The fact that the electric vector due to a moving point- charge is everywhere radial led Heaviside to conclude that the same solution is applicable when the charge is distributed over

The Followers of Maxwell. 343

a perfectly conducting sphere whose centre is at the point, the only chaftge being that E and H would now vanish inside the sphere. This inference was subsequently found* to be incorrect : a distribution of electric charge on a moving sphere could in fact not be in equilibrium if the electric force were radial, since there would then be nothing to balance the mechanical force exerted on the moving charge (which is equivalent to a current) by the magnetic field. The moving system which gives rise to the same field as a moving point-charge is not a sphere, but an oblate spheroid whose polar axis (which is in the direction of motion) bears to its equatorial axis the ratio (1 - tf/c*)^ : !.•)•

The energy of the field surrounding a charged sphere is greater when the sphere is in motion than when it is at rest. To determine the additional energy quantitatively (retaining only the lowest significant powers of v/c), we have only to integrate, throughout the space outside the sphere, the expression H2/87r, which represents the electrokinetic energy per unit volume : the result is ezv~/3a, where e denotes the charge, v the velocity, and a the radius of the sphere.

It is evident from this result that the work required to be done in order to communicate a given velocity to the sphere is greater when the sphere is charged than when it is uncharged ; that is to say, the virtual mass of the sphere is increased by an amount 2e2/3a, owing to the presence of the charge. This may be regarded as arising from the self-induction of the convection- current which is formed when the charge is set in motion. It was suggested by J. LarmorJ and by W. Wien§ that the inertia of ordinary ponderable matter may ultimately prove to be of this nature, the atoms being constituted of systems of electrons. ||

» By G. F. C. Searle.

t Cf. Searle, Phil. Trans, clxxxvii (1896), p. 675, and Phil. Ma?. xliv (1897), p. 329. On the theory of the moving electrified sphere, cf. also J. J. Thomson, Recent Researches in Elect, and Mag., p. 16; 0. Heaviside, Electrical Papers, ii, p. 514; Electromag. Theory, i, p. 269; W. B. Morton, Phil. Mag, xli (1896), p. 488 ; A. Schuster, Phil. Mag. xliii (1897), p. 1.

• Phil. Trans, clxxxvi (1895), p. 697. § Arch. Neerl (3) v (1900), p. 96.

|| Experimental evidence that the inertia of electrons is purely electromagnetic was afterwards furnished hy W. Kaufmann, Gott. Nach., 1901, p. 143 ; 1902, p 291.

344 The Followers of Maxwell.

It may, however, be remarked that this view of th«>rigin of mass is not altogether consistent with the principle^that the electron is an indivisible entity. For the so-called self-induction of the spherical electron is really the mutual induction of the convection-currents produced by the elements of electric charge which are distributed over its surface ; and the calculation of this quantity presupposes the divisibility of the total charge into elements capable of acting severally in all respects as ordinary electric charges ; a property which appears scarcely consistent with the supposed fundamental nature of the electron.

After the first attempt of J. J. Thomson to determine the field produced by a moving electrified sphere, the mathematical development of Maxwell's theory proceeded rapidly. The problems which admit of solution in terms of known functions are naturally those in which the conducting surfaces involved have simple geometrical forms — planes, spheres, and cylinders.*

A result which was obtained by Horace Lamb,f when investigating electrical motions in a spherical conductor, led to interesting consequences. Lamb found that if a spherical conductor is placed in a rapidly alternating field, the induced currents are almost entirely confined to a superficial layer ; and his result was shortly afterwards generalized by Oliver Heavi- side,| who showed that whatever be the form of a conductor rapidly alternating currents do not penetrate far into its sub- stance.§ The reason for this may be readily understood : it is virtually an application of the principle|| that a perfect conductor is impenetrable to magnetic lines of force. No perfect conductor is known to exist ; butU if the alternations of magnetic force to which a good conductor such as copper is exposed are very

• Cf., e.g., C. Niven, Phil. Trans, clxxii (1881), p. 307 ; H. Lamb, Phil. Trans, clxxiv (1883), p. 519 ; J. J. Thomson, Proc. Lond. Math. Soc. xv (1884), p. 197 : H. A. Rowland, Phil. Mag. xvii (1884), p. 413 ; J. J. Thomson, Proc. Lond. Math. Soc. xvii (1886), p. 310; xix (1888), p. 520; and many investigations of Oliver Heaviside, collected in his Electrical Papers.

t Loc. cit. % Electrician, Jan. 1885.

§ The mathematical theory was given hy Lord Rayleigh, Phil. Mag. xxi. (18S6), p. 381. Cf. Maxwell's Treatise, § 689. || Cf. p. 313.

H As was first remarked by Lord Rayleigh, Phil. Mag. xiii (1882), p. 344.

The Followers of Maxwell. 345

rapid, the. conductor has not time (so to speak) to display the impSfection of its conductivity, and the magnetic field is therefore unable to extend far below the surface.

The same conclusion may be reached by different reasoning.* When the alternations of the current are very rapid, the ohmic resistance ceases to play a dominant part, and the ordinary equations connecting electromotive force, induction, and current are equivalent to the conditions that the currents shall be so distributed as to make the electrokinetic or magnetic energy a minimum. Consider now the case of a single straight wire of circular cross-section. The magnetic energy in the space outside the wire is the same whatever be the distribution of current in the cross-section (so long as it is symmetrical about the centre), since it is the same as if the current were flowing along the central axis ; so the condition is that the magnetic energy in the wire shall be a minimum ; and this is obviously satisfied when the current is concentrated in the superficial layer, since then the magnetic force is zero in the substance of the wire.

In spite of the advances which were effected by Maxwell and his earliest followers in the theory of electric oscillations, the gulf between the classical electrodynamics and the theory of light was not yet completely bridged. For in all the cases considered in the former science, energy is merely exchanged between one body and another, remaining within the limits of a given system ; while in optics the energy travels freely through space, unattached to any material body. The first discovery of a more complete connexion between the two theories was made by Fitz Gerald, who argued that if the unification which had been indicated by Maxwell is valid, it ought to be possible to generate radiant energy by purely electrical means; and in 1883f he described methods by which this could be done.

Fitz Gerald's system is what has since become known as the magnetic oscillator : it consists of a small circuit, in which

• Of. J. Stefan, Wiener, Situungsber. xcix (1890), p. 319 ; Ann. d. Phys. xli (1890), p. 400.

t Trans. Roy. Dublin Soc. iii (1883) ; Fitz Gerald's Scient. Writings, p. 122.

346 The Followers of Maxwell.

the strength of the current is varied according to the simple periodic law. The circuit will be supposed to be ft circle of small area S, whose centre is the origin and whose plane is the plane of xy ; and the surrounding medium will be supposed to be free aether. The current may be taken to be of strength A cos (2ni?/jF), so that the moment of the equivalent magnet is SA cos (2irt/T). Now in the older electrodynamics, the vector-potential due to a magnetic molecule of (vector) moment M at the origin is (l/47r) curl (M/r), where r denotes distance from the origin. The vector-potential due to Fitz Gerald's magnetic oscillator would therefore be (l/47r) curl K, where K denotes a vector parallel to the axis of z, and of magnitude (1/r) SA cos (2-n-t/ T). The change which is involved in replacing the assumptions of the older electrodynamics by those of Maxwell's theory is in the present case equivalent* to retarding the potential ; so that the vector-potential a due to the oscillator is (l/47r) curl K where K is still directed parallel to the axis of z, and is of magnitude

SA 27T/ r K = - — cos — [ t —

The electric force E at any point of space is - a, and the magnetic force H is curl a : so that these quantities may be calculated without difficulty. The electric energy per unit volume is E2/8?rc2 : performing the calculations, it is found that the value of this quantity averaged over a period of the oscillation and also averaged over the surface of a sphere of radius r is

The part of this which is radiated is evidently that which is proportional to the inverse square of the distance,! so the

• Cf. pp. 298, 299.

tThe other term, which is neglected, is very small compared to the term retained, at great distances from the origin ; it is what would be obtained if the effects of induction of the displacement-currents were neglected : i.e. it is the energy of the forced displacement-currents which are produced directly by the variation of the primary current, and which originate the radiating displacement- currents.

The Followers of Maxwell. 347