A History of the Theories of Aether and Electricity (1910) — part 18 of 29

The mechanism must next be extended so as to take account of the phenomena of electrostatics. For this purpose Maxwell assumed that the particles, when they are displaced from their equilibrium position in any direction, exert a tangential action on the elastic substance of the cells ; and that this gives rise to a distortion of the cells, which in turn calls into play a force arising from their elasticity, equal and opposite to the force which urges the particles away from the equilibrium position. When the exciting force is removed, the cells recover their form, and the electricity returns to its former position. The state of the medium, in which the electric particles are displaced in a definite direction, is assumed to represent an electrostatic field. Such a displacement does not itself con- stitute a current, because when it has attained a certain value it remains constant ; but the variations of displacement are to be regarded as currents, in the positive or negative direction according as the displacement is increasing or diminishing.

Maxwell. 279

The conception of the electrostatic state as a displacement of something from its equilibrium position was not altogether new, although it had not been previously presented in this form. Thomson, as we have seen, had compared electric force to the displacement in an elastic solid ; and Faraday, who had likened the particles of a ponderable dielectric to small con- ductors embedded in an insulating medium,* had supposed that when the dielectric is subjected to an electrostatic field, there is a displacement of electric charge on each of the small conductors. The motion of these charges, when the field is varied, is equivalent to an electric current ; and it was from this precedent that Maxwell derived the principle, which became of cardinal importance in his theory, that variations of displace- ment are to be counted as currents. But in adopting the idea, he altogether transformed it ; for Faraday's conception of displacement was applicable only to ponderable dielectrics, and was in fact introduced solely in order to explain why the specific inductive capacity of such dielectrics is different from that of free aether; whereas according to Maxwell there is displacement wherever there is electric force, whether material bodies are present or not.

The difference between the conceptions of Faraday and Maxwell in this respect may be illustrated by an analogy drawn from the theory of magnetism. When a piece of iron is placed in a magnetic field, there is induced in it a magnetic distribution, say of intensity I ; this induced magnetization exists only within the iron, being zero in the free aether outside. The vector I may be compared to the polarization or displacement, which according to Faraday is induced in dielectrics by an electric field; and the electric current con- stituted by the variation of this polarization is then analogous to dl/dt. But the entity which was called by Maxwell the electric displacement in the dielectric is analogous not to I, but to the magnetic induction B : the Maxwellian displace-

• Cf. p. 210.

280 Maxwell.

merit-current corresponds to d'B/dt, and may therefore have a value different from zero even in free aether.

It may be remarked in passing that the term displacement, which was thus introduced, and which has been retained in the later development of the theory, is perhaps not well chosen ; what in the early models of the aether was represented as an actual displacement, has in later investigations been conceived of as a change of structure rather than of position in the elements of the aether.

Maxwell supposed the electromotive' force acting on the electric particles to be connected with the displacement D which accompanies it, by an equation of the form

where c, denotes a constant which depends on the elastic properties of the cells. The displacement-current D must now be inserted in the relation which connects the current with the magnetic force ; and thus we obtain the equation

curl H = 47rS,

where the vector S, which is called the total current, is the sum of the convection-current i and the displacement-current D. By performing the operation div on both sides of this equation, it is seen that the total current is a circuital vector. In the model, the total current is represented by the total motion of the rolling particles ; and this is conditioned by the rotations of the vortices in such a way as to impose the kinematic relation

div S = 0.

Having obtained the equations of motion of his system of vortices and particles, Maxwell proceeded to determine the rate of propagation of disturbances through it. He considered in particular the case in which the substance represented is a dielectric, so that the conduction-current is zero. If, moreover,

Maxwell. 281

the constant fi be supposed to have the value unity, the equations may be written

div H = 0,

c,2 curl H = E,

• curl E = H. Eliminating E, we see* that H satisfies the equations

jdivH = 0,

•«•

But these are precisely the equations which the light- vector satisfies in a medium in which the velocity of propagation is c^ : it follows that disturbances are propagated through the model by waves which are similar to waves of light, the magnetic (and similarly the electric) vector being in the wave-front. For a plane-polarized wave propagated parallel to the axis of z, the equations reduce to

2y = x 2*^y y

"Cl dz '"' dt' Cl ~dz '' dt' dz dt' dz

whence we have

= Ex - c\Sx = E

these equations show that the electric and magnetic vectors are at right angles to each other.

The question now arises as to the magnitude of the constant Cj.f This may be determined by comparing different expressions for the energy of an electrostatic field. The work done by an electromotive force E in producing a displacement D is

fD

E . dD or JED

o

per unit volume, since E is proportional to D. But if it be assumed that the energy of an electrostatic field is resident in the dielectric, the amount of energy per unit volume may be

• For if a denote any vector, we have identically

V-a -f grad div a + curl curl a = 0.

t For criticisms on the procedure by which Maxwell determined the velocity of propagation of disturbance, cf. P. Duhem, Les Theories Electriqv.es de J. Clerk Maxwell, Paris, 1902.

282 Maxwell.

calculated by considering the mechanical force required in order to increase the distance between the plates of a condenser, so as to enlarge the field comprised between them. The result is that the energy per unit volume of the dielectric is fE/2/87r, where c denotes the specific inductive capacity of the dielectric and E' denotes the electric force, measured in terms of the electrostatic unit : if E denotes the electric force expressed in terms of the electrodynamic units used in the present investi- gation, we have E = cE', where c denotes the constant which* occurs in transformations of this kind. The energy is therefore fcE2/87TC2 per unit volume. Comparing this with the expression for the energy in terms of E and D, we have

D

and therefore the constant Ci has the value ct*. Thus the result is obtained that the velocity of propagation of dis- turbances in Maxwell's medium is ce~£, where £ denotes the specific inductive capacity and c denotes the velocity for which Kohlrausch and Weber had foundf the value 31 x 1010 cm./sec. Now by this time the velocity of light was known, not only from the astronomical observations of aberration and of Jupiter's satellites, but also by direct terrestrial experiments. In 1849 Hippolyte Louis FizeauJ had ' determined it by rotating a toothed wheel so rapidly that a beam of light transmitted through the gap between two teeth and reflected back from a mirror was eclipsed by one of the teeth on its return journey. The velocity of light was calculated from the dimensions and angular velocity of the wheel and the distance of the mirror ; the result being 315 x 1010 cm. /sec. §

• Cf. pp. 227, 259. | Cf. p. 260.

| Comptes Rendus, xxix (1849), p. 90. A determination made by Cornu in 1874 was on this principle.

§ A different experimental method was employed in 1862 hy Leon Foucault (Comptes Rendus, Iv, pp. 501, 792) ; in this a ray from an origin 0 was reflected by a revolving mirror M to a fixed mirror, and so reflected back to J/, and again to O. It is evident that the returning ray ?dO must be deviated by twice the angle through which M turns while the light passes from M to the fixed mirror and back. The value thus obtained by Foucault for the velocity of light was

Maxwell. 283

Maxwell was impressed, as Kirchhoff had been before him, by the close agreement between the electric ratio c and the velocity of light* ; and having demonstrated that the propaga- tion of electric disturbance resembles that of light, he did not hesitate to assert the identity of the two phenomena. "We can scarcely avoid the inference," he said, " that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena." Thus was answered the question which Priestley had asked almost exactly a hundred years before :f "Is there any electric fluid sui generis at all, distinct from the aether ? "

The presence of the dielectric constant e in the expression ct -i, which Maxwell had obtained for the velocity of propaga- tion of electromagnetic disturbances, suggested a further test of the identity of these disturbances with light: for the velocity of light in a medium is known to be inversely proportional to the refractive index of the medium, and therefore the refractive index should be, according to the theory, proportional to the square root of the specific inductive capacity. At the time, however, Maxwell did not examine whether this relation was confirmed by experiment.

In what has preceded, the magnetic permeability //, has been supposed to have the value unity. If this is not the case, the

2-98 x 1010 cm./sec. Subsequent determinations by Michelson in 187'.) (Ast. Papers of the Amer. Ephemeris, i), and by Newcomb in 1882 (ibid., ii) depended on the same principle.

As was shown afterwards by Lord Rayleigh (Nature, xxiv, p. 382, xxv, p. 52) and by Gibbs (Nature, xxxiii, p. 582), the value obtained for the velocity of light by the methods of Fizeau and Foucault represents the group-velocity, not the wave- velocity ; the eclipses of Jupiter's satellites also give the group-velocity, while the value deduced from the coefficient of aberration is the wave- velocity. In a non- dispersive medium, the group- velocity coincides with the wave- velocity ; and the agreement of the values of the velocity of light obtained by the two astronomical methods seems to negative the possibility of any appreciable dispersion in free aether.

The velocity of light in dispersive media was directly investigated by Michelson in 1883-4, with results in accordance with theory.

• He had "worked out the formulae in the country, before seeing Weber's result." Cf. Campbell and Garnett's Life of Maxwell, p. 244.

f Priestley's Eistory, p. 488.

284 Maxwell.

velocity of propagation of disturbance may be shown, by the same analysis, to be ct~i^~i ; so that it is diminished when /u is greater than unity, i.e., in paramagnetic bodies. This inference had been anticipated by Faraday : " Nor is it likely," he wrote,* " that the paramagnetic body oxygen can exist in the air and not retard the transmission of the magnetism."

It was inevitable that a theory so novel and so capacious as that of Maxwell should involve conceptions which his contempo- raries understood with difficulty and accepted with reluctance. Of these the most difficult and unacceptable was the principle that the total current is always a circuital vector ; or, as it is generally expressed, that " all currents are closed." According to the older electricians, a current which is employed in charging a condenser is not closed, but terminates at the coatings of the condenser, where charges are accumulating. Maxwell, on the other hand, taught that the dielectric between the coatings is the seat of a process — the displacement-current — which is proportional to the rate of increase of the electric force in the dielectric ; and that this process produces the same magnetic effects as a true current, and forms, so to speak, a continuation, through the dielectric, of the charging current, so that the latter may be regarded as flowing in a closed circuit.

Another characteristic feature of Maxwell's theory is the conception — for which, as we have seen, he was largely indebted to Faraday and Thomson — that magnetic energy is the kinetic energy of a medium occupying the whole of space, and that electric energy is the energy of strain of the same medium. By this conception electromagnetic theory was brought into such close parallelism with the elastic- solid theories of the aether, that it was bound to issue in an electromagnetic theory of light.

Maxwell's views were presented in a more developed form in a memoir entitled "A Dynamical Theory of the Electro- magnetic Field," which was read to the Koyal Society in 1864 ;f

• Faraday's laboratory note-book for 1857 : of. Bence Jones's Life of Faraday, ii, p. 380.

t Phil. Trans, civ (1865), p. 459 : Maxwell's Scient. Papers, i, p. 526

Maxwell. 285

in this the architecture of his system was displayed, stripped of the scaffolding by aid of which it had been first erected.

As the equations employed were for the most part the same as had been set forth in the previous investigation, they need only be briefly recapitulated. The magnetic induction juH, being a circuital vector, may be expressed in terms of a vector-potential

A by the equation

luiK = curl A.

The electric displacement D is connected with the volume- density p of free electric charge by the electrostatic equation

div D = p.

The principle of conservation of electricity yields the equation div i = - dp/dt,

where i denotes the conduction-current.

The law of induction of currents — namely, that the total electromotive force in any circuit is proportional to the rate of decrease of the number of lines of magnetic induction which pass through it — may be written

• curl E = /LtH ;

from which it follows that the electric force E must be expressible

in the form

E = - A + grad i//,

where ^ denotes some scalar function. The quantities A and ;// which occur in this equation are not as yet completely deter- minate ; for the equation by which A is defined in terms of the magnetic induction specifies only the circuital part of A ; and as the irrotational part of A is thus indeterminate, it is evident that \p also must be indeterminate. Maxwell decided the matter by assuming* A to be a circuital vector ; thus

divA = 0, and therefore div E = -

• This is the effect of the introduction of (F1, G', H'} in § 98 of the memoir ; cf. also Maxwell's Treitise on Electricity and Magnetism, § 616.

286 Maxwell.

from which equation it is evident that ^ represents the electro- static potential.

The principle which is peculiar to Maxwell's theory must now be introduced. Currents of conduction are not the only kind of currents ; even in the older theory of Faraday, Thomson, and Mossotti, it had been assumed that electric charges are set in motion in the particles of a dielectric when the dielectric is subjected to an electric field ; and the prede- cessors of Maxwell would not have refused to admit that the motion of these charges is in some sense a current. Suppose, then, that S denotes the total current which is capable of generating a magnetic field : since the integral of the magnetic force round any curve is proportional to the electric current which flows through the gap enclosed by the curve, we have in suitable units

curl H = 4;rS.

In order to determine S, we may consider the case of a con- denser whose coatings are supplied with electricity by a conduction-current i per unit-area of coating. If ± o- denote the surface-density of electric charge on the coatings, we have

i = d(r/dtt and o- = D,

where D denotes the magnitude of the electric displacement D in the dielectric between the coatings ; so i = D. But since the total current is to be circuital, its value in the dielectric must be the same as the value i which it has in the rest of the circuit ; that is, the current in the dielectric has the value D. We shall assume that the current in dielectrics always has this value, so that in the general equations the total current must be understood to be i + D.

The above equations, together with those which express the proportionality of E to D in insulators, and to i in conductors, constituted Maxwell's system for a field formed by isotropic bodies which are not in motion. When the magnetic field is .due entirely to currents (including both conduction-currents

Maxwell. 287

and displacement-currents), so that there is no magnetization, we have

V2A = - curl curl A = - curl H

= - 47TS,

so that the vector-potential is connected with the total current by an equation of the same form as that which connects the scalar potential with the density of electric charge. To these potentials Maxwell inclined to attribute a physical significance ; he supposed i// to be analogous to a pressure subsisting in the mass of particles in his model, and A to be the measure of the electrotonic state. The two functions are, however, of merely analytical interest, and do not correspond to physical entities. For let two oppositely-charged conductors, placed close to each other, give rise to an electrostatic field throughout all space. In such a field the vector-potential A is everywhere zero, while the scalar potential $ has a definite value at every point. Now let these conductors discharge each other ; the electrostatic force at any point of space remains unchanged until the point in question is reached by a wave of disturbance, which is propagated outwards from the conductors with the velocity of light, and which annihilates the field as it passes over it. But this order of events is not reflected in the behaviour of Maxwell's functions ;// and A ; for at the instant of discharge, ^ is everywhere annihilated, and A suddenly acquires a finite value throughout all space.

As the potentials do not possess any physical significance, it is desirable to remove them from the equations. This was afterwards done by Maxwell himself, who* in 1868- proposed to base the electromagnetic theory of light solely on the equations

curl H = 47rS,

• curl E = B,

together with the equations which define S in terms of E, and B in terms of H.

• Phil. Trans, clviii (1868), p. 643 : Maxwell's Scient. Papers, ii, p. 125.

288 Maxwell.

The memoir of 1864 contained an extension of the equations to the case of bodies in motion ; the consideration of which naturally revives the question as to whether the aether is in any degree carried along with a body which moves through it. Maxwell did not formulate any express doctrine on this subject ; but his custom was to treat matter as if it were merely a modification of the aether, distinguished only by altered values of such constants as the magnetic permeability and the specific inductive capacity ; so that his theory may be said to involve the assumption that matter and aether move together. In deriving the equations which are applicable to moving bodies, he made use of Faraday's principle that the electromotive force induced in a body depends only on the relative motion of the body and the lines of magnetic force, whether one or the other is in motion absolutely. From this principle it may be inferred that the equation which determines the electric force* in terms of the potentials, in the case of a body which is moving with velocity w, is

E = [w . /zH] - A + grad ^.

Maxwell thought that the scalar quantity -fy in this equation represented the electrostatic potential; but the researches of other investigators-)- have indicated that it represents the sum of the electrostatic potential and the quantity (A . w).

The electromagnetic theory of light was moreover extended in this memoir so as to account for the optical properties of crystals. For this purpose Maxwell assumed that in crystals the values of the coefficients of electric and magnetic induction depend on direction, so that the equation

fjbK = curl A is replaced by

= curl A ;

• It may be here remarked that later writers have distinguished between the electric force in a moving body and the electric force in the aether through which the body is moving, and that E in the present equation corresponds to the former of these vectors.

t Helmholtz, Journ. fiir Math., Ixxviii (1874), p. 309; H. W. Watson, Phil. Mag. (5), xxv (1888), p. 271.

Maxwell. 289

and similarly the equation

E = 47rcO>/6

is replaced by

E = 4;r (c?D.xt c?Dy, cjDz\

The other equations are the same as in isotropic media ; so that

the propagation of disturbance is readily seen to depend on the

equation

(/i J?» ft.ffy, HZHZ} = - curl [c,2 (curl 5),, tf(cuilH}y, Ca2 (curl -#)*)•

Now, if jui, ju2, A3 are supposed equal to each other, this equation is the same as the equation of motion of MacCullagh's aether in crystalline media, the magnetic force H corresponding to MacCullagh's elastic displacement ; and we may therefore immediately infer that Maxwell's electromagnetic equations yield a satisfactory theory of the propagation of light in crystals, provided it is assumed that the magnetic permeability is (for optical purposes) the same in all directions, and pro- vided the plane of polarization is identified with the plane which contains the magnetic vector. It is readily shown that the direction of the ray is at right angles to the magnetic vector and the electric force, and that the wave-front is the plane of the magnetic vector and the electric displacement.f

After this Maxwell proceeded to investigate the propagation of light in metals. The difference between metals and dielectrics, so far as electricity is concerned, is that the former are con- ductors ; and it was therefore natural to seek the cause of the optical properties of metals in their ohmic conductivity. This idea at once suggested a physical reason for the opacity of metals — namely, that within a metal the energy of the light vibrations is converted into Joulian heat in the same way as the energy of ordinary electric currents.

• Cf. pp. 154 et sqq.

f In the memoir of 1864 Maxwell left open the choice between the above theory and that which is obtained by assuming that in crystals the specific inductive rapacity is (for optical purposes) the same in all directions, while the magnetic permeability is aeolotropic. In the latcer case the plane of polarization must be identified with the plane which contains the electric displacement. Nine years later, in his Treatise (§ 794), Maxwell definitely adopted the former alternative.

U

290 Maxwell.

The equations of the electromagnetic field in the metal may be written

curl H = 47rS,

• curl E = H,

S = i + D = KE +

where K denotes the ohmic conductivity ; whence it is seen that the electric force satisfies the equation

=c2V2E.

This is of the same form as the corresponding equation in the elastic-solid theory* ; and, like it, furnishes a satisfactory general explanation of metallic reflexion. It is indeed correct in all details, so long as the period of the disturbance is not too short — i.e., so long as the light- waves considered belong to the extreme infra-red region of the spectrum ; but if we attempt to apply the theory to the case of ordinary light, we are confronted by the difficulty which Lord Eayleigh indicated in the elastic- solid theory,f and which attends all attempts to explain the peculiar properties of metals by inserting a viscous term in the equation. The difficulty is that, in order to account for the properties of ideal silver, we must suppose the coefficient of E negative — that is, the dielectric constant of the metal must be negative, which would imply instability of electrical equilibrium in the metal. The problem, as we have already remarked,:}: was solved only when its relation to the theory of dispersion was rightly understood.

At this time important developments were in progress in the last-named subject. Since the time of Fresnel, theories of dispersion had proceeded! from the assumption that the radii of action of the particles of luminiferous media are so large as to be comparable with the wave-length of light. It was generally supposed that the aether is loaded by the molecules

• Cf. p. iso.

t Cf. p. 181. Cf. also Rayleigh, Phil. Mag. (5) xii (1881), p. 81, and H. A. Lorentz, Over de Theorie de Terugkaatsing, Arnhem, 1875.

• Cf. p. 181. § Cf. p. 182.

Maxwell. 291

of ponderable matter, and that the amount of dispersion depends on the ratio of the wave-length to the distance between adjacent molecules. This hypothesis was, however, seen to be inadequate, when, in 1862, F. P. Leroux* found that a prism filled with the vapour of iodine refracted the red rays to a greater degree than the blue rays; for in all theories which depend on the assumption of a coarse-grained lumini- f erous medium, the refractive index increases with the frequency of the light.

Leroux's phenomenon, to which the name anomalous dis- persion was given, was shown by later investigators-)- to be generally associated with " surface-colour." i.e., the property of brilliantly reflecting incident light of some particular frequency. Such an association seemed to indicate that the dispersive property of a substance is intimately connected with a certain frequency of vibration which is peculiar to that substance, and which, when it happens to fall within the limits of the visible spectrum, is apparent in the surface-colour. This idea of a frequency of vibration peculiar to each kind of ponderable matter is found in the writings of Stokes as far back as the year 1852 ;£ when, discussing fluorescence, he remarked: — " Nothing seems more natural than to suppose that the incident vibrations of the luminiferous aether produce vibratory move- ments among the ultimate molecules of sensitive substances, and that the molecules in turn, swinging on their own account, produce vibrations in the luminiferous aether, and thus cause the sensation of light. The periodic times of these vibrations depend on the periods in which the molecules are disposed to swing, not upon the periodic time of the incident vibrations."

The principle here introduced, of considering the molecules as dynamical systems which possess natural free periods, and which interact with the incident vibrations, lies at the basis of

• Comptes Rendus, Iv (1862), p. 126. In 1870 C. Christiansen (Ann. d. Phys. cxli, p. 479 ; cxliii, p. 250) observed a similar effect in a solution of fuchsin.

r Especially by Kundt, in a series of papers in the Annalen d. Phys., from vol. cxlii (1871) onwards.

j Phil. Trans., 1852, p. 463. Stokes's Coll. Papers, iii., p. 267.

U 2

292 Maxwell.

all modern theories of dispersion. The earliest of these was devised by Maxwell, who, in the Cambridge Mathematical Tripos for 1869,* published the results of the following investigation : —

A model of a dispersive medium may be constituted by embedding systems which represent the atoms of ponderable matter in a medium which represents the aether. We may picture each atomj- as composed of a single massive particle supported symmetrically by springs from the interior face of a massless spherical shell : if the shell be fixed, the particle will be capable of executing vibrations about the centre of the sphere, the effect of the springs being equivalent to a force on the particle proportional to its distance from the centre. The atoms thus constituted may be supposed to occupy small spherical cavities in the aether, the outer shell of each atom being in contact with the aether at all points and partaking of its motion. An immense number of atoms is supposed to exist in each unit volume of the dispersive medium, so that the medium as a whole is fine-grained.

Suppose that the potential energy of strain of free aether per unit volume is

where »j denotes the displacement and E an elastic constant ; so that the equation of wave-propagation in free aether is

3*1 a2,,

''a? = K&

where p denotes the aethereal density.

Then if <r denote the mass of the atomic particles in unit volume, (TJ + £) the total displacement of an atomic particle at the place x at time t, and <rp2£ the attractive force, it is evident that for the compound medium the kinetic energy per unit volume is

• Cambridge Calendar, 1869 ; republished by Lord Kayleigh, Phil. Mag. xlviii (1899), p. 151. t This illustration is due to "W. Thomson.

Maxwell. 293

and the potential energy per unit volume is

The equations of motion, derived by the process usual in dynamics, are

Consider the propagation, through the medium thus constituted, of vibrations whose frequency is n, and whose velocity of pro- pagation in the medium is v ; so that r\ and £ are harmonic functions of n(t - x/v). Substituting these values in the differential equations, we obtain

1 o oil?2

Now, p/^7 has the value 1/c2, where c denotes the velocity of light in free aether; and c/v is the refractive index ju of the medium for vibrations of frequency n. So the equation, which may be written

determines the refractive index of the substance for vibrations of any frequency n. The same formula was independently obtained from similar considerations three years later by W. Sellmeier *

If the oscillations are very slow, the incident light being in the extreme infra-red part of the spectrum, n is small, and the equation gives approximately ju2 = (p + a)jp : for such oscilla- tions, each atomic particle and its shell move together as a rigid body, so that the effect is the same as if the aether were simply loaded by the masses of the atomic particles, its rigidity remaining unaltered.

• Ann. d. Phys. oxlv (1872), pp. 399, 520 : cxlvii (1872), pp. 386, 525. Cf. also Helmholtz, Ann. d. Phys. cliv (1875), p. 582.

294 Maxwell.

The dispersion of light within the limits of the visible spectrum is for most substances controlled by a natural frequency p which corresponds to a vibration beyond the violet end of the visible spectrum : so that, n being smaller than p, we may expand the fraction in the formula of dispersion, and obtain the equation

(T I nz n*

fJL2 = 1 + - (1 + - + -+... f>\ P* P*

which resembles the formula of dispersion in Cauchy's theory* ; indeed, we may say that Cauchy's formula is the expansion of Maxwell's formula in a series which, as it converges only when n has values within a limited range, fails to represent the phenomena outside that range.

The theory as given above is defective in that it becomes meaningless when the frequency n of the incident light is equal to the frequency p of the free vibrations of the atoms. This defect may be remedied by supposing that the motion of an atomic particle relative to the shell in which it is contained is opposed by a dissipative force varying as the relative velocity ; such a force suffices to prevent the forced vibration from becoming indefinitely great as the period of the incident light approaches the period of free vibration of the atoms ; its introduction is justified by the fact that vibrations in this part of the spectrum suffer absorption in passing through the medium. When the incident vibration is not in the same region of the spectrum as the free vibration, the absorption is not of much importance, and may be neglected.

It is shown by the spectroscope that the atomic systems which emit and absorb radiation in actual bodies possess more than one distinct free period. The theory already given may, however, readily be extended-)- to the case in which the atoms have several natural frequencies of vibration ; we have only to suppose that the external massless rigid shell is connected by springs to an interior massive rigid shell, and that this again

• Cf. p. 183.

t This subject was developed by Lord Kelvin in the' Baltimore Lectures.

Maxwell. 295

is connected by springs to another massive shell inside it, and so on. The corresponding extension of the equation for the refractive index is

where p^ p2, . . . denote the frequencies of the natural periods of vibration of the atom.

The validity of the Maxwell- Sellmeier formula of disper- sion was strikingly confirmed by experimental researches in the closing years of the nineteenth century. In 1897 Rubens* showed that the formula represents closely the refractive indices of sylvin (potassium chloride) and rock-salt, with respect to light and radiant heat of wave-lengths between 4,240 A.U. and 223,000 A.U. The constants in the formula being known from this comparison, it was possible to predict the dispersion for radiations of still lower frequency ; and it was found that the square of the refractive index should have a negative value (indicating complete reflexion) for wave- lengths 370,000 A.U. to 550,000 A.U. in the case of rock-salt, and for wave-lengths 450,000 ^to 670,000 A.U. in the case of sylvin. This inference was verified experimentally in the following year.f