# A History of the Theories of Aether and Electricity/Chapter 8

CHAPTER VIII.
MAXWELL.
Since the time of Descartes, natural philosophers have never ceased to speculate on the manner in which electric and magnetic influences are transmitted through space. About the middle of the nineteenth century, speculation assumed a definite form, and issued in a rational theory.
Among those who thought much on the matter was Karl Friedrich Gauss (b. 1777, d. 1855). In a letter to Weber of date March 19, 1845, Gauss remarked that he had long ago proposed to himself to supplement the known forces which act between electric charges by other forces, such as would cause electric actions to be propagated between the charges with a finite velocity. But he expressed himself as determined not to publish his researches until he should have devised a mechanism by which the transmission could be conceived to be effected; and this he had not succeeded in doing.
More than one attempt to realize Gauss's aspiration was made by his pupil Riemann. In a fragmentary note, t which appears to have been written in 1853, but which was not published until after his death, Riemann proposed an aether whose elements should be endowed with the power of resisting compression, and also (like the elements of MacCullagh's aether) of resisting changes of orientation. The former property he conceived to be the cause of gravitational and electrostatic effects, and the latter to be the cause of optical and magnetic phenomena. The theory thus outlined was apparently not developed further by its author; but in a short investigation which was published posthumously in 1867, he ​returned to the question of the process by which electric action is propagated through space. In this memoir he proposed to replace Poisson's equation for the electrostatic potential, namely,
$\nabla ^{2}V+4\pi \rho =0$,
by the equation
$\nabla ^{2}V-{\frac {1}{c^{2}}}{\frac {\partial ^{2}V}{\partial t^{2}}}+4\pi \rho =0$,
according to which the changes of potential due to changing electrification would be propagated outwards from the charges with a velocity c. This, so far as it goes, is in agreement with the view which is now accepted as correct; but Riemann's hypothesis was too slight to serve as the basis of a complete theory. Success came only when the properties of the intervening medium were taken into account.
In that power to which Gauss attached so much importance, of devising dynamical models and analogies for obscure physical phenomena, perhaps no one has ever excelled W. Thomson; and to him, jointly with Faraday, is due the credit of having initiated the theory of the electric medium. In one of his earliest papers, written at the age of seventeen, Thomson compared the distribution of electrostatic force, in a region containing electrified conductors, with the distribution of the flow of heat in an infinite solid: the equipotential surfaces in the one case correspond to the isothermal surfaces in the other, and an electric charge corresponds to a source of heat.
​It may, perhaps, seem as if the value of such an analogy as this consisted merely in the prospect which it offered of comparing, and thereby extending, the mathematical theories of heat and electricity. But to the physicist its chief interest lay rather in the idea that formulae which relate to the electric field, and which had been deduced from laws of action at a distance, were shown to be identical with formulae relating to the theory of heat, which had been deduced from hypotheses of action between contiguous particles.
In 1846—the year after he had taken his degree as second wrangler at Cambridge—Thomson investigated the analogies of electric phenomena with those of elasticity. For this purpose he examined the equations of equilibrium of an incompressible elastic solid which is in a state of strain, and showed that the distribution of the vector which represents the elastic displacement might be assimilated to the distribution of the electric force in an electrostatic system. This, however, as he went on to show, is not the only analogy which may be perceived with the equations of elasticity; for the elastic displacement may equally well be identified with a vector a, defined in terms of the magnetic induction B by the relation
$\mathrm {curl} \ \mathbf {a} =\mathbf {B}$.
The vector a is equivalent to the vector-potential which had been used in the memoirs of Neumann, Weber, and Kirchhoff, on the induction of currents, but Thomson arrived at it independently by a different process, and without being at the time aware of the identification.
The results of Thomson's memoir seemed to suggest a picture of the propagation of electric or magnetic force: might it not take place in somewhat the same way as changes in the elastic displacement are transmitted through an elastic solid? These suggestions were not at the time pursued further by their author; but they helped to inspire another young ​Cambridge man to take up the matter a few years later. James Clerk Maxwell, by whom the problem was eventually solved, was born in 1831, the son of a landed proprietor in Dumfriesshire. He was educated at Edinburgh, and at Trinity College, Cambridge, of which society he became in 1855 a Fellow; and not long after his election to Fellowship, he communicated to the Cambridge Philosophical Society the first of his endeavours to form a mechanical conception of the electro-magnetic field.
Maxwell had been reading Faraday's Experimental Researches; and, gifted as he was with a physical imagination akin to Faraday's, he had been profonudly impressed by the theory of lines of force. At the same time, he was a trained mathematician; and the distinguishing feature of almost all his researches was the union of the imaginative and the analytical faculties to produce results partaking of both natures. This first memoir may be regarded as an attempt to connect the ideas of Faraday with the mathematical analogies which had been devised by Thomson.
Maxwell considered first the illustration of Faraday's lines of force which is afforded by the lines of flow of a liquid. The lines of force represent the direction of a vector; and the magnitude of this vector is everywhere inversely proportional to the cross-section of a narrow tube formed by such lines. This relation between magnitude and direction is possessed by any circuital vector; and in particular by the vector which represents the velocity at any point in a fluid, if the fluid be incompressible. It is therefore possible to represent the magnetic induction B, which is the vector represented by Faraday's lines of magnetic force, as the velocity of an incompressible fluid. Such an analogy had been indicated some years previously by Faraday himself, who had suggested that along the lines of magnetic force there may be a "dynamic condition," analogous to that of the electric current, and ​that, in fact, "the physical lines of magnetic force are currents."
The comparison with the lines of flow of a liquid is applicable to electric as well as to magnetic lines of force. In this case the vector which corresponds to the velocity of the fluid is, in free aether, the electric force E. But when different dielectrics are present in the field, the electric force is not a circuital vector, and therefore cannot be represented by lines of force; in fact, the equation
$\mathrm {div} \ \mathbf {E} =0$,
is now replaced by the equation
$\mathrm {div} (\epsilon \mathbf {E} )=0$,
where ε denotes the specific inductive capacity or dielectric constant at the place (x, y, z). It is, however, evident from this equation that the vector εE is circuital; this vector, which will be denoted by D, bears to E a relation similar to that which the magnetic induction B bears to the magnetic force H. It is the vector D which is represented by Faraday's lines of electric force, and which in the hydrodynamical analogy corresponds to the velocity of the incompressible fluid.
In comparing fluid motion with electric fields it is necessary to introduce sources and sinks into the fluid to correspond to the electric charges; for D is not circuital at places where there. is free charge. The magnetic analogy is therefore somewhat the simpler.
In the latter half of his memoir Maxwell discussed how Faraday's "electrotonic state" might be represented in mathematical symbols. This problem be solved by borrowing from Thomson's investigation of 1847 the vector a, which is defined in terms of the magnetic induction by the equation
$\mathrm {curl} \ \mathbf {a} =\mathbf {B}$;
if, with Maxwell, we call a the electrotonic intensity, the. equation is equivalent to the statement that "the entire electrotonic intensity round the boundary of any surface measures the number of lines of magnetic force which pass ​through that surface." The electromotive force of induction at the place (x, y, z) is -∂a/∂t: as Maxwell said, "the electromotive force on any element of a conductor is measured by the instantaneous rate of change of the electrotonic intensity on that element." From this it is evident that a is no other than the vector-potential which had been employed by Neumann, Weber, and Kirchhoff, in the calculation of induced currents; and we may take for the electrotonic intensity due to a current i′ flowing in a circuit s′ the value which results from Neumann's theory, namely,
$\mathbf {a} =i'\int {\frac {\mathbf {ds'} }{r}}$.
It may, however, be remarked that the equation
$\mathrm {curl} \ \mathbf {a} =\mathbf {B}$,
taken alone, is insufficient to determine a uniquely; for we can choose a so as to satisfy this, and also to satisfy the equation
$\mathrm {div} \ \mathbf {a} =\psi$,
where ψ denotes any arbitrary scalar. There are, therefore, an infinite number of possible functions a. With the particular value of a which has been adopted, we have
          ${\begin{matrix}\mathrm {div} \ \mathbf {a} &=&{\frac {\partial }{\partial x}}i'\int _{s'}{\frac {dx'}{r}}+{\frac {\partial }{\partial y}}i'\int _{s'}{\frac {dy'}{r}}+{\frac {\partial }{\partial z}}i'\int _{s'}{\frac {dz'}{r}}\\\ &=&-i'\int _{s'}\left(dx'.{\frac {\partial }{\partial x'}}+dy'.{\frac {\partial }{\partial y'}}+dz'.{\frac {\partial }{\partial z'}}\right)\left({\frac {1}{r}}\right)\\\ &=&-i'\int _{s'}d\left({\frac {1}{r}}\right)\\\ &=&0;\\\end{matrix}}$
so the vector-potential a which we have chosen is circuital.
In this memoir the physical importance of the operators curl and div first became evident; for, in addition to those applications which have been mentioned, Maxwell showed that ​he connexion between the strength ι of a current and the magnetic field H, to which it gives rise, may be represented by the equation
$4\pi \iota =\mathrm {curl} \ \mathbf {H}$;
this equation is equivalent to the statement that "the entire magnetic intensity round the boundary of any surface measures the quantity of electric current which passes through that surface."
In the same year (1856) in which Maxwell's investigation was published, Thomson put forward an alternative interpretation of magnetism. He had now come to the conclusion, from a study of the magnetic rotation of the plane of polarization of light, that magnetism possesses a rotatory character: and suggested that the resultant angular momentum of the thermal motions of a body might be taken as the measure of the magnetic moment. "Tho explanation," he wrote, "of all phenomena of electromagnetic attraction or repulsion, or of electromagnetic induction, is to be looked for simply in the inertia or pressure of the matter of which the motions constitute heat. Whether this matter is or is not electricity, whether it is a continuous fluid interpermeating the spaces between molecular nuclei, or is itself molecularly grouped: or whether all matter is continuous, and molecular heterogeneousness consists in finite vortical or other relative motions of contiguous parts of a body: it is impossible to decide, and, perhaps, in vain to speculate, in the present state of science."
The two interpretations of magnetism, in which the linear and rotatory characters respectively are attributed to it, occur frequently in the subsequent history of the subject. The former was amplified in 1858, when Helmholtz published his researches on vortex motion; for Helmholtz showed that if a ​magnetic field produced by electric currents is compared to the flow of an incompressible fluid, so that the magnetic vector is represented by the fluid velocity, then the electric currents correspond to the vortex-filaments in the fluid. This analogy correlates many theorems in hydrodynamics and electricity; for instance, the theorem that a re-entrant vortex-filament is equivalent to a uniform distribution of doublets over any surface bounded by it, corresponds to Ampère's theorem of the equivalence of electric currents and magnetic shells.
In his memoir of 1855, Maxwell had not attempted to construct a mechanical model of electrodynamic actions, but had expressed his inteution of doing so. "By a careful study," he wrote, "of the laws of elastic solids, and of the motions of viscous fluids, I hope to discover a method of forming a mechanical conception of this electrotonic state adapted to general reasoning", and in a foot-note he referred to the effort which Thomson had already made in this direction. Six years elapsed, however, before anything further on the subject was published. In the meantime, Maxwell became Professor of Natural Philosophy in King's College, London—a position in which he had opportunities of personal contact with Faraday, whom he had long reverenced. Faraday had now concluded the Experimental Researches, and was living in retirement at Hampton Court; but his thoughts frequently recurred to the great problem which he had brought so near to solution. It appears from his note-book that in 1857 he was speculating whether the velocity of propagation of magnetic action is of the same order as that of light, and whether it is affected by the susceptibility to induction of the bodies through which the action is transmitted.
The answer to this question was furnished in 1861-2, when Maxwell fulfilled his promise of devising a mechanical conception of the electromagnetic field.
​In the interval since the publication of his previous memoir Maxwell had become convinced by Thomson's arguments that magnetism is in its nature rotatory. "The transference of electrolytes in fixed directions by the electric current, and the rotation of polarized light in fixed directions by magnetic force, are," he wrote, "the facts the consideration of which has induced me to regard magnetism as a phenomenon of rotation, and electric currents as phenomena of translation." This conception of magnetism he brought into connexion with Faraday's idea, that tubes of force tend to contract longitudinally and to expand laterally. Such a tendency may be attributed to centrifugal force, if it be assumed that each tube of force contains fluid which is in rotation about the axis of the tube. Accordingly Maxwell supposed that, in any magnetic field, the median whose vibrations constitute light is in rotation about the lines of magnetic force; each unit tube of force may for the present be pictured as an isolated vortex.
The energy of the motion per unit volume is proportional to μH$^2$, where μ denotes the density of the medium, and H denotes the linear velocity at the circumference of each vortex. But, as we have seen, Thomson had already shown that the energy of any magnetic field, whether produced by magnets or by electric currents, is
${\frac {1}{8\pi }}\iiint \mu \mathbf {H} ^{2}\ dx\ dy\ dz$
where the integration is taken over all space, and where u denotes the magnetic permeability, and H the magnetic force. It was therefore natural to identify the density of the medium at any place with the magnetic permeability, and the circumferential velocity of the vortices with the magnetic force.
But an objection to the proposed analogy now presents itself. Since two neighbouring vortices rotate in the same direction, the particles in the circumference of one vortex must be moving in the opposite direction to the particles contiguous ​to them in the circumference of the adjacent vortex; and it seems, therefore, as if the motion would be discontinuous. Maxwell escaped from this difficulty by imitating a well-known mechanical arrangement. When it is desired that two wheels should revolve in the same sense, an "idle" wheel is inserted between them so as to be in gear with both. The model of the electromagnetic field to which Maxwell arrived by the introduction of this device greatly resembles that proposed by Bernoulli in 1736. He supposed a layer of particles, acting as idle wheels, to be interposed between each vortex and the next, and to roll without sliding on the vortices; so that each vortex tends to make the neighbouring vortices revolve in the same direction as itself. The particles were supposed to be not otherwise constrained, so that the velocity of the centre of any particle would be the mean of the circumferential velocities of the vortices between which it is placed. This condition yields (in suitable units) the analytical equation
$4\pi \mathbf {l} =\mathrm {curl} \ \mathbf {H}$,
where the vector l denotes the flux of the particles, so that its x-component i$_x$, denotes the quantity of particles transferred in unit time across unit area perpendicular to the x-direction. On comparing this equation with that which represents Oersted's discovery, it is seen that the flux l of the movable particles interposed between neighbouring vortices is the analogue of the electric current.
It will be noticed that in Maxwell's model the relation between electric current and magnetic force is secured by a connexion which is not of a dynamical, but of a purely kinematical character. The above equation simply expresses the existence of certain non-holonomic constraints within the system.
If from any cause the rotatory velocity of some of the cellular vortices is altered, the disturbance will be propagated from that part of the model to all other parts, by the mutual ​action of the particles and vortices. This action is determined, as Maxwell showed, by the relation
$\mu \mathbf {H} =-\mathrm {curl} \ \mathbf {E}$
which connects E, the force exerted on a unit quantity of particles at any place in consequence of the tangential action of the vortices, with Ḣ, the rate of change of velocity of the neighbouring vortices. It will be observed that this equation is not kinematical but dynamical. On comparing it with the electromagnetic equations
${\begin{cases}\mathrm {curl} \ \mathbf {a} &=&\mu \mathbf {H} ,\\\mathrm {Induced\ electromotive\ force} &=&-{\overset {.}{\mathbf {a} }},\end{cases}}$
it is seen that E must be interpreted electromagnetically as the induced electromotive force. Thus the motion of the particles constitutes an electric current, the tangential force with which they are pressed by the matter of the vortex-cells constitutes electromotive force, and the pressure of the particles on each other may be taken to correspond to the tension or potential of the electricity.
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 eclls; 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.
​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 conductors 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 displacement 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 constituted by the variation of this polarization is then analogous to ∂I/∂t. 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​ment-current corresponds to ∂B/∂t, 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
$={\frac {1}{4\pi c_{1}^{2}}}\mathbf {E}$,
where c$_1$ denotes a constant which depends on the elastic properties of the cells. The displacement-current Ḋ must now be inserted in the relation which connects the current with the magnetic force; and thus we obtain the equation
$\mathrm {curl} \ \mathbf {H} =4\pi \mathbf {s}$,
where the vector S, which is called the total current, is the sum of the convection-current i and the displacement-current Ḋ. 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
$\mathrm {div} \ \mathbf {S} =0$.
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, ​the constant μ be supposed to have the value unity, the equations may be written
${\begin{cases}\mathrm {div} \ \mathbf {H} &=&0,\\c_{1}^{2}\mathrm {curl} \ \mathbf {H} &=&{\overset {.}{\mathbf {E} }},\\-\mathrm {curl} \ \mathbf {H} &=&{\overset {.}{\mathbf {H} }}.\end{cases}}$
Eliminating E, we see that H satisfies the equations
${\begin{cases}\mathrm {div} \ \mathbf {H} &=&0,\mathrm {div} \ {\overset {.}{\mathbf {H} }}&=&c_{1}^{2}\nabla ^{2}\mathbf {H} .\end{cases}}$
But these are precisely the equations which the light-vector satisfies in a medium in which the velocity of propagation is c$_1$: 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
$-c_{1}^{2}{\frac {\partial H_{y}}{\partial z}}={\frac {\partial E_{x}}{\partial t}},c_{1}^{2}{\frac {\partial H_{x}}{\partial z}}={\frac {\partial E_{y}}{\partial t}},{\frac {\partial E_{y}}{\partial z}}={\frac {\partial H_{x}}{\partial t}},-{\frac {\partial E_{x}}{\partial z}}={\frac {\partial H_{y}}{\partial t}}$
whence we have
$c_{1}H_{y}=E_{x}$, $-c_{1}H_{z}=E_{y}$
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 c$_1$. 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
$\int _{0}^{\mathbf {D} }\mathbf {E.} d\mathbf {D}$ or ${\tfrac {1}{2}}\mathbf {ED}$.
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 ​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 tho energy per unit volume of the dielectric is εE′$^2$/8π, where ε 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 investigation, we have E = cE′, where e denotes the constant which occurs in transformations of this kind. The energy is therefore εE$^2$/8πc$^2$ per unit volume. Comparing this with the expression for the energy in terms of E and D, we have
$\mathbf {D} =\epsilon \mathbf {E} /4\pi c^{2}$,
and therefore the constant c$_1$ has the value cε$^{-1/2}$. Thus the result is obtained that the velocity of propagation of disturbances in Maxwell's medium is cε$^{-1/2}$, where ε denotes the specific inductive capacity and c denotes the velocity for which Kohlrausch and Weber had found the value 3·1 x 10$^{10}$ 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 Fizeau 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. Tho velocity of light was calculated from the dimensions and angular velocity of the wheel and the distance of the mirror; the result being 3·15 10$^{10}$ cm./sec.
​ 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 propagation 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: "Is there any electric fluid sui generis at all, distinct from the aether?"
The presence of the dielectric constant ε in the expression cε$^{-1/2}$, which Maxwell had obtained for the velocity of propagation 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 μ has been supposed to have the value unity. If this is not the case, the ​velocity of propagation of disturbance may be shown, by the same analysis, to be cε$^{-1/2}$μ$^{-1/2}$; so that it is diminished when μ 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 contemporaries 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, lie 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 Electromagnetic Field," which was read to the Royal Society in 1864; ​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 μH, being a circuital vector, may be expressed in terms of a vector-potential A by the equation
μH = curl A.
The electric displacement D is connected with the volume-density ρ of free electric charge by tho electrostatic equation
div D = ρ.
The principle of conservation of electricity yields the equation
div ι = -∂ρ/∂t,
where ι 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 = μḢ;
from which it follows that the electric force E must be expressible in the form
E = - Å + grad,
where ψ denotes some scalar function. The quantities A and ψ which occur in this equation are not as yet completely determinate; 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 ψ also must be indeterminate, Maxwell decided the matter by assuming A to be a circuital vector; thus div A = 0, and therefore div E = -∇$^2$ψ, ​from which equation it is evident that represents the electrostatic 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 predecessors 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 = 4πS. In order to determine S, we may consider the case of a condenser whose coatings are supplied with electricity by a conduction-current ι per unit-area of coating. If ± σ denote the surface-density of electric charge on the coatings, we have
i = ∂σ/∂t, and σ = D,
where D denotes the magnitude of the electric displacement D in the dielectric between the coatings; so ι = Ḋ. But since the total current is to be circuital, its value in the dielectric must be the same as the value ι which it has in the rest of the circuit; that is, the current in the dielectric has the value Ḋ. 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 ι + Ḋ.
The above equations, together with those which express the proportionality of E to D in insulators, and to ι 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 ​and displacement-currents), so that there is no magnetization, we have
${\begin{matrix}\nabla ^{2}\mathbf {A} &=&-\mathrm {curl\ curl} \ \mathbf {A} =-\mathrm {curl} \ \mathbf {H} \\\ &=&-4\pi \mathbf {S} ,\end{matrix}}$
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 ψ 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 ψ 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 ψ and A; for at the instant of discharge, ψ 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
${\begin{matrix}\mathrm {curl} \ \mathbf {H} &=&4\pi \mathbf {S} \\-\mathrm {curl} \ \mathbf {E} &=&\mathbf {\overset {.}{B}} ,\end{matrix}}$
together with the equations which define S in terms of E, and B in terms of H.
​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
$\mathbf {E} =[\mathbf {w.} \mu \mathbf {H} ]-{\overset {.}{\mathbf {A} }}+\mathrm {grad} \ \psi$.
Maxwell thought that the scalar quantity ψ 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
$\mu \mathbf {H} =\mathrm {curl} \ \mathbf {A}$
is replaced by
$(\mu _{1}H_{x},\mu _{2}H_{y},\mu _{3}H_{z})=\mathrm {curl} \ \mathbf {A}$;
$\mathbf {E} =4\pi c^{2}\mathbf {D} /\epsilon$
is replaced by
$\mathbf {E} =4\pi (c_{1}^{2}D_{x},c_{2}^{2}D_{y},c_{3}^{2}D_{z})$.
The other equations are the same as in isotropio media; so that the propagation of disturbance is readily seen to depend on the equation
$(\mu _{1}{\overset {..}{H}}_{x},\mu _{2}{\overset {..}{H}}_{y},\mu _{3}{\overset {..}{H}}_{z})=-\mathrm {curl} \{c_{1}^{2}(\mathrm {curl} H)_{x},c_{2}^{2}(\mathrm {curl} H)_{y},c_{3}^{2}(\mathrm {curl} H)_{z}\}$.
Now, if μ$_1$, μ$_2$, μ$_3$ 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 provided 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.
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 conductors; 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.
​The equations of the electromagnetic field in the metal may be written
${\begin{cases}\ &\mathrm {curl} &\mathbf {H} &=&4\pi \mathbf {S} ,\\-&\mathrm {curl} &\mathbf {H} &=&{\overset {.}{\mathbf {H} }},\\\ &\ &\mathbf {S} &=&\mathbf {\iota } +{\overset {.}{\mathbf {D} }}=\kappa \mathbf {E} +\epsilon {\overset {.}{\mathbf {E} }}/4\pi c^{2},\end{cases}}$
whore k denotes the ohmic conductivity; whence it is seen that the electric force satisfies the equation
$\epsilon {\overset {..}{\mathbf {E} }}+4\pi \kappa c^{2}{\overset {.}{\mathbf {E} }}=c^{2}\nabla ^{2}\mathbf {E}$.
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 Rayleigh indicated in the elastic-solid theory. 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 Ë 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 ​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. ferous medium,, the refractive index increases with the frequency of the light.
Leroux's phenomenon, to which the name anomalous dispersion 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 movements 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 ​We may 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. picture each atom 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
${\tfrac {1}{2}}\mathbf {E} \left({\frac {\partial \eta }{\partial x}}\right)^{2}$,
where η denotes the displacement and E an elastic constant; so that the equation of wave-propagation in free aether is
$\rho {\frac {\partial ^{2}\eta }{\partial t^{2}}}=E{\frac {\partial ^{2}\eta }{\partial x^{2}}}$,
where ρ denotes the aethereal density.
Then if σ denote the mass of the atomic particles in unit volume, (η + ζ) the total displacement of an atomic particle at the place x at tine t, and σp$^2$ζ the attractive force, it is evident that for the compound medium the kinetic energy per unit volume is
${\tfrac {1}{2}}\rho {\frac {\partial \eta }{\partial t}}^{2}+{\tfrac {1}{2}}\sigma \left({\frac {\partial \eta }{\partial t}}+{\frac {\partial \zeta }{\partial t}}\right)^{2}$,
${\tfrac {1}{2}}E(\partial \eta /\partial x)^{2}+{\tfrac {1}{2}}\sigma p^{2}\zeta ^{2}$.
The equations of motion, derived by the process usual in dynamics, are
${\begin{cases}\rho {\frac {\partial ^{2}\eta }{\partial t^{2}}}+\sigma \left({\frac {\partial ^{2}\eta }{\partial t^{2}}}+{\frac {\partial ^{2}\zeta }{\partial t^{2}}}\right)-E{\frac {\partial ^{2}\eta }{\partial x^{2}}}=0,\\\sigma \left({\frac {\partial ^{2}\eta }{\partial t^{2}}}+{\frac {\partial ^{2}\zeta }{\partial t^{2}}}\right)+\sigma p^{2}\zeta =0.\end{cases}}$
Consider the propagation, through the medium thus constituted, of vibrations whose frequency is n, and whose velocity of propagation in the medium is v; so that η and ζ are harmonic functions of n(t - x/v). Substituting these values in the differential equations, we obtain
${\frac {1}{v^{2}}}={\frac {\rho }{E}}+{\frac {\sigma p^{2}}{E(p^{2}-n^{2})}}$.
Now, ρ/E has the value 1/c$^2$, where c denotes the velocity of light in free aether; and c/v is the refractive index μ of the medium for vibrations of frequency n. So the equation, which may be written
$\mu ^{2}=1+{\frac {\sigma p^{2}}{\rho (p^{2}-n^{2})}}$,
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 μ$^2$ = (ρ + σ)/ρ: for such oscillations, 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.
​The dispersion of light within the limits of the visible spectrum is for most substances controlled by a natural frcquency 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
$\mu ^{2}=1+{\frac {\sigma }{\rho }}\left(1+{\frac {n^{2}}{p^{2}}}+{\frac {n^{4}}{p^{4}}}+...\right)$,
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 a 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 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 ​is connected by springs to another massive shell inside it, and so on. The corresponding extension of the equation for the refractive index is
$\mu ^{2}-1={\frac {c_{1}}{p_{1}^{2}-n^{2}}}+{\frac {c_{2}}{p_{2}^{2}-n^{2}}}+$…,
where p$_1$, p$_2$, … denote the frequencies of the natural periods of vibration of the atom.
The validity of the Maxwell-Sellmeier formula of dispersion 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 tho 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 wavelengths 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.V. in the case of sylvin. This inference was verified experimentally in the following year.
It may seem strange that Maxwell, having successfully employed his electromagnetic theory to explain the propagation of light in isotropic media, in crystals, and in metals, should have omitted to apply it to the problem of reflexion and refraction. This is all the more surprising, as the study of the optics of crystals had already revealed a close analogy between the electromagnetic theory and MacCullagh's elastic-solid theory; and in order to explain reflexion and refraction eloctromagnetically, nothing more was necessary than to transcribe MacCullagh's investigation of the same problem, interpreting ė (the time-flux of the displacement of MacCullagh's aether) as the magnetic force, and curl e as the electric displacement. As ​in MacCullagh's theory the difference between the contiguous media is represented by a difference of their elastic constants, 80 in the electromagnetic theory it may be represented by a difference in their specific inductive capacities. From a letter which Maxwell wrote to Stokes in 1864, and which has been preserved, it appears that the problem of reflexion and refraction was engaging Maxwell's attention at the time when he was preparing his Royal Society memoir on the electromagnetic field; but he was not able to satisfy himself regarding the conditions which should be satisfied at the interface between the media. He seems to have been in doubt which of the rival elastic-solid theories to take as a pattern; and it is not unlikely that he was led astray by relying too much on the analogy between the electric displacement and an elastic displacement. For in the elastic-solid theory all three components of the displacement must be continuous across the interface between two contiguous media; but Maxwell found that it was impossible to explain reflexion and refraction if all three components of the electric displacement were supposed to be continuous across the interface; and, unwilling to give up the analogy which had hitherto guided him aright, yet unable to disprove the Greenian conditions at bounding surfaces, he seems to have laid aside the problem until some new light should dawn upon it.
This was not the only difficulty which beset the electromagnetic theory. The theoretical conclusion, that the specific inductive capacity of a medium should be equal to the square of its refractive index with respect to waves of long period, was not as yet substantiated by experiment; and the theory of displacement-currents, on which everything else depended, was ​unfavourably received by the most distinguished of Maxwell's contemporaries. Helmholtz indeed ultimately accepted it, but only after many years; and W. Thomson (Kelvin) seems never to have thoroughly believed it to the end of his long life. In 1888 he referred to it as a "curious and ingenious, but not wholly tenable hypothesis," and proposed to replace it by an extension of the older potential theories. In 1896 he had some inclination to speculate that alterations of electrostatic force due to rapidly-changing electrification are propagated by condensational waves in the luminiferous aether. In 1904 he admitted that a bar-magnet rotating about an axis at right angles to its length is equivalent to a lamp emitting light of period equal to the period of the rotation, but gave his final judgment in the sentence:—"The so-called electromagnetic theory of light has not helped us hitherto."
Thomson appears to have based his ideas of the propagation of electric disturbance on the case which had first become familiar to him—that of the transmission of signals along a wire. He clung to the older view that in such a disturbance the wire is the actual medium of transmission; whereas in Maxwell's theory the function of the wire is merely to guide the disturbance, which is resident in the surrounding dielectric.
This opinion that conductors are the media of propagation of electric disturbance was entertained also by Ludwig Lorenz (b. 1829, d. 1891), of Copenhagen, who independently developed an electromagnetic theory of light a few years after the publication of Maxwell's memoirs. The procedure which Lorenz followed was that which Riemann had suggested in 1858—namely, to modify the accepted formulae of electrodynamics by introducing terms which, though too small to be ​appreciable in ordinary laboratory experiments, would be capable of accounting for the propagation of electrical effects through space with a finite velocity. We have seen that in Neumann's theory the electric force E was determined by the equation
$\mathbf {E} =c^{2}\mathrm {grad} \ \phi -{\overset {.}{\mathbf {a} }}$, (1)
where φ denotes the electrostatic potential defined by the equation
$\phi =\iiint (\rho ^{\prime }/r)dx^{\prime }\ dy^{\prime }\ dz^{\prime }$,
ρ′ being the density of electric charge at the point (x′, y′, z′), and where a denotes the vector-potential, defined by the equation
$\mathbf {a} =\iiint (\mathbf {\iota ^{\prime }} /r)dx^{\prime }\ dy^{\prime }\ dz^{\prime }$,
ι′ being the conduction-current at (x′, y′, z′). We suppose the specific inductive capacity and the magnetic permeability to be everywhere unity.
Lorenz proposed to replace these by the equations
$\mathbf {a} =\iiint \{\mathbf {\iota ^{\prime }} (t-r/c)/r\}dx^{\prime }\ dy^{\prime }\ dz^{\prime }$,
$\mathbf {a} =\iiint \{\mathbf {\iota ^{\prime }} (t-r/c)/r\}dx^{\prime }\ dy^{\prime }\ dz^{\prime }$;
the change consists in replacing the values which ρ′ and ι′ have at the instant t by those which they have at the instant (t - r/c), which is the instant at which a disturbance travelling with velocity c must leave the place (x′, y′, z′) in order to arrive at the place (x, y, z) at the instant t. Thus the values of the potentials at (x, y, z) at any instant t would, according to Lorenz's theory, depend on the electric state at the point (x′, y′, z′) at the previous instant (t - r/c): as if the potentials were propagated outwards from the charges and currents with velocity c. The functions φ and a formed in this way are generally known as the retarded potentials.
​The equations by which φ and a have been defined are equivalent to the equations
$\nabla ^{2}\phi ={\overset {..}{\phi }}/c^{2}=-4\pi \rho$, (2)
$\nabla ^{2}\mathbf {a} ={\overset {..}{\mathbf {a} }}/c^{2}=-4\pi \mathbf {\iota }$, (3)
while the equation of conservation of electricity,
$\mathrm {div} \ \mathbf {\iota } +{\overset {.}{\rho }}=0$
gives
$\mathrm {div} \ \mathbf {a} +{\overset {.}{\phi }}=0$. (4)
From equations (1), (2), (4), we may readily derive the equation
$\mathrm {div} \ \mathbf {E} =4\pi c^{2}\rho$; (I)
and from (1), (3), (4), we have
$\mathrm {curl} \ \mathbf {H} ={\overset {.}{\mathbf {E} }}/c^{2}+4\pi \mathbf {\iota }$, (II)
where H or curl a denotes the magnetic force: while from (1) we have
$\mathrm {curl} \ \mathbf {E} =-{\overset {.}{\mathbf {H} }}$. (III)
The equations (I), (II), (III) are, however, the fundamental equations of Maxwell's theory; and therefore the theory of L. Lorenz is practically equivalent to that of Maxwell, so far as concerns the propagation of electromagnetic disturbances through free aether. Lorenz himself, however, does not appear to have clearly perceived this; for in his memoir he postulated the presence of conducting matter throughout space, and was consequently led to equations resembling those which Maxwell had given for the propagation of light in metals. Observing that his equations represented periodic electric currents at right angles to the direction of propagation of the disturbance, he suggested that all luminous vibrations might be constituted by electric currents, and hence that there was "no longer any reason for maintaining the hypothesis of an aether, since we can admit that space contains sufficient ponderable matter to enable the disturbance to be propagated."
Lorenz was unable to derive from his equations any explanation of the existence of refractive indiecs, and his theory lacks ​the rich physical suggestiveness of Maxwell's; the value of his memoir lies chiefly in the introduction of the retarded potentials. It may be remarked in passing that Lorenz's retarded potentials are not identical with Maxwell's scalar and vector potentials; for Lorenz's a is not a circuital vector, and Lorenz's φ is not, like Maxwell's, the electrostatic potential, but depends on the positions occupied by the charges at certain previous instants.
For some years no progress was made either with Maxwell's theory or with Lorenz's. Meanwhile, Maxwell had in 1865 resigned his chair at King's College, and had retired to his estate in Dumfricsshire, where he occupied himself in writing a connected account of electrical theory. In 1871 he returned to Cambridge as Professor of Experimental Physics; and two years later published his Treatise on Electricity and Magnetism.
In this celebrated work is comprehended almost every branch of electric and magnetic theory; but the intention of the writer was to discuss the whole as far as possible from a single point of view, namely, that of Faraday; so that little or no account was given of the hypotheses which had been propounded in the two preceding decades by the great German electricians. So far as Maxwell's purpose was to disseminate the ideas of Faraday, it was undoubtedly fulfilled; but the Treatise was less successful when considered as the exposition of its author's own views. The doctrines peculiar to Maxwell—the existence of displacement-currents, and of electromagnetic vibrations identical with light—were not introduced in the first volume, or in the first half of the second volume; and the account which was given of them was scarcely more complete, and was perhaps less attractive, than that which had been furnished in the original memoirs.
Some matters were, however, discussed more fully in the Treatise than in Maxwell's previous writings, and among these was the question of stress in the electromagnetic field.
It will be remembered that Faraday, when studying the ​curvature of lines of force in electrostatic fields, had noticed an apparent tendency of adjacent lines to repel each other, as if each tube of force were inherently disposed to distend laterally; and that in addition to this repellent or diverging force in the transverse direction, he supposed an attractive or contractile force to be exerted at right angles to it, that is to say, in the direction of the lines of force.
Of the existence of these pressures and tensions Maxwell was fully persuaded; and he determined analytical expressions suitable to represent them. The tension along the lines of force must be supposed to maintain the ponderomotive force which acts on the conductor on which the lines of force terminate; and it may therefore be measured by the force which is exerted on unit area of the conductor, i.e., εE$^2$/8πc$^2$ or 1/2DE. The pressure at right angles to the lines of force must then be determined so as to satisfy the condition that the aether is to be in equilibrium.
For this purpose, consider a thin shell of aether included between two equipotential surfaces. The equilibrium of the portion of this shell which is intercepted by a tube of force: requires (as in the theory of the equilibrium of liquid films), that the resultant force per unit area due to the abovementioned normal tensions on its two faces shall have the value T(1/ρ$_1$ + 1/ρ$_2$), where ρ$_1$ and ρ$_2$ denote the principal radii of curvature of the shell at the place, and where T denotes. the lateral stress across unit length of the surface of the shell, I being analogous to the surface-tension of a liquid film.
Now, if t denote the thickness of the shell, the area intercepted on the second face by the tube of force bears to the area intercepted on the first face the ratio (ρ$_1$ + t) (ρ$_2$ + t)/ρ$_1$ρ$_2$; and by the fundamental property of tubes of force, D and E vary inversely as the cross-section of the tube, so the total force on the second face will bear to that on the first face the ratio
ρ$_1$ρ$_2$/(ρ$_1$ + t) (ρ$_2$ + t),
or approximately
(1 - t/ρ$_1$ - tρ$_2$);
therefore
$-{\tfrac {1}{2}}\mathbf {DE} .t.(1/\rho _{1}+1/\rho _{2})$,
and so we have
$T=-{\tfrac {1}{2}}\mathbf {DE} .t$;
or the pressure at right angles to the lines of force is 1/2DE per unit area—that is, it is numerically equal to the tension along the lines of force.
The principal stresses in the medium being thus determined, it readily follows that the stress across any plane, to which the unit vector N is normal, is
$(\mathbf {D.N} )\mathbf {E} -{\frac {1}{2}}(\mathbf {D.E} )\mathbf {N}$.
Maxwell obtained a similar formula for the case of magnetic fields; the pouderomotive forces on magnetized matter and on conductors carrying currents may be accounted for by assuming a stress in the medium, the stress across the plane N being represented by the vector
${\frac {1}{4\pi }}(\mathbf {B.N} ).\mathbf {H} -{\frac {1}{8\pi }}(\mathbf {B.H} ).\mathbf {N}$.
This, like the corresponding electrostatic formula, represents a tension across planes perpendicular to the lines of force, and a pressure across planes parallel to them.
It may be remarked that Maxwell made no distinction between stress in the material dielectric and stress in the aether: indeed, so long as it was supposed that material bodies when displaced carry tho contained aether along with them, no distinction was possible. In the modifications of Maxwell's theory which were developed many years afterwards by his followers, stresses corresponding to those introduced by Maxwell were assigned to the aether, as distinct from ponderable matter; and it was assumed that the only stresses set up in material bodies by the electromagnetic field are produced indirectly: they may be calculated by the methods of the theory of elasticity, from & knowledge of the ponderomotive forces exerted on the electric charges connected with the bodies.
​Another remark suggested by Maxwell's theory of stress in the medium is that he considered the question from the purely statical point of view. He determined the stress so that it might produce the required forces on ponderable bodies, and be self-equilibrating in free aether. But if the electric and magnetic phenomena are not really statical, but are kinetic in their nature, the stress or pressure need not be self-equilibrating, This may be illustrated by reference to the hydrodynamical models of the aether shortly to be described, in which perforated solids are immersed in a moving liquid: the ponderomotive forces exerted on the solids by the liquid correspond to those which act on conductors carrying currents in a magnetic field, and yet there is no stress in the medium beyond the pressure of the liquid.
Among the problems to which Maxwell applied his theory of stress in the medium was one which had engaged the attention of many generations of his predecessors. The adherents of the corpuscular theory of light in the eighteenth century believed that their hypothesis would be decisively confirmed if it could be shown that rays of light possess momentum: to determine the matter, several investigators directed powerful beams of light on delicately-suspended bodies, and looked for evidences of a pressure due to the impulse of the corpuscles. Such an experiment was performed in 1708 by Homberg, who imagined that he actually obtained the effect in question; but Mairan and Du Fay in the middle of the century, having repeated his operations, failed to confirm his conclusion.
The subject was afterwards taken up by Michell, who "some years ago," wrote Priestley in 1772, "endeavoured to ascertain the momentum of light in a much more accurate manner than those in which M. Homberg and M. Mairan had attempted it." He exposed a very thin and delicately-suspended copper plate ​to the rays of the sun concentrated by a mirror, and observed a deflexion. He was not satisfied that the effect of the heating of the air had been altogether excluded, but "there seems to be no doubt," in Priestley's opinion, "but that the motion above mentioned is to be ascribed to the impulse of the rays of light."
A similar experiment was made by A. Bennet, who directed the light from the focus of a large lens on writing-paper delicately suspended in an exhausted receiver, but "could not perceive any motion distinguishable from the effects of heat." "Perhaps," he concluded, " sensible heat and light may not be caused by the influx or rectilineal projections of fine particles, but by the vibrations made in the universally diffused caloric or matter of heat, or fluid of light." Thus Bennet, and after him Young, regarded the non-appearance of light-repulsion in this experiment as an argument in favour of the undulatory system of light. "For," wrote Young, "granting the utmost imaginable subtility of the corpuscles of light, their effects might naturally be expected to bear some proportion to the effects of the much less rapid notions of the electrical fluid, which are so very easily perceptible, even in their weakest states."
This attitude is all the more remarkable, because Euler many years before had expressed the opinion that light-pressure might be expected just as reasonably on the undulatory as on the corpuscular hypothesis. "Just as," he wrote, "a vehement sound excites not only a vibratory motion in the particles of the air, but there is also observed a real movement of the small particles of dust which are suspended therein, it is not to be doubted but that the vibratory motion set up by the light causes a similar effect." Euler not only inferred the existence of light-pressure, but even (adopting a suggestion of Kepler's) accounted for the tails of comets by supposing that the solar rays, impinging on the atmosphere of a comet, drive off from it the more subtle of its particles.
​The question was examined by Maxwell from the point of view of the electromagnetic theory of light; which readily furnishes reasons for the existence of light-pressure. For suppose that light falls on a metallic reflecting surface at perpendicular incidence. The light may be regarded as constituted of a rapidly-alternating magnetic field, and this must induce electric currents in the surface layers of the metal. But. a metal carrying currents in a magnetic field is acted on by a ponderomotive force, which is at right angles to both the magnetic force and the direction of the current, and is therefore, in the present case, normal to the reflecting surface: this ponderomotive force is the light-pressure. Thus, according to Maxwell's theory, light-pressure is only an extended case of effects which may readily be produced in the laboratory.
The magnitude of the light-pressure was deduced by Maxwell from his theory of stresses in the medium. We have seen that the stress across a plane whose unit-normal is N is represented by the vector
$(\mathbf {D.N} ).\mathbf {E} -{\frac {1}{2}}(\mathbf {D.E} ).\mathbf {N} +{\frac {1}{4\pi }}(\mathbf {B.N} ).\mathbf {H} -{\frac {1}{8\pi }}(\mathbf {B.H} ).\mathbf {N}$.
Now, suppose that a plane wave is incident perpendicularly on a perfectly reflecting metallic sheet: this sheet must support the mechanical stress which exists at its boundary in the aether. Owing to the presence of the reflected wave, D is zero at the surface; and B is perpendicular to N, so (B.N) vanishes. Thus the stress is a pressure of magnitude (1/8π) (B.H) normal to the surface: that is, the light-pressure is equal to the density of the aethereal energy in the region immediately outside the metal. This was Maxwell's result.
This conclusion has been reached on the assumption that the light is incident normally to the reflecting surface. If, on the other hand, the surface is placed in an enclosure completely surrounded by a radiating shell, so that radiation falls on it from all directions, it may be shown that the light-pressure is measured by one-third of the density of aethereal energy.
​A different way of inferring the necessity for light-pressure was indicated in 1876 by A. Bartoli, who showed that, when radiant energy is transported from a cold body to a hot one by means of a moving mirror, the second law of thermodynamics would be violated unless a pressure were exerted on the mirror by the light.
The thermodynamical ideas introduced into the subject by Bartoli have proved very fruitful. If a hollow vessel be at a definite temperature, the aether within the vessel must be full of radiation crossing from one side to the other: and hence the aether, when in radiative equilibrium with matter at a given temperature, is the seat of a definite quantity of energy per unit volume.
If U denote this energy per unit volume, and P the light-pressure on unit area of a surface exposed to the radiation, we may apply the equation of available energy
$U=T{\frac {dP}{dT}}-P$.
Since, as we have seen, $P={\frac {1}{3}}U$, this equation gives $4U=T{\frac {dll}{dT}}$, and therefore U must be proportional to T$^4$. From this it may be inferred that the intensity of emission of radiant energy by a body at temperature T is proportional to the fourth power of the absolute temperature—a law which was first discovered experimentally by Stefan in 1879.
In the year in which Maxwell's treatise was published, Sir William Crookes obtained experimental evidence of a pressure accompanying the incidence of light; but this was ​soon found to be due to thermal effects; and the existence of a true light-pressure was not confirmed experimentally until 1899. Since then the subject has been considerably developed, especially in regard to the part played by the pressure of radiation in cosmical physics.
Another matter which received attention in Maxwell's Treatise was the influence of a magnetic field on the propagation of light in material substances. We have already seen that the theory of magnetic vortices had its origin in Thomson's speculations on this phenomenon; and Maxwell in his memoir of 1861–2 had attempted by the help of that theory to arrive at some explanation of it. The more complete investigation which is given in the Treatise is based on the same general assumptions, namely, that in a medium subjected to a magnetic field there exist concealed vortical motions, the axes of the vortices being in the direction of the lines of magnetic force; and that waves of light passing through the medium disturb the vortices, which thereupon react dynamically on the luminous motion, and so affect its velocity of propagation.
The manner of this dynamical interaction must now be more closely examined. Maxwell supposed that the magnetic vortices are affected by the light-waves in the same way as vortex-filaments in a liquid would be affected by any other coexisting motion in the liquid. The latter problem had been already discussed in Helmholtz's great memoir on vortex-motion; adopting Helmholtz's results, Maxwell assumed for the additional term introduced into the magnetic force by the displacement of the vortices the value ∂e/∂θ, where e denotes the displacement of the medium (i.e. the light vector), and the operator ∂/∂θ denotes H$_x$∂/∂x + H$_y$∂/∂y + H$_z$∂/∂z, H denoting the imposed magnetic field. Thus the luminous motion, by disturbing the vortices, gives rise to an electric current in the medium, proportional to curl ∂e/∂θ.
​Maxwell further assumed that the current thus produced interacts dynamically with the luminous motion in such a manner that the kinetic energy of the medium contains a term proportional to the scalar product of ė and curl ∂e/∂θ. The total kinetic energy of the medium may therefore be written
${\tfrac {1}{2}}\rho \mathbf {\overset {.}{e}} ^{2}+{\tfrac {1}{2}}(\mathbf {{\overset {.}{e}}.} \mathrm {curl} \ \partial \mathbf {e} /\partial \theta )$,
where p denotes the density of the medium, and o denotes a constant which measures the capacity of the medium to rotate the plane of polarization of light in a magnetic field.
The equation of motion may now be derived as in the elastic-solid theories of light: it is
$\rho \mathbf {\overset {..}{e}} +n\nabla ^{2}e-\sigma {\frac {\partial ^{2}}{\partial t\partial \theta }}\mathrm {curl} \ \mathbf {e}$.
When the light is transmitted in the direction of the lines of force, and the axis of x is taken parallel to this direction, the equation reduces to
${\begin{cases}\rho {\frac {\partial ^{2}e_{y}}{\partial t^{2}}}=n{\frac {\partial ^{2}e_{y}}{\partial x^{2}}}+\sigma \mathbf {H} {\frac {\partial ^{3}e_{z}}{\partial t\partial x^{2}}},\\\rho {\frac {\partial ^{2}e_{z}}{\partial t^{2}}}=n{\frac {\partial ^{2}e_{z}}{\partial x^{2}}}-\sigma \mathbf {H} {\frac {\partial ^{3}e_{y}}{\partial t\partial x^{2}}};\end{cases}}$,
and these equations, as we have seen, furnish an explanation of Faraday's phenomenon.
It may be remarked that the term
${\tfrac {1}{2}}\sigma (\mathbf {{\overset {.}{e}}.} \mathrm {curl} \ \partial \mathbf {e} /\partial \theta )$
in the kinetic energy may by partial integration be transformed into a term
${\tfrac {1}{2}}\sigma (\mathrm {curl} \ \mathbf {{\overset {.}{e}}.} \partial \mathbf {e} /\partial \theta )$,
together with surface-terms; or, again, into
$-{\tfrac {1}{2}}\sigma (\mathrm {curl} \ \mathbf {e.} \partial \mathbf {\overset {.}{e}} /\partial \theta )$,
together with surface-terms. These different forms all yield ​the same equation of motion for the medium; but, owing to the differences in the surface-terms, they yield different conditions at the boundary of the medium, and consequently give rise to different theories of reflexion.
The assumptions involved in Maxwell's treatment of the magnetic rotation of light were such as might scarcely be justified in themselves; but since the discussion as a whole proceeded from sound dynamical principles, and its conclusions were in harmony with experimental results, it was fitted to lead to tho more perfect explanations which were afterwards devised by his successors. At the time of Maxwell's death, which happened in 1879, before he had completed his fortyninth year, much yet remained to be done both in this and in the other investigations with which his name is associated; and the energies of the next generation were largely spent in extending and refining that conception of electrical and optical phenomena whose origin is correctly indicated in its name of Maxwell's Theory.
* ↑ Gauss' Werke, y, p. 629.
* ↑ Riemann's Werke, 2e Aufl., p. 526.
* ↑ Ann. d. Phys. (xxxi (1867), p. 237; Riemann's Werke, 2e Aufl., p. 288; Phil. Mag. xxxiv (1867), p. 368.
* ↑ It had been presented to the Göttingen Academy in 1868, but afterwards withdrawn.
* ↑ As will appear from the present chapter, Maxwell had the same power in a very marked degree. It has always been cultivated by the "Cambridge school" of natural philosophers.
* ↑ Camb. Math. Journal, iii (Feb. 1842). p. 71; reprinted in Thomson's Papers on Electrostatics and Magnetism, p. 1. Also Camb. and Dub. Math. Journal, Nov., 1845; reprinted in Papers, p. 15.
* ↑ As regards this comparison, Thomson had been anticipated by Chasles, Journal de I'Éc. Polyt. xv (1837), p. 266, who had shown that attraction accord. ing to Newton's law gives rise to the same fields as the steady conduction of heat, both depending on Laplace's equation ∇$^2$V = 0. It will be remembered that Ohm had used an analogy between thermal conduction and galvanic phenomena.
* ↑ Camb. and Dub. Math. Journ. ii (1847), p. 61: Thomson's Math, and Phys. Papers, i, p. 76.
* ↑ Trans, Camb. Phil, Soe, x, p. 27; Maxwell's Scientific Papers, i, p. 155.
* ↑ Exp. Res., § 3269 (1852).
* ↑ Cf. p. 224.
* ↑ These operators had, however, occurred frequently in the writings of Stokes, especially in his memoir of 1849 on the Dynamical Theory of Diffraction.
* ↑ Proc. Roy. Soc. viii (1856), p. 150; xi (1861), p. 827, foot-note: Phil. Mag. xiii (1867), p. 198; Baltimore Lectures, Appendix F.
* ↑ This was written shortly before the kinetic theory of gases was developed by Clausius and Maxwell.
* ↑ Journal für Math. lv (1858), p. 25; Helmholtz's Wiss. Abh, i, p. 101; translated Phil. Mag. xxxiii (1867), p. 485.
* ↑ Maxwell's Scientific Papers, i, p. 188.
* ↑ Bence Jones's Life of Faraday ii, p. 379.
* ↑ Phil. Mag, axi (1861), pp. 161, 281, 338; xxiii (1862), pp. 12, 85; Maxwell's Scientific Papers, i, p. 451.
* ↑ Cf. pp. 248, 250.
* ↑ Ci, p. 100.
* ↑ Cf. p. 210.
* ↑ For if a denote any vector, we have identically $\nabla ^{2}\mathbf {a} +\mathrm {grad\ div} \ \mathbf {a} +\mathrm {curl\ curl} \ \mathbf {a} =0$
* ↑ For criticisms on the procedure by which Maxwell determined the velocity of propagation of disturbance, cf. P. Duhem, Les Théories Electriques de J. Clerk Maxwell, Paris, 1902.
* ↑ Cf. pp. 227, 259.
* ↑ Cf. p. 260.
* ↑ Comptes Rendis, xxix (1849), p. 90. A determination made by Cornu in 1874 was on this principle.
* ↑ A different experimental method was employed in 1862 by Léon Foucault (Comptes Rendus, lv, pp. 501, 792); in this a ray from an origin O was reflected by a revolving mirror M to a fixed mirror, and so reflected back to M, and again to O. It is evident that the returning ray MO 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 2·98 * 10$^{10}$ cm./sec. Subsequent determinations by Michelson in 1879 (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. 682), the value obtained for the velocity of light by the methods of Fizesu 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 nondispersive 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.
* ↑ Priestley's History, p. 488.
* ↑ Faraday's laboratory note-book for 1857: ef. Bence Jones's Life of Faraday, ii, p. 380.
* ↑ Phil. Trans. clv (1865), p. 459: Maxwell's Scient. Papers, i, p. 526
* ↑ This is the effect of the introduction of (F′, G′, H′) in § 98 of the memoir; cf. also Maxwell's Treatise on Electricity and Magnetism, § 616.
* ↑ Phil. Trans. clviii (1868), p. 643: Maxwell's Scient. Papers, ii, p. 125.
* ↑ 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.
* ↑ Helmholtz, Journ. für Muth., lxxviii (1874), p. 309; H. W. Watson, Phil. Mag. (6), XIV (1888), p. 271.
* ↑ Cf. pp. 164 et sqq.
* ↑ 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 capacity is (for optical purposes) the same in all directions, while the magnetic permeability is aeolotropic. In the latter 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.
* ↑ Cf. p. 180.
* ↑ Cf. p. 181. Cf. also Rayleigh, Phil. Mag. (5) xii (1881), p. 81, and H. A. Lorentz, Over de Theorie de Terugkaatsing, Arnhem, 1870.
* ↑ Cf. p. 181.
* ↑ Cf. p. 182.
* ↑ Comptes Rendus, ly (1862), p. 126. In 1870 C. Christiansen (Ann. d. Phys. exli, p. 479; cxliii, p. 250) observed a similar effect in a solution of fuchsin.
* ↑ Especially by Kundt, in a series of papers in the analen d. Phys., from vol. cxlii (1871) onwards.
* ↑ Phil. 'Trans., 1852, p. 463. Stokes's Coll. Papers, iii., p. 267.
* ↑ Cambridge: Calendar, 1869; republished by Lord Rayleigh, Phil. Mag. xlviii (1899), p. 151.
* ↑ This illustration is due to W. Thomson.
* ↑ Ann. d. Phys. cxlv (1872), pp. 399, 520: cxlvii (1872), pp. 385, 525. Cf. also Helmholtz, Ann. d. Phys. cliv (1875), p. 582.
* ↑ Cf. p. 183.
* ↑ This subject was developed by Lord Kelvin in the Baltimore Lectures.
* ↑ Ann. d. Phys. Is (1897), p. 454.
* ↑ Rubens and Aschkinass, Ann. d. Phys. lxiv (1898).
* ↑ Stokes's Scientific Correspondence, ii, pp. 25, 26.
* ↑ It must be remembered that Maxwell pictured the electric displacement as a real displacement of a medium. "My theory of electrical forces," he wrote, "is that they are called into play in insulating media by slight electric displacements, which put certain small portions of the medium into a state of distortion, which, being resisted by the elasticity of the medium, produces an electromotive force." Campbell and Garnett's Life of Maxwell, p. 244.
* ↑ The letter to Stokes already mentioned appears to indicate that Maxwell for a time doubted the correctness of Green's conditions.
* ↑ Nature, xxxviii (1888) p. 571.
* ↑ Brit. Assoc. Report, 1883, p. 567.
* ↑ Cf. Bottomley, in Nature, liii (1896), p. 268: Kelvin, ib., p. 316; J. Willard Gibbs, ib., p. 509.
* ↑ Baltimore Lectures (ed. 1904), p. 376.
* ↑ Ibid., preface, p. 7.
* ↑ Oversigt over det K. danske vid. Selskaps Forhandlinger, 1867, p. 26; Annal. der Phys, cxxxi (1867), p. 243: Phil. Mag., xxxiv (1867), p. 287.
* ↑ Cf, p. 268. Riemann's memoir was, however, published only in the same year (1867) as Lorenz's.
* ↑ Cf. p. 209.
* ↑ Maxwell's Treatise in Electricity and Magnetism, § 643.
* ↑ Cf. V. Bjerkues, Phil. Mag. ix (1905), p. 491.
* ↑ Histoire de l'Acad., 1708, p. 21.
* ↑ J. J. de Mairan, Traité de l' Aurore boréale, p. 370.
* ↑ History of Vision, i, p. 387.
* ↑ Phil. Trans., 1792, p. 81.
* ↑ Ibid., 1802, p. 46.
* ↑ Histoire de l'Aoitd. de Berlin, ii (1748), p. 117.
* ↑ Maxwell's Treatise on Electricity and magnetism, § 792.
* ↑ Bartoli, Sopra i morimenti prodotti dalla luce e dal calore e sopra il radiometro di Crookes. Firenze, 1876. Also Nuovo Cimento (3) xv (1884), p. 193; and Exner's Rep, xxi (1885), p. 198.
* ↑ Boltzmann, Ann. d. Phys. xxii (1884), p. 31. Cf. also B. Galitzine, Ann, d. Phys. xlvii (1892), p. 479.
* ↑ Cf. p. 240.
* ↑ Wien, Ber. lxxix (1879), p. 391.
* ↑ Phil. Truns. clxiv (1874), p. 501. The radiometer was discovered in 1875.
* ↑ P. Lebedew, Archives des Sciences Phys. et Nat. (4) viii (1899), p. 184. Ann. d. Phys. vi (1901), p. 433. E. F. Nichols and G. F. Hull, Phys. Rev. xiii (1901), p. 293; Astrophys. Jour., xvii (1903), p. 315.
* ↑ Cf. p. 274.
* ↑ Cf. p. 215.
* ↑ This form was suggested by FitzGerald six years later, Phil. Trans., 1880, p. 691: FitzGerald's Scientific Writings, p. 45.