A History of the Theories of Aether and Electricity (1910) — part 26 of 29

when transmitting light : the orientation of the wave-fronts of the light will consequently in general be altered ; and the direc- tion in which a heavenly body is seen, being normal to the wave- fronts will thereby be affected. But if the aethereal motion is irrotational, so that the elements of the aether do not rotate, it is easily seen that the direction of propagation of the light in space is unaffected ; the luminous disturbance is still propagated in straight lines from the star, while the normal to the wave-front at any point deviates from this line of propagation by the small angle ujc, where u denotes the component of the aethereal velocity at the point, resolved at right angles to the line of propagation, and c denotes the velocity of light. If it be supposed that the aether near the earth is at rest relatively to the earth's surface, the star will appear to be displaced towards the direction in which the earth is moving, through an angle measured by the ratio of the velocity of the earth to the velocity of light, multiplied by the sine of the angle between the direction of the earth's motion and the line joining the earth and star. This is precisely the law of aberration.

An objection to Stokes's theory has been pointed out by several writers, amongst others by H. A. Lorentz.* This is, that the irrotational motion of an incompressible fluid is completely determinate when the normal component of the velocity at its boundary is given : so that if the aether were supposed to have the same normal component of velocity as the earth, it would not have the same tangential component of velocity. It follows that no motion will in general exist which satisfies Stokes's conditions ; and the difficulty is not solved in any very satisfactory fashion by either of the suggestions which, have been proposed to meet it. One of these is to suppose that the moving earth does generate a rotational disturbance, which, however, being radiated away with the velocity of light, does not affect the steadier irrotational motion ; the other, which was

• Archives Neerl, xxi (1896), p. 103.

Closing Years of the Nineteenth Century. 413-

advanced by Planck,* is that the two conditions of Stokes's theory — namely, that the motion of the aether is to be irrotational and that at the earth's surface its velocity is to be the same as that of the earth — may both be satisfied if the aether is supposed to be compressible in accordance with Boyle's law, and subject to gravity, so that round the earth it is compressed like the atmosphere ; the velocity of light being supposed independent of the condensation of the aether.

Lorentz,f in calling attention to the defects of Stokes's theory, proposed to combine the ideas of Stokes and Fresnel, by assuming that the aether near the earth is moving irrotationally (as in Stokes's theory), but that at the surface of the earth the aethereal velocity is not necessarily the same as that of ponder- able matter, and that (as in Fresiiel's theory) a material body imparts the fraction (ju2 - l}/ju2 of its own motion to the aether within it. Fresnel's theory is a particular case of this new theory, being derived from it by supposing the velocity -potential to be zero.

Aberration is by no means the only astronomical phenomenon which depends on the velocity of propagation of light ; we have indeed seent that this velocity was originally determined by observing the retardation of the eclipses of Jupiter's satellites. It was remarked by Maxwell§ in 1879 that these eclipses furnish, theoretically at least, a means of determining the velocity of the solar system relative to the aether. For if the distance from the eclipsed satellite to the earth be divided by the observed retardation in time of the eclipse, the quotient represents the velocity of propagation of light in this direction, relative to the solar system; and this will differ from the velocity of propagation of light relative to the aether by the component, in this direction, of the sun's velocity relative to the aether. By taking observations when Jupiter is in different signs of the

• Of. Lorentz, Proc. Amsterdam Acad. (English ed.), i (1899), p. 443. t Archives Neerl. xxi (1886), p. 103 : cf. also Zittinsgsversl. Kon. Ak. Amster- dam, 1897-98, p. 266.

J Cf. p. 22. § Proc. R. S. xxx (1880), p. 108.

414 The Theory of Aether and Electrons in the

zodiac, it should therefore be possible to determine the sun's velocity relative to the aether, or at least that component of it which lies in the ecliptic.

The same principles may be applied to the discussion of other astronomical phenomena. Thus the minimum of a variable star of the Algol type will be retarded or accelerated by an interval of time which is found by dividing the projection of the radius from the sun to the earth on the direction from the sun to the Algol variable by the velocity, relative to the solar system, of propagation of light from the variable ; and thus the latter quantity may be deduced from observations of the retardation.*

Another instance in which the time taken by light to cross an orbit influences an observable quantity is afforded by the astronomy of double stars. Savaryf long ago remarked that when the plane of the orbit of a double star is not at right angles to the line of sight, an inequality in the apparent motion must be caused by the circumstance that the light from the remoter star has the longer journey to make. Yvon VillarceauJ showed that the effect might be represented by a constant alteration of the elliptic elements of the orbit (which alteration is of course beyond detection), together with a periodic inequality, which may be completely specified by the following statement : the apparent coordinates of one star relative to the other have the values which in the absence of this effect they would have at an earlier or later instant, differing from the actual time by the amount

m, - >)iz z m} + mz ' c'

where ml and m2 denote the masses of the stars, c the velocity of light, and z the actual distance of the two stars from each

• The velocity of light was found from observations of Algol, by C. V. L. Charlier, Of versigt af K. Vet.-Ak. Forhandl. xivi (1889), p. 523.

t Conn, des Temps, 1830.

J Additions a la Connaissance des Temps, 1878 : an improved deduction was given by H. Seeliger, Sitzungsberichte d. K. Ak. zu Miinchen, xix (1889), p. 19.

Closing Years of the Nineteenth Century. 415

other at the time when the light was emitted, resolved along the line of sight. In the existing state of double-star astronomy, this effect would be masked by errors of observation.

Villarceau also examined the consequences of supposing that the velocity of light depends on the velocity of the source by which it is emitted. If, for instance, the velocity of light from a star occulted by the moon were less than the velocity of light reflected by the moon, then the apparent position of the lunar disk would be more advanced in its movement than that of the star, so that at emersion the star would first appear at some distance outside the lunar disk, and at immersion the star would be projected on the interior of the disk at the instant of its disappearance. The amount by which the image of the star could encroach on that of the disk on this account could not be so much as 0"'71 ; encroachment to the extent of more than 1" has been observed, but is evidently to be attributed for the most part to other causes.

Among the consequences of the finite velocity of propagation of light which are of importance in astronomy, a leading place must be assigned to the principle enunciated in 1842 by Christian Doppler,* that the motion of a source of light relative to an observer modifies the period of the disturbance which is received by him. The phenomenon resembles the depression of the pitch of a note when the source of sound is receding from the observer. In either case, the period of the vibrations perceived by the observer is (c + v) / c x the natural period, where v denotes the velocity of separation of the source and observer, and c denotes the velocity of propagation of the disturbance. If, e.g., the velocity of separation is equal to the orbital velocity of the earth, the D lines of sodium in the spectrum of the source will be displaced towards the red, as compared with lines derived from a terrestrial sodium flame, bs about one-tenth of the distance between them. The application of this principle to the determination of the relative velocity of

• Abhandl. der K. Hohm. Ges. der Wissensch. (5) ii (1842), p. 465.

416 The Theory of Aether and Electrons in the

stars in the line of sight, which has proved of great service in astrophysical research, was suggested by Fizeau in 1848.*

Passing now from the astronomical observatory, we must examine the information which has been gained in the physical laboratory regarding the effect of the earth's motion on optical phenomena. We have alreadyf referred to the investigations by which the truth of Fresnel's formula was tested. An experiment of a different type was suggested in 1852 by FizeauJ who remarked that, unless the aether is carried along by the earth, the radiation emitted by a terrestrial source should have different intensities in different directions. It was, how- ever, shown long afterwards by Lorentz§ that such an experiment would not be expected on theoretical grounds to yield a positive result ; the amount of radiant energy imparted to an absorbing body is independent of the earth's motion. A few years later Fizeau investigatedll another possible effect. If a beam of polarized light is sent obliquely through a glass plate, the azimuth of polarization is altered to an extent which depends, amongst other things, on the refractive index of the glass. Fizeau performed this experiment with sunlight, the light being sent through the glass in the direction of the terrestrial motion, and in the opposite direction ; the readings seemed to differ in the two cases, but on account of experimental difficulties the result was indecisive.

Some years later, the effect of the earth's motion on the rotation of the plane of polarization of light propagated along the axis of a quartz crystal was investigated by Mascart.^f The result was negative, Mascart stating that the rotation could not have been altered by more than the (l/40,000)th part when the orientation of the apparatus was reversed from that of

• An apparatus for demonstrating the Doppler-Fizeau effect in the laboratory was constructed by Belopolsky, Astrophys. Journal xiii (1901), p. 15.

t Of. pp. 117-120. + Ann. d. Phys. xcii (1854), p. 652.

\ Proc. Amsterdam Acad. (English edition), iv (1902) p. 678.

|| Annales de Chim. (3) Ixviii (1860), p. 129; Ann. d. Phys. cxiv (1861), p. 554.

H Annales de 1'Ec. Norm. (2) i (1872), p. 157.

Closing Years of the Nineteenth Century. 417

the terrestrial motion to the opposite direction. This was afterwards confirmed by Lord Kayleigh,* who found that the alteration, if it existed, could not amount to (l/100,000)th part.

In terrestrial methods of determining the velocity of light the ray is made to retrace its path, so that any velocity which the earth might possess with respect to the luminiferous medium would affect the time of the double passage only by an amount proportional to the square of the constant of aberration.f In 1881, however, A. A. MichelsonJ remarked that the effect, though of the second order, should be manifested by a measur- able difference between the times for rays describing equal paths parallel and perpendicular respectively to the direction of the earth's motion. He produced interference-fringes between two pencils of light which had traversed paths perpendicular to each other ; but when the apparatus was rotated through a right angle, so that the difference would be reversed, the expected displacement of the fringes could not be perceived. This result was regarded by Michelson himself as a vindication of Stokes's theory^ in which the aether in the neighbourhood of the dearth is supposed to be set in motion. Lorentzj), however, showed that the quantity to be measured had only half the value supposed by Michelson, and suggested that the negative result of the experiment might be explained by that combina- tion of Fresnel's and Stokes's theories which was developed in his own memoirIF ; since, if the velocity of the aether near the earth were (say) half the earth's velocity, the displacement of Michelson's fringes would be insensible.

• Phil. Mag. iv. (1902), p. 215.

t The constant of aberration is the ratio of the earth's orbital velocity to the velocity of light ; cf. supra, p. 100.

£ Amer. Journ. Sci. xxii (1881), p. 20. His method was afterwards improved : cf. Michelson and Morley, Amer. Journ. Sci. xxxiv (1887), p. 333; Phil. Mag. xxiv (1887), p. 449.

§ Cf. p. 411.

|| Arch. Xeerl. xxi (1886), p. 103. On the Micbelson-Morley experiment cf. also Hicks, Phil. Mag. iii (1902), p. 9.

U Cf. p. 413.

2 E

418 The Theory of Aether and Electrons in the

A sequel to the experiment of Michelson and Morley was performed in 1897, when Michelson* attempted to determine by experiment whether the relative motion of earth and aether varies with the vertical height above the terrestrial surface. No result, however, could be obtained to indicate that the velocity of light depends on the distance from the centre of the earth ; and Michelson concluded that if there were no choice <"but between the theories of Fresnel and Stokes, it would be necessary to adopt the latter, and to suppose that the earth's influence on the aether exends to many thousand kilometres above its surface. By this time, however, as will subsequently appear, a different explanation was at hand.

Meanwhile the perplexity of the subject was increased by experimental results which pointed in the opposite direction to that of Michelson. In 1892 Sir Oliver Lodgef observed the interference between the two portions of a bifurcated beam of light, which were made to travel in opposite directions round a closed path in the space * between two rapidly rotating steel disks. The observations showed that the velocity of light is not affected by the motion of adjacent matter to the extent of (l/200)th part of the velocity of the matter. Continuing his investigations, Lodge} strongly magnetized the moving matter (iron in this experiment), so that the light was propagated across a moving magnetic field ; and electrified it so that the path of the beams lay in a moving electrostatic field ; but in no case was the velocity of the light appreciably affected.

We must now trace the steps by which theoretical physicists not only arrived at a solution of the apparent contradictions furnished by experiments with moving bodies, but so extended the domain of electrical science that it became necessary to enlarge the boundaries of space and time to contain it.

The first memoir in which the new conceptions were unfolded-j was published by H. A. Lorentzg in 1892. The

• Amer. Journ. Sci. (4) iii (1897), p. 475.

t Phil. Trans, clxxxiv (1893), p. 727. J Ibid., clxxxix (1897), p. 149.

§ Archives Neerl. xxv (1892), p. 363 : the theory is given in eh. iv, pp. 432 et sqq.

Closing Years of the Nineteenth Century. 419

theory of Lorentz was, like those of Weber, Kiemann, and Clausius,* a theory of electrons ; that is to say, all electro- dynamical phenomena were ascribed to the agency of moving electric charges, which were supposed in a magnetic field to experience forces proportional to their velocities, and to com- municate these forces to the ponderable matter with which they might be associated.t

In spite of the fact that the earlier theories of electrons had failed to fulfil the expectations of their authors, the assumption that all electric and magnetic phenomena are due to the presence or motion of individual electric charges was one to which physicists were at this time disposed to give a favourable consideration ; for, as we have seen,* evidence of the atomic nature of electricity was now contributed by the study of the conduction of electricity through liquids and gases. Moreover, the discoveries of Hertz § had shown that a molecule which is emitting light must contain some system resembling a Hertzian vibrator; and the essential process in a Hertzian vibrator is the oscillation of electricity to and fro. Lorentz himself from the outset of his career! | had supposed the inter- action of ponderable matter with the electric field to be effected by the agency of electric charges associated with the material atoms.

The principal difference by which the theory now advanced by Lorentz is distinguished from the theories of Weber,

• Cf. pp. 226, 231, 262.

• Some writers have inclined to use the term ' electron-theory ' as if it were specially connected with Sir Joseph Thomson's justly celebrated discovery (cf . p. 407, supra) that all negative electrons have equal charges. But Thomson's discovery, though undoubtedly of the greatest importance as a guide to the structure of the universe, has hitherto exercised hut little influence on general electromagnetic theory. The reason for this is that in theoretical investigations it is customary to denote the changes of electrons by symbols, e, e-z, . . . ; and the equality or non-equality of these makes no difference to the equations. To take an illustration from Celestial Mechanics, it would clearly make no difference in the general equations of the planetary theory if the masses of the planets happened to be all equal.

• Cf. chapter xi.

§ Cf. pp. 357-363.

|| Verb. d. Ak. v. Wetenschappen, Amsterdam, Deel xviii (1878).

2 E 2

420 The Theory of Aether and Electrons in the

Kiemann, and Clausing, and from Lorentz' own earlier work, lies in the conception which is entertained of the propagation of influence from one electron to another. In the older writ- ings, the electrons were assumed to be capable of acting on each other at a distance, with forces depending on their charges, mutual distances, and velocities ; in the present memoir, on the other hand, the electrons were supposed to interact not directly with each other, but with the medium in which they were embedded. To this medium were ascribed the properties characteristic of the aether in Maxwell's theory.

The only respect in which Lorentz' medium differed from Maxwell's was in regard to the effects of the motion of bodies. Impressed by the success of Fresnel's beautiful theory of the propagation of light in moving transparent substances,* Lorentz designed his equations so as to accord with that theory, and showed that this might be done by drawing a distinction between matter and aether, and assuming that a moving ponderable body cannot communicate its motion to the aether which surrounds it, or even to the aether which is entangled in its own particles ; so that no part of the aether can be in motion relative to any other part. Such an aether simply space endowed with certain dynamical properties.

The general plan of Lorentz' investigation was to reduce all the complicated cases of electromagnetic action to one simple and fundamental case, in which the field contains only free aether with solitary electrons dispersed in it ; the theory which he adopted in this fundamental case was a combination of Clausius' theory of electricity with Maxwell's theory of the aether.

Suppose that e (x, y, z) and e(x, y', z) are two electrons. In the theory of Clausius,f the kinetic potential of their mutual action is

ee'

— (xx + yy + ss' - c2) ;

so when any number of electrons are present, the part of the

*Cf. pp. 116 etxqq. t Cf. p. 262.

Closing Years of the Nineteenth Century. 421

kinetic potential which concerns any one of them — say, e — may be written

Le = e (axx + ayy + azz - c2<£),

where a and c£ denote potential functions, defined by the

equations

• / f c r r f

£— dxdy'dz, </> = \\p- dx'dy'dz' ;

p denoting the volume-density of electric charge, and v its velocity, and the integration being taken over all space.

We shall now reject Clausius' assumption that electrons act instantaneously at a distance, and replace it by the assumption that they act on each other only through the mediation of an aether which fills all space, and satisfies Maxwell's equations. This modification may be effected in Clausius' theory without difficulty ; for, as we have seen,* if the state of Maxwell's aether at any point is defined by the electric vector d and magnetic vector h,f these vectors may be expressed in terms of potentials a and ^ by the equations

d = c" grad <£ - a, h = curl a ;

and the functions a and <£ may in turn be expressed in terms of the electric charges by the equations

a - JTJ ((**)'lr\ dx'dy'dz', </> = J/J |(J5)» dxdtfdsf,

where the bars indicate that the values of (pvr)' and (p)' refer to the instant (t - r/c). Comparing these formulae with those given above for Clausius' potentials, we see that the only change which it is necessary to make in Clausius' theory is that of retarding the potentials in the way indicated by L. Lorenz.J The electric and magnetic forces, thus defined in terms of the

• Cf. pp. 298, 299.

t We shall use the small letters d and h. in place of E and H, when MC are concerned with Lorentz' fundamental case, in which the system consists solely of free aether and isolated electrons.

% Cf. p. 298.

422 The Theory of Aether and Electrons in the

position and motion of the charges, satisfy the Maxwellian equations

div d = 47rc2/o,

div h * 0,

curl d = - K,

curl h = d/c2 + 47r/ov.

The theory of Lorentz is based on these four aethereal equations of Maxwell, together with the equation which deter- mines the ponderomotive force on a charged particle ; this, which we shall now derive, is the contribution furnished by Clausius' theory.

The Lagrangian equations of motion of the electron e are

^-0

fa-

and two similar equations, where L denotes the total kinetic otential due to all causes, electric and mechanical. The ponderomotive force exerted on the electron by the electro- magnetic field has for its ^-component

dx ~ dt\ dx or

fdax . dav . daz . d<t>\ dax

e{ —— x + — - it H z — c*—} — e — '-

\dx dx* dx dxj dt

which, since

reduces to

• e I & -^

or edx + e (yhz - z

so that the force in question is

ed + e [v . h]. This was Lorentz' expression for the ponderomotive force on an

Closing Years of the Nineteenth Century. 423

electrified corpuscle of charge e moving with velocity v in a field defined by the electric force d and magnetic force h.

In Lorentz' fundamental case, which has thus been examined, account has been taken only of the ultimate constituents of which the universe is supposed to be composed, namely, cor- puscles and the aether. We must now see how to build up from these the more complex systems which are directly presented to our experience.

The electromagnetic field in ponderable bodies, which to our senses appears in general to vary continuously, would present a different aspect if we were able to discern molecular structure ; we should then perceive the individual electrons by which the field is produced, and the rapid fluctuations of electric and magnetic •force between them. As it is, the values furnished by our instruments represent averages taken over volumes which, though they appear small to us, are large compared with molecular dimensions.* We shall denote an average value of this kind by a bar placed over the corresponding symbol.

Lorentz supposed that the phenomena of electrostatic charge and of conduction-currents are due to the presence or motion of simple electrons such as have been considered above. The part of p arising from these is the measurable density of electrostatic charge ; this we shall denote by pi. If w denote the velocity of the ponderable matter, and if the velocity v of the electrons be written w + u, then the quantity pv, so far as it arises from electrons of this type, may be written ^ w + pu. The former of these terms represents the convection-current, and the latter the conduction-current.

Consider next the phenomena of dielectrics. Following Faraday, Thomson, and Mossotti,f Lorentz supposed that each dielectric molecule contains corpuscles charged vitreously and also corpuscles charged resinously. These in the absence of an

• These principles had been enunciated, and to some extent developed, by J. Willard Gibbs in 1882-3 : Amer. Journ. Sci. xxiii, pp. 262, 460, xxv, p. 107 ; Gibbs' Scientific Papei-s, ii, pp. 182, 195, 211.

t Cf. pp. 210, 211.

424 The Theory of Aether and Electrons in the

external field are so arranged as to neutralize each other's electric fields outside the molecule. For simplicity we may suppose that in each molecule only one corpuscle, of charge e, is capable of being displaced from its position ; it follows from what has been assumed that the other corpuscles in the molecule exert the same electrostatic action as a charge - e situated at the original position of this corpuscle. Thus if e is displaced to an adjacent position, the entire molecule becomes equivalent to an electric doublet, whose moment is measured by the- product of e and the displacement of e. The molecules in unit volume, taken together, will in this way give rise to a (vector) electric moment per unit volume, P, which may be compared to the (vector) intensity of magnetization in Poisson's theory of magnetism.* As in that theory, we may replace the doublet -distribution P of the scalar quantity p by a volume-distribution of p, determined by the equationf

p = - div P.

This represents the part of jo due to the dielectric molecules.

Moreover, the scalar quantity pwx has also a doublet-distri- bution, to which the same theorem may be applied ; the average value of the part of pwx, due to dielectric molecules, is therefore determined by the equation

pwx = - div (W.J.?) = - wx div P - (P . V) wx, or

/ow = - div P . w - (P . V) w.

We have now to find that part of j»u which is due to dielectric molecules. For a single doublet of moment p we have, by differentiation,

f JJ pM dx dy dz = dp/dt,

where the integration is taken throughout the molecule; so that

/// PM dxdydz = (d/dt) ( FP),

where the integration is taken throughout a volume V, which *Cf. p. 64.

t We assume all transitions gradual, so as to avoid surface-distributions.

Closing Years of the Nineteenth Century. 425

encloses a large number of molecules, but which is small com- pared with measurable quantities; and this equation may be written

Now, if P refers to differentiation at a fixed point of space (as opposed to a differentiation which accompanies the moving body),

we have

(£/&)*-? + (w.V)P,

and (d/dt) V = Fdiv w; so that

/ou = P + (w . V) P + div w . P

= P + curl [P . w] + div P . w + (P . V) w, and therefore

pu + pw = P + curl [P . w].

This equation determines the part of f>v which arises from the dielectric molecules.

The general equations of the aether thus become, when the averaging process is performed,

div d = 4>!r<?pi ~ 4-Trc2 div P, div h = 0, curl d = - h,

curl h =- (1/c2) d + 47r ,

( + P + curl [P . w] I

In order to assimilate these to the ordinary electromagnetic equations, we must evidently write

d = E, the electric force; (1/4-7TC2) E + P = D, the electric induction ;

h = H, the magnetic vector.

The equations then become (writing p for plt as there is no longer any need to use the subscript),

div D = p, - curl E = H,

where div H = °' curl H = 4lrS'

S = conduction-current + convection-current + D + curl [P . w].

convection-current + conduction-current .

426 The Theory of Aether and Electrons in the

The term D in S evidently represents the displacement- current of Maxwell ; and the term curl [P . w] will be recognized as a modified form of the term curl [D . w], which was first introduced into the equations by Hertz.* It will be remembered that Hertz supposed this term to repre- sent the generation of a magnetic force within a dielectric which is in motion in an electric field ; and that Heaviside,f by adducing considerations relative to the energy, showed that the term ought to be regarded as part of the total current, and inferred from its existence that a dielectric which moves in an electric field is the seat of an electric current, which produces a magnetic field in the surrounding space. The modification introduced by Lorentz consisted in replacing D by P in the vector-product ; this implied that the moving dielectric does not carry along the aethereal displacement, which is represented by the term E/4?rc2 in D, but only carries along the charges which exist at opposite ends of the molecules of the ponderable dielectric, and which are represented by the term P. The part of the total current represented by the term curl [P . w] is generally called the current of dielectric convection.

That a magnetic field is produced when an uncharged dielectric is in motion at right angles to the lines of force of a constant electrostatic field had been shown experimentally in 1888 by Rontgen.J His experiment consisted in rotating a dielectric disk between the plates of a condenser ; a magnetic field was produced, equivalent to that which would be produced by the rotation of the " fictitious charges " on the two faces of the dielectric, i.e., charges which bear the same relation to the dielectric polarization that Poisson's equivalent surface- density of magnetism§ bears to magnetic polarization. If U denote the difference of potential between the opposite coatings of the condenser, and * the specific inductive capacity of the dielectric, the surf ace -density of electric charge on the coatings

• Cf. p. 366. t Cf. p. 367.

I Ann. d. Phys. xxxv (1888), p. 264 ; xl (1890), p. 93. § Cf. p. 64.

Closing Years of the Nineteenth Century. 427

is proportional to ± t£7, and the fictitious charge on the sur- faces of the dielectric is proportional to + (a - 1) U. It is evident from this that if a plane condenser is charged to a given difference of potential, and is rotated in its own plane, the magnetic field produced is proportional to * if (as in Kowland's experiment*) the coatings are rotated while the dielectric remains at rest, but is in the opposite direction, and is propor- tional to (c - 1) if (as in Kontgen's experiment) the dielectric is rotated while the coatings remain at rest. If the coatings and dielectric are rotated together, the magnetic action (being the sum of these) should be independent of f — a conclusion which was verified later by Eichenwald.f

Hitherto we have taken no account of the possible mag- netization of the ponderable body. This would modify the equations in the usual manner,:}: so that they finally take the form

div D = p, (I)

div B = 0, (II)

curl H = 47rS, (III)

-curl.E = B, (IV),

where S denotes the total current formed of the displacement - current, the convection-current, the conduction-current, and the current of dielectric convection. Moreover, since

S =pv + d'/47rc2, we have

div S = div pv + (l/4;rc2) div (ad/80

= div v

*Cf. p. 339.

t Ann. d. Piiys. xi (1903), p. 421 ; xiii (1904), p. 919. Eichenwald performed other experiments of a similar character, e.g. he observed the magnetic field due to the changes of polarization in a dielectric which was moved in a non- homogeneous electric field.

J It is possible to construct a purely electronic theory of magnetization, a magnetic molecule being supposed to contain electrons in orbital revolution. It then appears that the vector which represents the average value of h. is not H, but B.

428 The Theory of Aether and Electrons in the which vanishes by virtue of the principle of conservation of

div 8 = 0, (V)

electricity. Thus

or the total current is a circuital vector. Equations (I) to (Y) are the fundamental equations of Lorentz' theory of electrons.

We have now to consider the relation by which the polari- zation P of dielectrics is determined. If the dielectric is moving with velocity w, the ponderomotive force on unit electric charge moving with it is (as in all theories)*

E' = E + [w . B ]. (1)

In order to connect P with E', it is necessary to consider the motion of the corpuscles. Let e denote the charge and m the mass of a corpuscle, (£, ?, £) its displacement from its position of equilibrium, k (£, 77, £) the restitutive force which retains it in the vicinity of this point ; then the equations of motion of the corpuscle are

ra£ + A-2£ = eEx't

and similar equations in 17 and £. When the corpuscle is set in motion by light of frequency n passing through the medium, the displacements and forces will be periodic functions of nt — say,

Substituting these values in the equations of motion, we obtain A(Jc* - mnz) -= eE«, and therefore ? (kz - tun*) = eE'x.

Thus, if N denote the number of polarizable molecules per unit volume, the polarization is determined by the equation

• = Ne (g, TJ, ?) = JVVE7(&2 - m?i2).

In the particular case in which the dielectric is at rest, this equatio^ gives

= (l/47rc2)E + P = (l/47rc2)E + Ne*E/(k2 - mw\ But, as we have seen,f D bears to E the ratio ^u2/47rc2, where ^

*Cf. p. 365. tCf. p. 281.

Closing Years of the Nineteenth Century. 429

denotes the refractive index of the dielectric ; and therefore the refractive index is determined in terms of the frequency by the equation

• mnz).