In studying this part of Maxwell’s work, it must clearly be remembered that he did not look upon the ether as a series of cog-wheels with idle wheels between, or anything of the kind. He devised a mechanical model of such cogs and idle wheels, the properties of which would in some respects closely resemble those of the ether; from this model he deduced, among other things, the important fact that electric waves would travel outwards with the velocity of light. Other such models have been devised since his time to illustrate the same laws. Prof. Fitzgerald has actually constructed one of wheels connected together by elastic bands, which shows clearly the kind of processes which Maxwell supposed to go on in a dielectric when under electric force. Professor Lodge, in his book, “Modern Views of Electricity,” has very fully developed a somewhat different arrangement of cog-wheels to attain the same result.

Maxwell’s predictions as to the propagation of electric waves have in recent days received their full verification in the brilliant experiments of Hertz and his followers; it remains for us, before dealing with these, to trace their final development in his hands.

The papers we have been discussing were perhaps too material to receive the full attention they deserved; the ether is not a series of cogs, and electricity is something different from material idle wheels. In his paper on “The Dynamical Theory of the Electro-magnetic Field,” Phil. Trans., 1864, Maxwell treats the same questions in a more general manner. On a former occasion he says, “I have attempted to describe a particular kind of motion and a particular kind of strain so arranged as to account for the phenomena. In the present paper I avoid any hypothesis of this kind; and in using such words as electric momentum and electric elasticity in reference to the known phenomena of the induction of currents and the polarisation of dielectrics, I wish merely to direct the mind of the reader to mechanical phenomena, which will assist him in understanding the electrical ones. All such phrases in the present paper are to be considered as illustrative and not as explanatory.” He then continues:—

In his introduction to the paper, he discusses in a general way the various explanations of electric phenomena which had been given, and points out that—

These general laws of dynamics, applicable to the motion of any connected system, had been developed by Lagrange, and are expressed in his generalised equations of motion. It is one of Maxwell’s chief claims to fame that he saw in the electric field a connected system to which Lagrange’s equations could be applied, and that he was able to deduce the mechanical and electrical actions which take place by means of fundamental propositions of dynamics.

The methods of the paper now under discussion were developed further in the “Treatise on Electricity and Magnetism,” published in 1873; in endeavouring to give some slight account of Maxwell’s work, we shall describe it in the form it ultimately took.

The task which Maxwell set himself was a double one; he had first to express in symbols, in as general a form as possible, the fundamental laws of electro-magnetism as deduced from experiments, chiefly the experiments of Faraday, and the relations between the various quantities involved; when this was done he had to show how these laws could be deduced from the general dynamical laws applicable to any system of moving bodies.

There are two classes of phenomena, electric and magnetic, which have been known from very early times, and which are connected together. When a piece of sealing-wax is rubbed it is found to attract other bodies, it is said to exert electric force throughout the space surrounding it; when two different metals are dipped in slightly acidulated water and connected by a wire, certain changes take place in the plates, the water, the wire, and the space round the wire, electric force is again exerted and a current of electricity is said to flow in the wire. Again, certain bodies, such as the lodestone, or pieces of iron and steel which have been treated in a certain manner, exhibit phenomena of action at a distance: they are said to exert magnetic force, and it is found that this magnetic force exists in the neighbourhood of an electric current and is connected with the current.

Again, when electric force is applied to a body, the effects may be in part electrical, in part mechanical; the electrical state of the body is in general changed, while in addition, mechanical forces tending to move the body are set up. Experiment must teach us how the electrical state depends on the electric force, and what is the connection between this electric force and the magnetic forces which may, under certain circumstances, be observed. Now, in specifying the electric and magnetic conditions of the system, various other quantities, in addition to the electric force, will have to be introduced; the first step is to formulate the necessary quantities, and to determine the relations between them and the electric force.

Consider now a wire connecting the two poles of an electric battery—in its simplest form, a piece of zinc and a piece of copper in a vessel of dilute acid—electric force is produced at each point of the wire. Let us suppose this force known; an electric current depending on the material and the size of the wire flows along it, its value can be determined at each point of the wire in terms of the electric force by Ohm’s law. If we take either this current or the electric force as known, we can determine by known laws the electric and magnetic conditions elsewhere. If we suppose the wire to be straight and very long, then, so long as the current is steady and we neglect the small effect due to the electrostatic charge on the wire, there is no electric force outside the wire. There is, however, magnetic force, and it is found that the lines of magnetic force are circles round the wire. It is found also that the work done in travelling once completely round the wire against the magnetic force is measured by the current flowing through the wire, and is obtained in the system of units usually adopted by multiplying the current by 4π. This last result then gives us one of the necessary relations, that between the magnetic force due to a current and the strength of the current.

Again, consider a steady current flowing in a conductor of any form or shape, the total flow of current across any section of the conductor can be measured in various ways, and it is found that at any time this total flow is the same for each section of the conductor. In this respect the flow of a current resembles that of an incompressible fluid through a pipe; where the pipe is narrow the velocity of flow is greater than it is where the pipe is broad, but the total quantity crossing each section at any given instant is the same.

Consider now two conducting bodies, two spheres, or two flat plates placed near together but insulated. Let each conductor be connected to one of the poles of the battery by a conducting wire. Then, for a very short interval after the contact is made, it is found that there is a current in each wire which rapidly dies away to zero. In the neighbourhood of the balls there is electric force; the balls are said to be charged with electricity, and the lines of force are curved lines running from one ball to the other. It is found that the balls slightly attract each other, and the space between them is now in a different condition from what it was before the balls were charged. According to Maxwell, Electric Displacement has been produced in this space, and the electric displacement at each point is proportional to the electric force at that point.

Thus, (i) when electric force acts on a conductor, it produces a current, the current being by Ohm’s law proportional to the force: (ii) when it acts on an insulator it produces electric displacement, and the displacement is proportional to the force; while (iii) there is magnetic force in the neighbourhood of the current, and the work done in carrying a magnetic pole round any complete circuit linked with the current is proportional to the current. The first two of these principles give us two sets of equations connecting together the electric force and the current in a conductor or the displacement in a dielectric respectively; the third connects the magnetic force and the current.

Now let us go back to the variable period when the current is flowing in the wires; and to make ideas precise, let the two conductors be two equal large flat plates placed with their faces parallel, and at some small distance apart. In this case, when the plates are charged, and the current has ceased, the electric displacement and the force are confined almost entirely to the space between the plates. During the variable period the total flow at any instant across each section of the wire is the same, but in the ordinary sense of the word there is no flow of electricity across the insulating medium between the plates. In this space, however, the electric displacement is continuously changing, rising from zero initially to its final steady value when the current ceases. It is a fundamental part of Maxwell’s theory that this variation of electric displacement is equivalent in all respects to a current. The current at any point in a dielectric is measured by the rate of change of displacement at that point.

Moreover, it is also an essential point that if we consider any section of the dielectric between the two plates, the rate of change of the total displacement across this section is at each moment equal to the total flow of current across each section of the conducting wire.

Currents of electricity, therefore, including displacement currents, always flow in closed circuits, and obey the laws of an incompressible fluid in that the total flow across each section of the circuit—conducting or dielectric—is at any moment the same.

It should be clearly remembered that this fundamental hypothesis of Maxwell’s theory is an assumption only to be justified by experiment. Von Helmholtz, in his paper on “The Equations of Motion of Electricity for Bodies at Rest,” formed his equations in an entirely different manner from Maxwell, and arrived at results of a more general character, which do not require us to suppose that currents flow always in closed circuits, but permit of the condensation of electricity at points in the circuit where the conductors end and the non-conducting part of the circuit begins. We leave for the present the question which of the two theories, if either, represents the facts.

We have obtained above three fundamental relations—(i) that between electric force and electric current in a conductor; (ii) that between electric force and electric displacement in a dielectric; (iii) that between magnetic force and the current which gives rise to it. And we have seen that an electric current—i.e. in a dielectric the variation of the strength of an electric field of force—gives rise to magnetic force. Now, magnetic force acting on a medium produces “magnetic displacement,” or magnetic induction, as it is called. In all media except iron, nickel, cobalt, and a few other substances, the magnetic induction is proportional to the magnetic force, and the ratio between the magnetic induction produced by a given force and the force is found to be very nearly the same for all such media. This ratio is known as the permeability, and is generally denoted by the symbol μ.

A relation reciprocal to that given in (iii) above might be anticipated, and was, in fact, discovered by Faraday. Changes in a field of magnetic induction give rise to electric force, and hence to displacement currents in a dielectric or to conduction currents in a conductor. In considering the relation between these changes and the electric force, it is simplest at first not to deal with magnetic matter such as iron, nickel, or cobalt; and then we may say that (iv) the work which at any instant would be done in carrying a unit quantity of electricity round a closed circuit in a magnetic field against the electric forces due to the field is equal to the rate at which the total magnetic induction which threads the circuit is being decreased. This law, summing up Faraday’s experiments on electro-magnetic induction, gives a fourth principle, leading to a fourth series of equations connecting together the electric and magnetic quantities involved.

The equations deduced from the above four principles, together with the condition implied in the continuity of an electric current, constitute Maxwell’s equations of the electro-magnetic field.

If we are dealing only with a dielectric medium, the reciprocal relation between the third and fourth principle may be made more clear by the following statement:—

(A) The work done at any moment in carrying a unit quantity of magnetism round a closed circuit in a field in which electric displacement is varying, is equal to the rate of change of the total electric displacement through the circuit multiplied by 4 π.62

(B) The work done at any moment in carrying a unit quantity of electricity round a circuit in a field in which the magnetic induction is varying, is equal to the rate of change of the total magnetic induction through the circuit.

From these two principles, combined with the laws connecting electric force and displacement, magnetic force and induction, and with the condition of continuity, Maxwell obtained his equations of the field.

Faraday’s experiments on electro-magnetic induction afford the proof of the truth of the fourth principle. It follows from those experiments that when the number of lines of magnetic induction which are linked with any closed circuit are made to vary, an induced electromotive force is brought into play round that circuit. This electromotive force is, according to Faraday’s results, measured by the rate of decrease in the number of lines of magnetic induction which thread the circuit. Maxwell applies this principle to all circuits, whether conducting or not.

In obtaining equations to express in symbols the results of the fourth principle just enunciated, Maxwell introduces a new quantity, to which he gives the name of the “vector potential.” This quantity appears in his analysis, and its physical meaning is not at first quite clear. Professor Poynting has, however, put Maxwell’s principles in a slightly different form, which enables us to see definitely the meaning of the vector potential, and to deduce Maxwell’s equations more readily from the fundamental statements.

We are dealing with a circuit with which lines of magnetic induction are linked, while the number of such lines linked with the circuit is varying. Now, let us suppose the variation to take place in consequence of the lines of induction moving outwards or inwards, as the case may be, so as to cut the circuit. Originally there are none linked with the circuit. As the magnetic field has grown to its present strength lines of magnetic induction have moved inwards. Each little element of the circuit has been cut by some, and the total number linked with the circuit can be found by adding together those cut by each element. Now, Professor Poynting’s statement of Maxwell’s fourth principle is that the electrical force in the direction of any element of the circuit is found by dividing by the length of the element the number of lines of magnetic induction which are cut in one second by it.

Moreover, the total number of lines of magnetic induction which have been cut by an element of unit length is defined as the component of the vector potential in the direction of the element; hence the electrical force in any direction is the rate of decrease of the component of the vector potential in that direction. We have thus a physical meaning for the vector potential, and shall find that in the dynamical theory this quantity is of great importance.

Professor Poynting has modified Maxwell’s third principle in a similar manner; he looks upon the variation in the electric displacement as due to the motion of tubes of electric induction,63 and the magnetic force along any circuit is equal to the number of tubes of electric induction cutting or cut by unit length of the circuit per second, multiplied by 4π.

From the equations of the field, as found by Maxwell, it is possible to derive two sets of symmetrical equations. The one set connects the rate of change of the electric force with quantities depending on the magnetic force; the other set connects in a similar manner the rate of change of the magnetic force with quantities depending on the electric force. Several writers in recent years adopt these equations as the fundamental relations of the field, establishing them by the argument that they lead to consequences which are found to be in accordance with experiment.

We have endeavoured to give some account of Maxwell’s historical method, according to which the equations are deduced from the laws of electric currents and of electro-magnetic induction derived directly from experiment.

While the manner in which Maxwell obtained his equations is all his own, he was not alone in stating and discussing general equations of the electro-magnetic field. The next steps which we are about to consider are, however, in a special manner due to him. An electrical or magnetic system is the seat of energy; this energy is partly electrical, partly magnetic, and various expressions can be found for it. In Maxwell’s theory it is a fundamental assumption that energy has position. “The electric and magnetic energies of any electro-magnetic system,” says Professor Poynting, “reside, therefore, somewhere in the field.” It follows from this that they are present wherever electric and magnetic force can be shown to exist. Maxwell showed that all the electric energy is accounted for by supposing that in the neighbourhood of a point at which the electric force is R there is an amount of energy per unit of volume equal to KR²/8π, K being the inductive capacity of the medium, while in the neighbourhood of a point at which the magnetic force is H, the magnetic energy per unit of volume is μH²/8π, μ being the permeability. He supposes, then, that at each point of an electro-magnetic system energy is stored according to these laws. It follows, then, that the electro-magnetic field resembles a dynamical system in which energy is stored. Can we discover more of the mechanism by which the actions in the field are maintained? Now the motion of any point of a connected system depends on that of other points of the system; there are generally, in any machine, a certain number of points called driving-points, the motion of which controls the motion of all other parts of the machine; if the motion of the driving-points be known, that of any other point can be determined. Thus in a steam engine the motion of a point on the fly-wheel can be found if the motion of the piston and the connections between the piston and the wheel be known.

In order to determine the force which is acting on any part of the machine we must find its momentum, and then calculate the rate at which this momentum is being changed. This rate of change will give us the force. The method of calculation which it is necessary to employ was first given by Lagrange, and afterwards developed, with some modifications, by Hamilton. It is usually referred to as Hamilton’s principle; when the equations in the original form are used they are known as Lagrange’s equations.

Now Maxwell showed how these methods of calculation could be applied to the electro-magnetic field. The energy of a dynamical system is partly kinetic, partly potential. Maxwell supposes that the magnetic energy of the field is kinetic energy, the electric energy potential. When the kinetic energy of a system is known, the momentum of any part of the system can be calculated by recognised processes. Thus if we consider a circuit in an electro-magnetic field we can calculate the energy of the field, and hence obtain the momentum corresponding to this circuit. If we deal with a simple case in which the conducting circuits are fixed in position, and only the current in each circuit is allowed to vary, the rate of change of momentum corresponding to any circuit will give the force in that circuit. The momentum in question is electric momentum, and the force is electric force. Now we have already seen that the electric force at any point of a conducting circuit is given by the rate of change of the vector potential in the direction considered. Hence we are led to identify the vector potential with the electric momentum of our dynamical system; and, referring to the original definition of vector potential, we see that the electric momentum of a circuit is measured by the number of lines of magnetic induction which are interlinked with it.

Again, the kinetic energy of a dynamical system can be expressed in terms of the squares and products of the velocities of its several parts. It can also be expressed by multiplying the velocity of each driving-point by the momentum corresponding to that driving-point, and taking half the sum of the products. Suppose, now, we are dealing with a system consisting of a number of wire circuits in which currents are running, and let us suppose that we may represent the current in each wire as the velocity of a driving-point in our dynamical system. We can also express in terms of these currents the electric momentum of each wire circuit; let this be done, and let half the sum of the products of the corresponding velocities and momenta be formed.

In maintaining the currents in the wires energy is needed to supply the heat which is produced in each wire; but in starting the currents it is found that more energy is needed than is requisite for the supply of this heat. This excess of energy can be calculated, and when the calculation is made it is found that the excess is equal to half the sum of the products of the currents and corresponding momenta. Moreover, if this sum be expressed in terms of the magnetic force, it is found to be equal to μ H²/8 π, which is the magnetic energy of the field. Now, when a dynamical system is set in motion against known forces, more energy is supplied than is needed to do the work against the forces; this excess of energy measures the kinetic energy acquired by the system.

Hence, Maxwell was justified in taking the magnetic energy of the field as the kinetic energy of the mechanical system, and if the strengths of the currents in the wires be taken to represent the velocities of the driving-points, this energy is measured in terms of the electrical velocities and momenta in exactly the same way as the energy of a mechanical system is measured in terms of the velocities and momenta of its driving-points.

The mechanical system in which, according to Maxwell, the energy is stored is the ether. A state of motion or of strain is set up in the ether of the field. The electric forces which drive the currents, and also the mechanical forces acting on the conductors carrying the currents, are due to this state of motion, or it may be of strain, in the ether. It must not be supposed that the term electric displacement in Maxwell’s mind meant an actual bodily displacement of the particles of the ether; it is in some way connected with such a material displacement. In his view, without motion of the ether particles there would be no electric action, but he does not identify electric displacement and the displacement of an ether particle.

His mechanical theory, however, does account for the electro-magnetic forces between conductors carrying currents. The energy of the system depends on the relative positions of the currents which form part of it. Now, any conservative mechanical system tends to set itself in such a position that its potential energy is least, its kinetic energy greatest. The circuits of the system, then, will tend to set themselves so that the electro-kinetic energy of the system may be as large as possible; forces will be needed to hold them in any position in which this condition is not satisfied.

We have another proof of the correctness of the value found for the energy of the field in that the forces calculated from this value agree with those which are determined by direct experiment.

Again, the forces applied at the various driving-points are transmitted to other points by the connections of the machine; the connections are thrown into a state of strain; stress exists throughout their substance. When we see the piston-rod and the shaft of an engine connected by the crank and the connecting-rod, we recognise that the work done on the piston is transmitted thus to the shaft. So, too, in the electro-magnetic field, the ether forms the connection between the various circuits in the field; the forces with which those circuits act on each other are transmitted from one circuit to another by the stresses set up in the ether.

To take another instance, consider the electrostatic attraction between two charged bodies. Let us suppose the bodies charged by connecting each to the opposite pole of a battery; a current flows from the battery setting up electric displacement in the space between the bodies, and throwing the ether into a state of strain. As the strain increases the current gets less; the reaction resulting from the strain tends to stop it, until at last this reaction is so great that the current is stopped. When this is the case the wires to the battery may be removed, provided this is done without destroying the insulation of the bodies; the state of strain will remain and shows itself in the attraction between the balls.

Looking at the problem in this manner, we are face to face with two great questions—the one, What is the state of strain in the ether which will enable it to produce the observed electrostatic attractions and repulsions between charged bodies? and the other, What is the mechanical structure of the ether which would give rise to such a state of strain as will account for the observed forces? Maxwell gives one answer to the first question; it is not the only answer which could be given, but it does account for the facts. He failed to answer the second. He says (“Electricity and Magnetism,” vol. i. p. 132):—

Faraday had pointed out that the inductive action between two bodies takes place along the lines of force, which tend to shorten along their length and to spread outwards in other directions. Maxwell compares them to the fibres of a muscle, which contracts and at the same time thickens when exerting force. In the electric field there is, on Maxwell’s theory, a tension along the lines of electric force and a pressure at right angles to those lines. Maxwell proved that a tension K R²/8 π along the lines of force, combined with an equal pressure in perpendicular directions, would maintain the equilibrium of the field, and would give rise to the observed attractions or repulsions between electrified bodies. Other distributions of stress might be found which would lead to the same result. The one just stated will always be connected with Maxwell’s name. It will be noticed that the tension along the lines of force and the pressure at right angles to them are each numerically equal to the potential energy stored per unit of volume in the field. The value of each of the three quantities is K R²/8 π.

In the same way, in a magnetic field, there is a state of stress, and on Maxwell’s theory this, too, consists of a tension along the lines of force and an equal pressure at right angles to them, the values of the tension and the pressure being each equal to that of the magnetic energy per unit of volume, or μH²/8π.

In a case in which both electric and magnetic force exists, these two states of stress are superposed. The total energy per unit of volume is KR²/8π + μH²/8π; the total stress is made up of tensions KR²/8π and μH²/8π along the lines of electric and magnetic force respectively, and equal pressures at right angles to these lines.

We see, then, from Maxwell’s theory, that electric force produced at any given point in space is transmitted from that point by the action of the ether. The question suggests itself, Does the transmission take time, and if so, does it proceed with a definite velocity depending on the nature of the medium through which the change is proceeding?

According to the molecular-vortex theory, we have seen that waves of electric force are transmitted with a definite velocity. The more general theory developed in the “Electricity and Magnetism” leads to the same result. Electric force produced at any point travels outwards from that point with a velocity given by 1/√(Kμ). At a distant point the force is zero, until the disturbance reaches it. If the disturbance last only for a limited interval, its effects will at any future time be confined to the space within a spherical shell of constant thickness depending on the interval; the radii of this shell increase with uniform speed 1/√(Kμ).

If the initial disturbance be periodic, periodic waves of electric force will travel out from the centre, just as waves of sound travel out from a bell, or waves of light from a candle flame. A wire carrying an alternating current may be such a source of periodic disturbance, and from the wire waves travel outwards into space.

Now, it is known that in a sound wave the displacements of the air particles take place in the direction in which the wave is travelling; they lie at right angles to the wave front, and are spoken of as longitudinal. In light waves, on the other hand, the displacements are, as Fresnel proved, in the wave front, at right angles, that is, to the direction of propagation; they are transverse.

Theory shows that in general both these waves may exist in an elastic solid body, and that they travel with different velocities. Of which nature are the waves of electric displacement in a dielectric? It can be shewn to follow as a necessary consequence of Maxwell’s views as to the closed character of all electric currents, that waves of electric displacement are transverse. Electric vibrations, like those of light, are in the wave front and at right angles to the direction of propagation; they depend on the rigidity or quasi-rigidity of the medium through which they travel, not on its resistance to compression.

Again, an electric current, whether due to variation of displacement in a dielectric or to conduction in a conductor, is accompanied by magnetic force. A wave of periodic electric displacement, then, will be also a wave of periodic magnetic force travelling at the same rate; and Maxwell shewed that the direction of this magnetic force also lies in the wave front, and is always at right angles to the electric displacement. In the ordinary theory of light the wave of linear displacement is accompanied by a wave of periodic angular twist about a direction lying in the wave front and perpendicular to the linear displacement.

In many respects, then, waves of electric displacement resemble waves of light, and, indeed, as we proceed we shall find closer connections still. Hence comes Maxwell’s electro-magnetic theory of light.

It is only in dielectric media that electric force is propagated by wave motion. In conductors, although the third and fourth of Maxwell’s principles given on page 185 still are true, the relation between the electric force and the electric current differs from that which holds in a dielectric. Hence the equations satisfied by the force are different. The laws of its propagation resemble those of the conduction of heat rather than those of the transmission of light.

Again, light travels with different velocities in different transparent media. The velocity of electric waves, as has been stated, is equal to 1/√(μK); but in making this statement it is assumed that the simple laws which hold where there is no gross matter—or, rather, where air is the only dielectric with which we are concerned—hold also in solid or liquid dielectrics. In a solid or a liquid, as in vacuo, the waves are propagated by the ether. We assume, as a first step towards a complete theory, that so far as the electric waves are concerned the sole effect produced by the matter shews itself in a change of inductive capacity or of permeability. It is not likely that such a supposition should be the whole truth, and we may, therefore, expect results deduced from it to be only approximation to the true result.

Now, electro-magnetic experiments show that, excluding magnetic substances, the permeability of all bodies is very nearly the same, and differs very slightly from that of air. The inductive capacity, however, of different bodies is different, and hence the velocity with which electro-magnetic waves travel differs in different bodies.

But the refraction of waves of light depends on the fact that light travels with different velocities in different media; hence we should expect to have waves of electric displacement reflected and refracted when they pass from one dielectric, such as air, to another, such as glass or gutta-percha; moreover, for light the refractive index of a medium such as glass is the ratio of the velocity in air to the velocity in the glass.

Thus the electrical refractive index of glass is the ratio of the velocity of electric waves in air to their velocity in glass.

Now let K₀ be the inductive capacity of air, K₁ that of glass, taking the permeability of air and glass to be the same, we have the result that—

Electrical refractive index = √(K₁/K₀).

But the ratio of the inductive capacity of glass to that of air is known as the specific inductive capacity of glass.

Hence, the specific inductive capacity of any medium is equal to the square of the electrical refractive index of that medium.

Since Maxwell’s time the mathematical laws of the reflexion and refraction of electric waves have been investigated by various writers, and it has been shewn that they agree exactly with those enunciated by Fresnel for light.

Hitherto we have been discussing the propagation of electric waves in an isotropic medium, one which has identical properties in all directions about a point. Let us now consider how these laws are modified if the dielectric be crystalline in structure.

Maxwell assumes that the crystalline character of the dielectric can be sufficiently represented by supposing the inductive capacity to be different in different directions; experiments have since shewn that this is true for crystals such as Iceland Spar and Aragonite; he assumes also, and this, too, is justified by experiment, that the magnetic permeability does not depend on the direction. It follows from these assumptions that a crystal will produce double refraction and polarisation of electric waves which fall upon it, and, further, that the laws of double refraction will be those given by Fresnel for light waves in a doubly refracting medium. There will be two waves in the crystal. The disturbance in each of these will be plane polarised; their velocity and the position of their plane of polarisation can be found from the direction in which they are travelling by Fresnel’s construction exactly.

Maxwell’s theory, then, would appear to indicate some close connection between electric waves and those of light. Faraday’s experiments on the rotation of the plane of polarisation by magnetic force shew one phenomenon in which the two are connected, and Maxwell endeavoured to apply his theory to explain this. Here, however, it became necessary to introduce an additional hypothesis—there must be some connection between the motion of the ether to which magnetic force is due and that which constitutes light. It is impossible to give a mechanical account of the rotation of the plane of polarisation without some assumption as to the relation between these two kinds of motion. Maxwell, therefore, supposes the linear displacements of a point in the ether to be those which give rise to light, while the components of the magnetic force are connected with these in the same way as the components of a vortex in a liquid in vortex motion are connected with the displacements of the liquid. He further assumes the existence of a term of special form in the expression for the kinetic energy, and from these assumptions he deduces the laws of the propagation of polarised light in a magnetic field. These laws agree in the main with the results of Verdet’s experiments.