A History of the Theories of Aether and Electricity (1910) — part 22 of 29

average value of the radiant energy of electric type at distance r from the oscillator is 2iTzAS2/3criT* per unit volume. The radiant energy of magnetic type ma}' be calculated in a similar way, and is found to have the same value ; so the total radiant energy at distance r is 47r3^42/S^/3cVT4 per unit volume; and therefore the energy radiated in unit time is 16ir4tA'iS2/3csT*. This is small, unless the frequency is very high ; so that ordinary alternating currents would give no appreciable radia- tion. Fitz Gerald, however, in the same year* indicated a method by which the difficulty of obtaining currents of sufficiently high frequency might be overcome: this was, to employ the alternating currents which are produced when a condenser is discharged.

The Fitz Gerald radiator constructed on this principle is closely akin to the radiator afterwards developed with such success by Hertz : the only difference is that in Fitz Gerald's arrangement the condenser is used merely as the store of energy (its plates being so close together that the electrostatic field due to the charges is practically confined to the space between them), and the actual source of radiation is the alternating magnetic field due to the circular loop of wire: while in Hertz's arrangement the loop of wire is abolished, the condenser plates are at some distance apart, and the source of radiation is the alternating electrostatic field due to their charges.

In the study of electrical radiation, valuable help is afforded by a general theorem on the transfer of energy in the electro- magnetic field, which was discovered in 1884 by John Henry Poynting.-)- We have seen that the older writers on electric currents recognized that an electric current is associated with the transport of energy from one place (e.g. the voltaic cell which maintains the current) to another (e.g. an electromotor which is worked by the current) ; but they supposed the energy to be conveyed by the current itself within the wire, in much

• Brit. Assoc. Rep., 1883 ; FitzGerald's Scientific Writings, p. 129. tPhil. Trans, clxxv (1884), p. 343.

348 The Followers of Maxwell.

the same way as dynamical energy is carried by water flowing in a pipe; whereas in Maxwell's theory, the storehouse and vehicle of energy is the dielectric medium surrounding the wire. What Poynting achieved was to show that the flux of energy at any place might be expressed by a simple formula in terms of the electric and magnetic forces at the place.

Denoting as usual by E the electric force, by D the electric displacement, by H the magnetic force, and by B the magnetic induction, the energy stored in unit volume of the medium is* l ED + (1/8*) BH ;

so the increase of this in unit time is (since in isotropic media D is proportional to E, and B is proportional to H)

ED + (1/4*) HB

or E (S - i) + (1/4*) HB,

where S denotes the total current, and i the current of

conduction ; or (in virtue of the fundamental electromagnetic

equations)

• (E . i) 4. (1/4*) (E . curl H) - (1/4*) (H . curl E}, or - (E . i) - (1/4*) div [E . H].

Now (E . i) is the amount of electric energy transformed into heat per unit volume per second; and therefore the quantity

• (1/4*) div [E . H] must represent the deposit of energy in unit volume per second due to the streaming of energy; which shows that the flux of energy is represented by the vector (1/4*) [E.HJ.f This is Poynting's theorem: that the flux of energy at any place is represented by the vector-product of the electric and magnetic forces, divided by 4*.*

• Cf. pp. 248, 250, 282.

t Of course any circuital vector may be added. II. M. Macdonald, Electric Waves, p. 72, propounded a form which differs from Poynting's by a non-circuital vector.

J The analogue of Poynting's theorem in the theory of the vibrations of an isotropic elastic solid may be easily obtained ; for from the equation of motion of an elastic solid,

p& = - (k + 4«/3) gnid div e — n curl curl e, it follows that

• tot* + i (* + $») (div e)» + in (curl e)'} = - div W,

The Followers of Maxwell. 349

In the special case of the field which surrounds a straight wire carrying a continuous current, the lines of magnetic force are circles round the axis of the wire, while the lines of electric force are directed along the wire ; hence energy must be flowing in the medium in a direction at right angles to the axis of the wire. A current in any conductor may therefore be regarded as consisting essentially of a convergence of electric and magnetic energy from the medium upon the conductor, and its trans- formation there into other forms.

This association of a current with motions at right angles to the wire in which it flows doubtless suggested to Poynting the conceptions of a memoir which he published* in the following year. When an electric current flowing in a straight wire is gradually increased in strength from zero, the surrounding space becomes filled with lines of magnetic force, which have the form of circles round the axis of the wire. Poynting, adopting Faraday's idea of the physical reality of lines of force, assumed that these lines of force arrive at their places by moving out- wards from the wire ; so that the magnetic field grows by a con- tinual emission from the wire of lines of force, which enlarge and spread out like the circular ripples from the place where a. stone is dropped into a pond. The electromotive force which is- associated with a changing magnetic field was now attributed directly to the motion of the lines of force, so that wherever electromotive force is produced by change in the magnetic field,, or by motion of matter through the field, the electric intensity is equal to the number of tubes of magnetic force intersected by unit length in unit time.

A similar conception was introduced in regard to lines of electric force. It was assumed that any change in the total

where W denotes the vector

• (k + 4w/3) div e . e + n [curl e . e] ;

and since the expression which is differentiated with respect to t represents the sum of the kinetic and potential energies per unit volume of the solid (save for terms which give only surface-integrals), it is seen that W is the analogue of the Poynting vector. Cf. L. Donati, Bologna Mem. (5) vii (1899), p. 633.

• Phil. Trans, clxxvi (1885), p. 277.

350 The Followers of Maxwell.

electric induction through a curve is caused by the passage of tubes of force in or out across the boundary ; so that whenever magnetomotive force is produced by change in the electric field, or by motion of matter through the field, the magnetomotive force is proportional to the number of tubes of electric force intersected by unit length in unit time.

Poynting, moreover, assumed that when a steady current C flows in a straight wire, C tubes of electric force close in upon the wire in unit time, and are there dissolved, their energy appearing as heat. If E denote the magnitude of the electric force, the energy of each tube per unit length is \E, so the amount of energy brought to the wire is \CE per unit length per unit time. This is, however, only half the energy actually transformed into heat in the wire : so Poynting further assumed that E tubes of magnetic force also move in per unit length per unit time, and finally disappear by contraction to infinitely small rings. This motion accounts for the existence of- the electric field ; and since each tube (which is a closed ring) contains energy of amount J(7, the disappearance of the tubes accounts for the remaining \GE units of energy dissipated in the wire.

The theory of moving tubes of force has been extensively developed by Sir Joseph Thomson.* Of the two kinds of tubes — magnetic and electric — which had been introduced by Faraday and used by Poynting, Thomson resolved to discard the former and employ only the latter. This was a distinct departure from Faraday's conceptions, in which, as we have seen, great significance was attached to the physical reality of the magnetic lines ; but Thomson justified his choice by inferences drawn from the phenomena of electric conduction in liquids and gases. As will appear subsequently, these phenomena indicate that molecular structure is closely connected with tubes of electro- static force — perhaps much more closely than with tubes of magnetic force ; and Thomson therefore decided to regard

• Phil. Mag. xxxi (1891), p. 149; Thomson's Recent Researches in Elect, and Mag. (1893), chapter i.

The Followers of Maxwell. 351

magnetism as the secondary effect, and to ascribe magnetic fields, not to the presence of magnetic tubes, but to the motion of electric tubes. In order to account for the fact that magnetic fields may occur without any manifestation of electric force, he assumed that tubes exist in great numbers everywhere in space, either in the form of closed circuits or else terminating on atoms, and that electric force is only perceived when the tubes have a greater tendency to lie in one direction than in another. In a steady magnetic field the positive and negative tubes might be conceived to be moving in opposite directions with equal velocities.

A beam of light might, from this point of view, be regarded simply as a group of tubes of force which are moving with the velocity of light at right angles to their own length. Such a conception almost amounts to a return to the corpuscular theory ; but since the tubes have definite directions per- pendicular to the direction of propagation, there would now be no difficulty in explaining polarization.

The energy accompanying all electric and magnetic pheno- mena was supposed by Thomson to be ultimately kinetic energy of the aether ; the electric part of it being represented by rota- tion of the aether inside and about the tubes, and the magnetic part being the energy of the additional disturbance set up in the aether by the movement of the tubes. The inertia of this latter motion he regarded as the cause of induced electromotive force.

There was, however, one phenomenon of the electromagnetic field as yet unexplained in terms of these conceptions — namely, the ponderomotive force which is exerted by the field on a conductor carrying an electric current. Now any pondero- motive force consists in a transfer of mechanical momentum from the agent which exerts the force to the body which experiences it ; and it occurred to Thomson that the pondero- motive forces of the electromagnetic field might be explained if the moving tubes of force, which enter a conductor carrying a current and are there dissolved, were supposed to possess

352 The Followers of Maxwell.

mechanical momentum, which could be yielded up to the conductor. It is readily seen that such momentum must be directed at right angles to the tube and to the magnetic induction — a result which suggests that the momentum stored in unit volume of the aether may be proportional to the vector- product of the electric and magnetic vectors.

For this conjecture reasons of a more definite kind may be given.* We have already seenf that the ponderomotive forces on material bodies in the electromagnetic field may be accounted for by Maxwell's supposition that across any plane in the aether whose unit normal is N, there is a stress represented by

PN = (D . N) E - J (D .E)N + (l/47r) (B . H)H - (I/Sir) (B. H) N.

So long as the field is steady (i.e. electrostatic or magnetostatic) the resultant of the stresses acting on any element of volume of the aether is zero, so that the element is in equilibrium. But when the field is variable, this is no longer the case. The resultant stress on the aether contained within a surface S is

JJ PN . dS

integrated over the surface : transforming this into a volume- integral, the term (D . N) E gives a term div D . E + (D . V) E, where V denotes the vector operator (9/9a?, d/dy, d/dz) ; and the first of these terms vanishes, since D is a circuital vector; the term - J (D . E) N gives in the volume-integral a term J grad (D . E) ; and the magnetic terms give similar results. So the resultant force on unit- volume of the aether is

(D . V) E + J grad (D . E) + (l/4ir) (B . V) E + (I /Sir) grad (B . H), which may be written

[curl E . D] + (l/47r) [curl H . B] ;

• The hypothesis that the aether is a storehouse of mechanical momentum, which was first advanced by ,T. J. Thomson (Recent Researches in Elect, and Mag. (1893), p. 13), was afterwards developed by H. Poincare, Archives Neerl. (2) v (1900), p. 252, and by M. Abraham, Gott, Nach., 1902, p. 20.

tCf. p. 302.

The Followers of Maxwell. 353

or, by virtue of the fundamental equations for dielectrics, [- B . D] + [D . B] , or (a/ft) [D . B].

This result compels us to adopt one of three alternatives: either to modify the theory so as to reduce to zero the resultant force on an element of free aether ; this expedient has not met with general favour ;* or to assume that the force in question sets the aether in motion: this alternative was chosen by Helmholtz,f but is inconsistent with the theory of the aether which was generally received in the closing years of the century; or lastly, with Thomson^ to accept the principle that the aether is itself the vehicle of mechanical momentum, of amount [D . B] per unit volume.

Maxwell's theory was now being developed in ways which could scarcely have been anticipated by its author. But although every year added something to the superstructure, the founda- tions remained much as Maxwell had laid them ; the doubtful argument by which he had sought to justify the introduction of displacement- currents was still all that was offered in their defence. In 1884, however, the theory was established§ on a different basis by a pupil of Helmholtz', Heinrich Hertz (b. 1857, d. 1894).

The train of Hertz' ideas resembles that by which Ampere, on hearing of Oersted's discovery of the magnetic field produced by electric currents, inferred that electric currents should exert ponderomotive forces on each other. Ampere argued that a current, being competent to originate a magnetic field, must be equivalent to a magnet in other respects ; and therefore that currents, like magnets, should exhibit forces of mutual attraction and repulsion.

• It was, however, adopted by G. T. "Walker, Aberration and the Electromagnetic Field, Camb., 1900.

t Berlin Sitzungsberichte, 1893, p. 649; Ann. d. Phys. liii (1894), p. 135. Helmholtz supposed the aether to behave as a frictionless incompressible fluid.

• Loc. cit.

§ Ann. d. Phys. xxiii (1884), p. 84: English version in Hertz's Miscellaneous Papers, translated by D. E. Jones and G. A. Schott, p. 273.

2 A

354 The Followers of Maxwell.

Ampere's reasoning rests on the assumption that the mag- netic field produced by a current is in all respects of the same nature as that produced by a magnet ; in other words, that only one land of magnetic force exists. This principle of the " unity of magnetic force" Hertz now proposed to supplement by assert- ing that the electric force generated by a changing magnetic field is identical in nature with the electric force due to electro- static charges; this second principle he called the "unity of electric force." Suppose, then, that a system of electric currents i exists in otherwise empty space. According to the older theory, these currents give rise to a vector-potential a, , equal to Pot i ;* and the magnetic force Ht is the curl of at : while the electric force E! at any point in the field, produced by the variation of the currents, is — ai.

It is now assumed that the electric force so produced is indistinguishable from the electric force which would be set up by electrostatic charges, and therefore that the system of varying currents exerts ponderomobive forces on electrostatic charges ; the principle of action and reaction then requires that electrostatic charges should exert ponderomotive forces on a system of varying currents, and consequently (again appealing to the principle of the unity of electric force) that two systems of varying currents should exert on each other ponderomotive forces due to the variations.

But just as Helmholtz,f by aid of the principle of conser- vation of energy, deduced the existence of an electromotive force of induction from the existence of the ponderomotive forces between electric currents (Le. variable electric systems), so from the existence of ponderomotive forces between variable systems of currents (i.e. variable magnetic systems) we may infer that variations in the rate of change of a variable magnetic system give rise to induced magnetic forces in the surrounding space. The analytical formulae which determine these forces

• a = Pot /3 is used to denote the solution of the equation V'a + 47r£ = 0. fCf. p. 243.

The Followers of Maxwell. 355

will be of the same kind as in the electric case ; so that the induced magnetic force H' is given by an equation of the form

where c denotes some constant, and bi, which is analogous to the vector-potential in the electric case, is a circuital vector whose curl is the electric force E! of the variable magnetic system. The value of bi is therefore (l/47r) curl Pot Et : so we have

H' = - J-. |, curl Pot a,

47TC" (jt~

This must be added to Hi. Writing H2 for the sum, Hi + H', we see that H2 is the curl of a2, where

and the electric force E2 will then be - a2.

This system is not, however, final ; for we must now perform the process again with these improved values of the electric and magnetic forces and the vector-potential ; and so we obtain for the magnetic force the value curl a3, and for the electric force the value - a3, where

1 r)z 1 ^*

= ax - - Pot ax + — — -- — Pot Pot

4rrc2 fit*

This process must again be repeated indefinitely ; so finally we obtain for the magnetic force H the value curl a, and for the electric force E the value - a, where

1 £}*>

• Pot Pot Pot a! +

(47TC2)3 2A2

356 The Followers of Maxwell.

It is evident that the quantity a thus defined satisfies the equation

or v*a - -— a = - 47ri.

c2 dt'

This equation may be written

while the equations H = curl a, E = - a give

curl E = - H.

These are, however, the fundamental equations of Maxwell's theory in the form given in his memoir of 1868,*

That Hertz's deduction is ingenious and interesting will readily be admitted. That it is conclusive may scarcely be claimed : for the argument of Helmholtz regarding the induc- tion of currents is not altogether satisfactory; and Hertz, in following his master, is on no surer ground.

In the course of a discussion^ on the validity of Hertz's assumptions, which followed the publication of his paper, E. AulingerJ brought to light a contradiction between the principles of the unity of electric and of magnetic force and the electrodynamics of Weber. Consider an electrostatically charged hollow sphere, in the interior of which is a wire carrying a variable current. According to Weber's theory, the sphere would exert a turning couple on the wire; but according to Hertz's principles, no action would be exerted, since charging the sphere makes no difference to either the electric or the magnetic force in its interior. The experiment thus suggested would be a crucial test of the correctness of Weber's theory ; it has the advantage of requiring nothing but closed currents and electrostatic charges at rest ; but the quantities to be observed would be on the limits of observational accuracy. »Cf. p. 287.

f Lorberg, Ann. d. Phys. xxvii (1886), p. 666; xxxi (1887), p. 131. Boltzmann, ibid, xxix (1886), p. 598. + Ann. d. Phys. xxvii (1886), p. 119.

The Followers of Maxwell. 357

After his attempt to justify the Maxwellian equations on theoretical grounds, Hertz turned his attention to the possibility of verifying them by direct experiment. His interest in the matter had first been aroused some years previously, when the Berlin Academy proposed as a prize subject " To establish experimentally a relation between electromagnetic actions and the polarization of dielectrics." Helmholtz suggested to Hertz that he should attempt the solution ; but at the time he saw no way of bringing phenomena of this kind within the limits of observation. From this time forward, however, the idea of electric oscillations was continually present to his mind ; and in the spring of 1886 he noticed an effect* which formed the starting- point of his later researches. When an open circuit was formed of a piece of copper wire, bent into the form of a rectangle, so that the ends of the wire were separated only by a short air- gap, and when this open circuit was connected by a wire with any point of a circuit through which the spark -discharge of an induction-coil was taking place, it was found that a spark passed in the air-gap of the open circuit. This was explained by supposing that the change of potential, which is propagated along the connecting wire from the induction-coil, reaches one end of the open circuit before it reaches the other, so that a spark passes between them; and the phenomenon therefore was regarded as indicating a finite velocity of propagation of electric potential along wires.!

• Ann. d. Phys. xxxi (1887), p. 421. Hertz's Electric Waves, translated by D. E. Jones, p. 29.

t Unknown to Hertz, the transmission of electric waves along wires had been observed in 1870 by Wilhelm von Bezold, Miinchen Sitzungsbericlite, i (1870), p. 113 ; Phil. Mag. xl (1870), p. 42. «* If," he wrote at the conclusion of a series of experiments, "electrical waves be sent into a wire insulated at the end, they will be reflected at that end. The phenomena which accompany this process in alternating discharges appear to owe their origin to the interference of the advancing and reflected waves," and, "an electric discharge travels with the «atne rapidity in wires of equal length, without reference to the materials of which these wires are made."

The subject was investigated by 0. J. Lodge and A. P. Chattock at almost the same time as Hertz's experiments were being carried out: mention was made of their researches at the meeting of the British Association in 1888.

358 The Followers of Maxwell.

Continuing his experiments, Hertz* found that a spark could be induced in the open or secondary circuit even when it was not in metallic connexion with the primary circuit in which the electric oscillations were generated; and he rightly inter- preted the phenomenon by showing that the secondary circuit was of such dimensions as to make the free period of electric oscillations in it nearly equal to the period of the oscillations in the primary circuit ; the disturbance which passed from one circuit to the other by induction would consequently be greatly intensified in the secondary circuit by resonance.

The discovery that sparks may be produced in the air-gap of a secondary circuit, provided it has the dimensions proper for resonance, was of great importance : for it supplied a method of detecting electrical effects in air at a distance from the primary disturbance ; a suitable detector was in fact all that was needed in order to observe the propagation of electric waves in free space, and thereby decisively test the Maxwellian theory. To this work Hertz now addressed himself.f

The radiator or primary source of the disturbances studied by Hertz may be constructed of two sheets of metal in the same plane, each sheet carrying a stiff wire which projects towards the other sheet and terminates in a knob ; the sheets are to be excited by connecting them to the terminals of an induction coil. The sheets may be regarded as the two coatings of a modified Leyden jar, with air as the dielectric between them ; the electric field is extended throughout the air, instead of being confined to the narrow space between the coatings, as in the ordinary Leyden jar. Such a disposition ensures that the system shall lose a large part of its energy by radiation at each oscillation.

• Loc. cit.

t Sir Oliver Lodge was about this time independently studying electric oscilla- tions in air in connexion with the theory of lightning-conductors : cf. Lodge, Phil. Mag. xxvi (1888), p. 217. So long before as 1842, Joseph Henry, of Washington, had noticed that the inductive effects of the Leyden jar discharge could be observed at considerable distances, and had even suggested a comparison with " a spark from flint and steel in the case of light."

The Followers of Maxwell. 359

As in the jar discharge,* the electricity surges from one sheet to the other, with a period proportional to (CL)l, where 0 denotes the electrostatic capacity of the system formed by the two sheets, and L denotes the self-induction of the connexion. The capacity and induction should be made as small as possible in order to make the period small. The detector used by Hertz was that already described, namely, a wire bent into an incompletely closed curve, and of such dimensions that its free period of oscillation was the same as that of the primary oscillation, so that resonance might take place.

Towards the end of the year 1887, when studying the sparks induced in the resonating circuit by the primary disturbance, Hertz noticedf that the phenomena were distinctly modified when a large mass of an insulating substance was brought into the neighbourhood of the apparatus ; thus confirming the principle that the changing electric polarization which is pro- duced when an alternating electric force acts on a dielectric is capable of displaying electromagnetic effects.

Early in the following year (1888) Hertz determined to verify Maxwell's theory directly by showing that electro- magnetic actions are propagated in air with a finite velocity .{ For this purpose he transmitted the disturbance from the primary oscillator by two different paths, viz., through the air and along a wire ; and having exposed the detector to the joint influence of the two partial disturbances, he observed inter- ference between them. In this way he found the ratio of the velocity of electric waves in air to their velocity when conducted by wires ; and the latter velocity he determined by observing the distance between the nodes of stationary waves in the wire, and calculating the period of the primary oscillation. The velocity of propagation of electric disturbances in air was in

• Cf. p. 253.

t Ann. d. Phys. xxxiv, p. 373. Electric Waves (English edition), p. 95.

J Ann. d. Phys. xxxiv (1888), p. 551. Electric Waves (English edition) p. 107.

360 The Followers of Maxwell.

this way shown to be finite and of the same order as the velocity of light.*

Later in 1888 Hertzf showed that electric waves in air are reflected at the surface of a wall ; stationary waves may thus be produced, and interference may be obtained between direct and reflected beams travelling in the same direction.

The theoretical analysis of the disturbance emitted by a Hertzian radiator according to Maxwell's theory was given by Hertz in the following year.J

The effects of the radiator are chiefly determined by the free electric charges which, alternately appearing at the two sides, generate an electric field by their presence and a magnetic field by their motion. In each oscillation, as the charges on the poles of the radiator increase from zero, lines of electric force, having their ends on these poles, move outwards into the surrounding space. When the charges on the poles attain their greatest values, the lines cease to issue outwards, and the existing lines begin to retreat inwards towards the poles; but the outer lines of force contract in such a way that their upper and lower parts touch each other at some distance from the radiator, and the remoter portion of each of these lines thus takes the form of a loop ; and when the rest of the line of force retreats inwards towards the radiator, this loop becomes detached and is propagated outwards as radiation. In this way the radiator emits a series of whirl-rings, which as they move grow thinner and wider; at a distance, the disturbance

• Hertz's experiments gave the value 45/28 for the ratio of the velocity of electric waves in air to the velocity of electric waves conducted by the wires, and 2 x 1010 cms. per sec. for the latter velocity. These numbers were afterwards found to be open to objection: Poincare (Comptes Rendus, cxi (1890), p. 322) showed that the period calculated by Hertz was V2 x the true period, which would make the velocity of propagation in air equal to that of light x v'2. Ernst Lecher (Wiener Berichte, May 8, 1890; Phil. Mag. xxx (1890), p. 128), experimenting on the velocity of propagation of electric vibrations in wires, found instead of Hertz's 2 x 1010 cms. per sec., a value within two per cent, of the velocity of light. E. Sarasin and L. De La Rive at Geneva (Archives des Sc. Phys. xxix (1893)) finally proved that the velocities of propagation in air and along wires are equal.

t Ann. d. Phys. xxxiv (1888), p. 610. Electric Waves (English edition), p. 124.

J Ibid., xxxvi (1889), p. 1. Electric Waves (English edition), p. 137.

The Followers of Maxwell. 361

is approximately a plane wave, the opposite sides of the ring representing the two phases of the wave. When one of these rings has become detached from the radiator, the energy con- tained may subsequently be regarded as travelling outwards with it.

To discuss the problem analytically* we take the axis of the radiator as axis of z, and the centre of the spark-gap as origin. The field may be regarded as due to an electric doublet formed of a positive and an equal negative charge, displaced from each other along the axis of the vibrator, and of

moment

Ae~p^ sin (2irct/),

the factor e~p^ being inserted to represent the damping.

The simplest method of proceeding, which was suggested by Fitz Gerald,f is to form the retarded potentials <£ and a of L. Lorenz.J These are determined in terms of the charges and their velocities by the equations

I. * = a<^, o.-s^,

whence it is readily shown that in the present case

4> = - dF/dz, a = (0, 0, where

Ae'KC-'P , 2;r . , F = sin — (ct - r).

T A*

The electric and magnetic forces are then determined by the equations

E = c2 grad <£ - a, H = curl a.

It is found that the electric force may be regarded as com- pounded of a force <£2, parallel to the axis of the vibrator and depending at any instant only on the distance from the vibrator, together with a force fa sin 0 acting in the meridian plane

• Cf. Karl Pearson and A. Lee, Phil. Trans, cxciii (1899), p. 165.

• Brit. Assoc. Rep., Leeds (1890), p. 755.

J Cf. p. 298. The use of retarded potentials was also recommended in the following year by Poincare, Comptes Rendus, cxiii (1891), p. 515.

362 The Followers of Maxwell.

perpendicular to the radius from the centre, where $1 depends at any instant only on the distance from the vibrator, and 0 denotes the angle which the radius makes with the axis of the oscillator. At points on the axis, and in the equatorial plane, the electric force is parallel to the axis. At a great distance from the oscillator, 02 is small compared with 0,, so the wave is purely transverse. The magnetic force is directed along circles whose centres are on the axis of the radiator ; and its magnitude may be represented in the form 03 sin 9, where 03 depends only on r and t ; at great distances from the radiator, c<£3 is approximately equal to 0,.

If the activity of the oscillator be supposed to be continually maintained, so that there is no damping, we may replace p{ by zero, and may proceed as in the case of the magnetic oscillator* to determine the amount of energy radiated. The mean out- ward flow of energy per unit time is found to be Jc3^2 (27T/X)4; from which it is seen that the rate of loss of energy by radiation increases greatly as the wave-length decreases.

The action of an electrical vibrator may be studied by the aid of mechanical models. In one of these, devised by Larmor,f the aether is represented by an incompressible elastic solid, in which are two cavities, corresponding to the conductors of the vibrator, filled with incompressible fluid of negligible inertia. The electric force is represented by the displacement of the solid. For such rapid alternations as are here considered, the metallic poles behave as perfect conductors; and the tangential components of electric force at their surfaces' are zero. This condition may be satisfied in the model by suppos- ing the lining of each cavity to be of flexible sheet-metal, so as to be incapable of tangential displacement ; the normal displace- ment of the lining then corresponds to the surface-density of electric charge on the conductor.

In order to obtain oscillations in the solid resembling those of an electric vibrator, we may suppose that the two cavities

• Cf. p. 346.

7 Proc. Camb. Phil. Soc. vii (1891), p. 165.

The Followers of Maxwell. 36li

have the form of semicircular tubes forming the two halves of a complete circle. Each tube is enlarged at each of its ends, so as to present a front of considerable area to the corresponding front at the end of the other tube. Thus at each end of one diameter of the circle there is a pair of opposing fronts, which are separated from each other by a thin sheet of the elastic solid.

The disturbance may be originated by forcing an excess of liquid into one of the enlarged ends of one of the cavities. This involves displacing the thin sheet of elastic solid, which separates it from the opposing front of the other cavity, and thus causing a corresponding deficiency of liquid in the enlarged end behind this front. The liquid will then surge backwards and forwards in each cavity between its enlarged ends ; and, the motion being communicated to the elastic solid, vibrations will be generated resembling those which are produced in the aether by a Hertzian oscillator.