These bodies serve the same purpose as “idle wheels” in machinery, which, coming between a driver and follower, transmit the motion of the former to the latter unchanged in direction. These minute spherical particles Maxwell supposed to constitute electricity. They roll upon the cells or vortices as if the surfaces in contact were perfectly rough, or provided with teeth gearing into one another, and thus, whatever forces may be applied, sliding is impossible. What we ordinarily consider as molecules of matter are supposed to be very large compared with the molecular vortices, and therefore à fortiori with the particles of electricity. In an insulator, or dielectric, it is supposed that the electric particles are unable to pass from molecule to molecule of the body, but in a conductor they can do so, the passage, however, being opposed by friction, so that heat is generated and energy dissipated in the transfer.

Now suppose that we have a current of electricity flowing through a conducting wire. Let us confine our attention at first to the central line of particles. These, as they flow, will cause all the cells they touch to rotate about axes perpendicular to the line of flow, so that the stream of particles will be surrounded by rings of vortices. Each ring of vortices will behave like an indiarubber umbrella ring when we pass it over the finger or the stick of an umbrella. Instead of sliding into its place it proceeds by a rolling motion, continually turning itself inside out, as it were, each circular section of the ring or tore rotating about its own centre. Now this motion of the vortices would tend to cause the layer of electric particles outside them to move in the opposite direction to the central stream, and this tendency, to which we shall again refer when we speak of induction, can only be overcome by causing the next ring of cells to rotate in the same direction as the inner ring, when the particles may simply roll round between the coaxial rings of vortices without moving backwards or forwards. But if the layer of particles be compelled to move forwards like the inner stream the layer of vortices surrounding it must rotate more rapidly that the layer within it, and so on, each successive shell of vortices rotating more rapidly until we reach the extreme layer contained within the conducting wire. The shell of vortices which bounds the conductor must by the same mechanism set up molecular vortices in the dielectric, the motion being communicated in ever-widening circles to an unlimited distance. It does not follow that this communication of motion is instantaneous. The cells may consist of elastic material which does not assume its final state of motion as soon as the tangential action of the electric particles is exerted upon it, but begins at first to undergo a deformation, the time taken to set up a given rotation in each depending on its density and elasticity. Hence electro-magnetic induction, which is the name given to the action we are now discussing, will be propagated through space with a finite velocity, but of this we must say more hereafter.

From what has been said it appears that when a steady (i.e. constant) current is flowing in a wire, molecular vortices will be set up in the surrounding dielectric, the axis of rotation of each vortex being perpendicular to the plane passing through the wire and the vortex. The axes about which the vortices turn will therefore form circles surrounding the wire, while the vortices themselves will constitute vortex rings, spinning with very great velocity in the same manner as the indiarubber ring above referred to, or the rings of smoke which are sometimes seen to emerge from a tobacco pipe. But the lines about which the molecular vortices rotate are magnetic lines of force, there being a tension in the medium along these lines, and a pressure everywhere at right angles to them.

Hence a straight line carrying an electric current will be surrounded with magnetic lines of force, forming circles with their centres on the axis of the wire, and since the direction of the magnetic force is that in which a right-handed screw would advance if rotating with the vortices, it follows that the direction of the magnetic force around the wire will be that in which a right-handed screw would rotate if advancing with the current. The medium will be subject to tension in circles around the wire, and to pressure in planes passing through the wire, reminding us of the cylinders of an Armstrong gun.

If the wire he bent the same will be true in kind, but the lines will no longer be accurately circles. All the magnetic lines of force pass through a closed circuit in the direction in which a right-handed screw would advance if rotating in the direction of the current. Fig. 11, taken from the paper in the Philosophical Magazine, shows the relations between the current, the lines of magnetic force, and the direction of motion of the vortices, the arrows normal upper E normal upper E prime representing the current, normal upper S normal upper N indicating the direction of the magnetic force, while the arrows normal upper V normal upper V prime show the direction of rotation of the vortices.

Now suppose a wire conveying a current to be placed in a magnetic field at right angles to the lines of force. Let normal upper S normal upper N (Fig. 12) represent the lines of force, normal upper A the section of the conductor, and let the current be travelling from the reader through the paper. In the space immediately above the wire, the molecular vortices due to the magnetic force originally in the field will be rotating in the direction in which the current in normal upper A urges, them, while in the space below the conductor the reverse will be the case. Hence the velocity of the vortices above the wire will be increased by the current, while that of the vortices below the wire is diminished. The pressure of the medium at right angles to the lines of force will therefore be greater above the wire than below it, and the wire will be urged downwards at right angles to the lines of force and to its own direction.

Again, suppose two parallel wires to be near together and to convey currents in opposite directions. The strength of a current determines the difference between the velocities of the molecular vortices on opposite sides of it, the electric particles being related to the vortices in the same way as a differential wheel in mechanism; but the vortices on one side of a moving stream of electric particles may be brought to rest if the velocity of those on the other side be doubled, the current remaining the same though the electric particles themselves will now have to spin round, but this makes no difference. Hence, when parallel wires convey currents in opposite directions, the vortices between them being made to spin in the same direction by both currents, will rotate faster than those on the opposite sides of the wires, and pressing as they do with force proportional to the squares of their circumferential velocities, the wires will be pushed apart as if they repelled one another.

When two parallel wires convey currents in the same direction, they tend to make the cells in the space between them spin in opposite directions, and the velocities of the molecular vortices there will consequently be less than on the other side of the wires. The pressure of the medium between the wires will therefore be less than in the space beyond, and the wires will be pushed together as if they attracted one another.

How, suppose that a current of electricity commences to flow in a wire. Molecular vortices will be set up in the immediate neighbourhood of the wire, and these vortices acting on the electric particles on the other side of them, remote from the wire, will endeavour to set them in motion in the direction opposite to the current in the wire. But if the medium be a dielectric, the particles cannot be displaced through a sensible distance. They will therefore be made to rotate and start another and larger layer of vortices surrounding the wire, and so the motion will be propagated as above explained. But suppose that at a certain distance there is placed another wire parallel to the first, and forming part of a closed circuit in which no current is flowing. The particles of electricity in this wire will be acted on in the same way as those in the dielectric, but meeting with very little resistance to their motion along the wire, they find it easier to move through the wire than at once to transmit the vortex motion to the elastic bodies on the other side of them. But when a driver and follower are connected by a differential wheel, if the follower he retarded only by its own inertia, however small a resistance the differential wheel may experience to its motion of translation it will at length cause the follower to turn at the same rate as the driver, and will itself cease to move. Hence the resistance of the conductor at length brings the electric particles to rest, and causes them to communicate the vortex motion to cells beyond them. Thus when a current is started in a wire transitory currents in the opposite direction will be induced in neighbouring conductors, while electric stress will be produced in the dielectric, the elastic cells whose motion constitutes the molecular vortices being at first deformed by the tangential stress of the electric particles, but both the induced currents and the stress will have entirely ceased as soon as all the molecular vortices are in full swing.

Before a current can be maintained in a primary wire, the molecular vortices in the surrounding field must be properly started, and this requires the expenditure of work in consequence of the mass of the bodies which constitute the vortices. It is therefore impossible for a finite electro-motive force to start a finite current in an indefinitely short time, in the same way as it is impossible for a finite force to produce instantaneously a finite velocity in a material body, and, just as in dynamics we sometimes speak of the reaction of a body against acceleration as though it were a force opposing the force applied, so we sometimes speak of the corresponding action in the case of the current as though it were a force opposing the battery or other electro-motor, and speak of it as the electro-motive force of self-induction. As, however, this depends not on the current in the wire simply, but on the molecular vortices in the surrounding medium, it is clear that the self-induction of a wire will depend on the energy of these vortices, and this must depend on the relations of the several portions of the wire to one another and to the medium, as well as on the density of the medium. The density of the medium Maxwell identified with its magnetic permeability. This is greater in (para-)magnetic substances than in air or vacuum; greatest of all in iron. In fact, it is so great in the case of iron, that Maxwell supposed the particles of the iron itself to take part in the vortex motion. Hence the energy of the field, and therefore the self-induction of the wire, is greater the greater the magnetic permeability of the surrounding medium, and the presence of an iron-core in a coil immensely increases its self-induction and the energy corresponding to a given current flowing in the coil.

If, after a current has been established in a wire, the circuit be broken or the electro-motive force removed, the molecular vortices refuse to come to rest till they have expended their energy. The only outlet for this energy is a current in the wire, since there is no opportunity of doing work in a non-conducting medium, where there can be no slipping between the elements of the mechanism. The vortices, therefore, keep the electricity moving in the wire after the battery has been removed, until they have expended all their energy in doing work against the resistance of the wire.

But if there be another conductor in the field parallel or slightly inclined to the first, there is another partial outlet for the energy of the system, and a “secondary” current will be set up in the second wire in the same direction as the current in the primary, while that in the primary will be less than it would have been if no secondary circuit had existed. In this way the hypothesis of molecular vortices affords an explanation both of the mutual induction of two circuits and the self-induction of one.

Suppose a wire to be placed in a magnetic field at right angles to the lines of force, and then to be moved so as to cut the lines at right angles, we should expect that in front of the moving wire the lines of force or threads of vortices would be squeezed together transversely, but extended in the direction of their length, somewhat in the same way as elastic strings would be affected by the wire before they broke and allowed it to pass through. Behind the wire the lateral pressure will be relieved, the vortices will contract in the direction of their axes and expand equatorially. But we have seen that the effect of stretching a rotating elastic body in the direction of its axis of rotation, and compressing it at right angles to this direction, increases the velocity of rotation so that the actual velocity of every point on the surface is increased; while the contraction of the body along the axis of rotation diminishes the velocity. Hence as long as the wire is moving across the lines of force the velocity of the vortices in front of the wire will be greater than that of the vortices behind, and the electric particles in the wire, coming, as they do, between two sets of vortices, which are rotating with different velocities, will flow in a stream along the wire. The direction of the current in the wire will be that which would cause the vortices in front to rotate more rapidly than those behind, and therefore to exert a greater pressure on the wire; in other words, there will be a current induced in such direction as to oppose the motion of the wire. We arrive at a similar result if we suppose the lines of force to be cut obliquely instead of orthogonally. Thus Lenz’s law is a consequence of the hypothesis of molecular vortices. If we suppose the magnetic force to act from south to north horizontally, the wire to be vertical and to move from west to east, we have magnetic force acting from south to north, mechanical force acting from east to west, and opposing the motion of the wire, and electro-motive force acting in the wire vertically upwards.

Suppose that all over a certain area the electricity, is pushed forwards through a very small distance along the normal, so that it does not pass from molecule to molecule of the substance, but in each molecule undergoes a displacement from back to front. The electric particles pressing tangentially on the walls of the elastic cells are unable to set them rotating, because each cell is acted upon equally all round in the direction in which the electricity tends to move, and the substance of the cell therefore undergoes a shearing strain which is resisted by its elasticity, and the state of strain of the cells is propagated through the dielectric by means of the displacement of the electric particles which behave like perfectly incompressible bodies. When the force producing the original displacement is removed the cells resume their original form in virtue of their elasticity, the electric particles return to their normal positions, and the energy of the strained elastic cells expends itself in the work done during the electric discharge. Thus the same medium which serves as the vehicle of magnetic force and produces all the phenomena of electromagnetism also serves for the transmission of the force between charges of statical electricity and as a reservoir of the energy due to electro-static charges. If the dielectric be divided into cells by unit tubes of force and equipotential surfaces drawn for every unit difference of potential, each cell will contain the same amount of energy.[257] The following quotations from the paper in the Philosophical Magazine explain the application of the hypothesis of molecular vortices to statical electricity in Maxwell’s own words:—

Suppose we have a body positively electrified. This means that a displacement of the electricity in the medium takes place in all directions around the body and away from its surface. The cells are thus exposed to a shearing strain, diminishing as the distance increases, because the surface over which the displacement takes place being increased the linear displacement of the electricity is proportionately diminished, the particles of electricity behaving like a perfectly incompressible fluid. The medium being isotropic the lines of electric displacement coincide with those of electric stress, which stress is everywhere proportional to the displacement. The distortion which the cells experience by the pressure of the electric particles induces an elastic pressure in all directions, at right angles to the direction of displacement, so that there is a pressure in the medium at right angles to the lines of force.

Now suppose that we have two positively charged bodies in the field, which we may suppose to possess equal charges. Each produces a displacement of the medium outwards from itself, but the electric particles behaving like an incompressible fluid, it is clear that there can be no lines of displacement from the one to the other, but that between the bodies the lines of displacement will be curved so as to avoid one another in the same way as the stream lines emanating from two pipes, each of which is supplying water to a tank, would he curved round, and would avoid one another. The lines of displacement, and consequently the lines of force which coincide with them, will therefore he bent in exactly the same manner as the magnetic lines of force represented in Fig. 10, p. 533, and the pressure in the medium at right angles to these lines will cause an apparent repulsion of the bodies.

For the same displacement, that is, for the same charges of the little bodies, the repulsion will be proportional to the elasticity of the medium. It is also proportional to the product of the charges, or, since they are equal, to the square of one of them. Suppose then that the medium is exchanged for one of greater elasticity. If we wish to keep the repulsion between the bodies the same, the displacements and therefore the charges must be diminished, the product of these charges, that is, the square of either charge, being made inversely proportional to the elasticity of the medium. The magnitude of each charge must therefore vary inversely as the square root of the elasticity of the medium when the dielectric is changed. Hence, if we define the electrostatic unit of electricity as “that quantity of positive electricity which, acting on an equal quantity at unit distance repels it with unit force,” it follows that the unit will vary with the character of the dielectric, being inversely proportional to the square root of its elasticity.

But the attraction or repulsion between two given charges of electricity varies inversely as the specific inductive capacity of the dielectric, so that the electrostatic unit of electricity varies directly as the square root of the specific inductive capacity, and thus the specific inductive capacity is a quantity which varies inversely as the elasticity of the medium.

Suppose we have two parallel wires conveying equal electric currents in the same direction. Other things remaining unchanged, the velocity of the molecular vortices at any point is proportional to the strength of the currents. The attraction between the wires we know to be proportional to the product of the strength of the currents, that is to the square of one of them. The pressure excited by the vortices is, cœteris paribus, proportional to their density and the square of their velocity. Suppose we keep the attraction between the wires the same, but change the density of the medium. Then the velocity of the vortices at any point must vary inversely as the square root of the density of the medium. But the velocity of the vortices is proportional to the strength of the currents. Hence the strength of each current must vary inversely as the square root of the density of the medium. If then the electromagnetic unit of current be defined as that current which, flowing in a certain wire, attracts an equal current in another given wire with unit force, the unit of current, and therefore the unit of electricity, which is the amount flowing per second across any section of a wire conveying a unit current, will vary inversely as the square root of the density of the medium.

The ratio of the electromagnetic to the electrostatic unit of electricity will therefore be proportional to the ratio of the square root of the elasticity to the square root of the density of the medium. But this is known to be the velocity with which a transverse vibration is propagated through the medium. Hence the ratio of these units is a concrete velocity, and is proportional to the velocity of propagation of an electromagnetic disturbance, or of the vortex motions above described, through the dielectric. If the units are chosen according to the ordinary system their ratio is not only proportional to but identical with this velocity.

In a paper published in the Phil. Trans. for 1868, Professor Maxwell gave an account of an experiment for determining the ratio of the electrostatic and electromagnetic units of electricity where air is the dielectric. The principle of the method lay in balancing the attraction between two electrified discs by the repulsion between two coils of wire in which currents were flowing in opposite directions. One of the discs and one coil was placed at one end of the beam of a torsion balance, the other disc and coil being fixed, but a third coil, conveying the same current as the other two, was placed at the other end of the beam in order to eliminate the magnetic action of the earth and the suspended coil. The apparatus is now in the Cavendish Laboratory. The result of the experiment gave for the ratio of the units a velocity of 288,000,000 metres, or 179,000 statute miles per second. The result obtained by another method by MM. Weber and Kohlrausch is 310,740,000 metres per second. The battery employed for the electrostatic charges was M. Gassiot’s battery of 2600 cells, charged with corrosive sublimate. The accuracy of this result depends on that of the B. A. unit of resistance, the velocity being in fact represented by 28·8 Ohms.

Now, according to the undulatory theory, light consists of transverse vibrations of an elastic substance pervading space and all bodies, and the velocity of light as determined by Foucault is 298,000,000 metres per second, or very near the mean of the values obtained by Maxwell, and by Weber and Kohlrausch, for the velocity of propagation of electro-magnetic disturbances. If this is found to be always the case, clearly the same medium will serve to account for the phenomena of electrostatics and electromagnetism, and for the propagation of light which must consequently be of the nature of an electromagnetic disturbance.

If an electromagnetic disturbance take place in a perfect insulator we have seen that it must be transmitted to an unlimited distance, for as no slipping can take place between the electric particles and the cells, and as the particles themselves cannot be displaced except by inducing a corresponding elastic stress in the medium, there is no outlet for the energy of the disturbance, which must therefore be communicated from cell to cell without limit. But if the medium be a conductor, that is, if the electric particles can undergo a permanent displacement passing from molecule to molecule against a frictional resistance and without any tendency to return, the energy of the electromagnetic disturbance will be gradually dissipated; for the electric particles, instead of communicating the whole of the motion of one layer of cells to the next, will themselves be set in motion, and part of the energy will be dissipated as heat instead of being imparted to the external layer of cells. The disturbance will therefore continually diminish as it is propagated, until it very soon becomes insensible. The behaviour is the same as that of a driver and follower connected by a differential wheel, whose epicyclic motion is retarded by forces of the nature of friction. Hence electromagnetic disturbances cannot be propagated in conductors of electricity, and we therefore infer that all true conductors are opaque to light.

The transparency of electrolytes, such as saline solutions and the like, offers no difficulty in the face of this conclusion, as the transference of electricity in them is by a process entirely different from true conduction and more allied to the convection of heat, but Maxwell pointed out that the transparency of gold leaf is much greater than the theory would indicate. Thus the resistance of a particular piece of gold leaf was such that it ought to transmit only 10 Superscript negative 50 of the light incident upon it, which would be totally imperceptible, while the amount of green light actually transmitted by it was easily perceived. This result Professor Maxwell could reconcile with the theory only by supposing “that there is less loss of energy when the electromotive forces are reversed with the rapidity of the vibrations of light than when they act for sensible times, as in our experiments.”

We have seen that the velocity of transmission of an electromagnetic disturbance in any medium is expressed by the quotient of the square root of the elasticity divided by the square root of the density of the dielectric. We have learned that the elasticity is inversely proportional to the specific inductive capacity of the medium while the density corresponds with the magnetic permeability. Hence we infer that the velocity of transmission of an electromagnetic disturbance varies inversely as the square root of the specific inductive capacity, and also inversely as the square root of the magnetic permeability of the dielectric, and this must be true for the velocity of light if light be an electromagnetic disturbance. Now the magnetic permeability of most transparent media, such as glass, quartz, sulphur, hydrocarbons, and the like, does not differ sensibly from that of a vacuum, and hence in these substances the velocity of light must be inversely proportional to the square root of their specific inductive capacity; or, since the index of refraction of a medium is the ratio of the velocity of light in a vacuum to its velocity in that medium, it follows that the refractive index must be directly proportional to the square root of the specific inductive capacity. As all our measurements of specific inductive capacity refer tb the action of electro-motive forces which continue for a much longer time than the duration of a luminous vibration, we should expect the last mentioned relation to agree most nearly with experiment the longer the wave length of the light, or, as it is sometimes stated, the specific inductive capacity of a dielectric should be equal to the square of its refractive index for “light of infinite wave length.”

The results of the measurements of the specific inductive capacity of certain liquids by Silow, and of gases, sulphur, paraffin, and resin, agree with this theory as well as can be expected. Boltzmann also finds that the specific inductive capacities of crystalline sulphur along its three crystallographic axes are different, these differences coinciding with the differences of the squares of the refractive indices for light transmitted along these three directions.

Dr. Hopkinson (Phil. Trans. Part II. 1881) has recently measured the specific inductive capacities of turpentine, benzol, petroleum, ozokerit lubricating oil, castor oil, sperm oil, olive oil, and neats’ foot oil. The hydrocarbons give results which are quite in accordance with Maxwell's theory, but the fatty oils, which are compounds of glycerine with fatty acids, have inductive capacities far too great. The same appears to be the case with all the varieties of glass tested by Hopkinson, the specific inductive capacities of which vary from 6·61 in the case of very light flint to 9·896 for “double extra-dense” flint. In the case of solid paraffin, Hopkinson’s result agrees very nearly with that of Boltzmann and with Maxwell’s theory. In the case of glass, as in that of the fatty oils, the high specific inductive capacity is associated with a complex chemical constitution, glass consisting essentially of metallic silicates, including silicates of the alkaline and alkaline-earthy metals.

The measurement of the specific inductive capacity of glass is attended with great difficulty on account of the phenomenon generally known as residual charge or electric absorption, that is the apparent soaking of the electricity into the substance of the glass. This is a subject in which Maxwell took very great interest, and in his work on electricity and magnetism he has given a mechanical illustration of the action on the supposition that it is due to a want of homogeneity in the glass, some parts of which, he supposed to conduct electricity better than others, though badly at the best. A form of experiment, very beautiful in its design, was devised by Maxwell for measuring specific inductive capacities, and was carried out by Mr. J. E. H. Gordon, who was able to reverse the electric stress in the glass 12,000 times per second; but this is of course no approximation to the rapid alternations of the “waves” of light. With the apparatus employed, however, the reduction of the observations involves great mathematical difficulties, and the results must therefore be received with caution whether we regard them as supporting the theory or as opposed thereto.

In applying the hypothesis of molecular vortices to the action of a magnetic field oh polarised light, Maxwell “found that the only effect which the rotation of the vortices will have on the light will be to make the plane of polarisation rotate in the same direction as the vortices, through an angle proportional—

(A) to the thickness of the substance.

(B) to the resolved part of the magnetic force parallel to the ray.

(C) to the index of refraction of the ray.

(D) inversely to the square of the wave length in air.

(E) to the mean radius of the vortices.

(F) to the capacity for magnetic induction.”

The relation (E) between the amount of rotation and the size of the vortices, shows that different substances may differ in rotating power independently of any observable difference in other respects. We know nothing of the absolute size of the vortices; and on our hypothesis the optical phenomena are probably the only data for determining their relative size in different substances.

Now, independently of the action of a magnetic field on polarised light, all the phenomena of diamagnetism can be accounted for on the hypothesis that the magnetic permeability of diamagnetic substances is less than that of a vacuum, so that they behave like a paramagnetic substance immersed in a medium more magnetic than itself. But Maxwell has pointed out that “since M. Verdet has discovered that magnetic substances have an effect on light opposite to that of diamagnetic substances, it follows that the molecular rotation must be opposite in the two classes of substances.”

Perhaps we cannot conclude this account of the hypothesis of molecular vortices better than by quoting Maxwell’s own words:—[258]

In a paper entitled “A Dynamical Theory of the Electro-magnetic Field,” read before the Royal Society on December 8, 1864, Maxwell deduced all the above results by purely mechanical reasoning, only assuming the existence of a medium capable of receiving and storing up potential and kinetic energy, and therefore capable of doing work in “recovering from displacement in virtue of its elasticity,” while the parts of the medium are connected by “a complicated mechanism capable of a vast variety of motion, but at the same time so connected that the motion of one part depends, according to definite relations, on the motion of other parts, these motions being communicated by forces arising from the relative displacements of the connected parts, in virtue of their elasticity.” For the existence of such a medium we have evidence independent of electrical actions. With regard to the mechanism no attempt is made in the paper to give to it any definite constitution. This paper has been regarded as Maxwell’s greatest contribution to electrical science, but most of the results obtained in it have been already mentioned.

The following is a good specimen of Maxwell’s humorous irony, of which there are many samples in his scientific works. He is discussing certain developments by Bernhard Riemann Lorenzo, of Weber and Neumann’s theory of Electro-magnetism, which is based on the assumption that the action between two quantities of electricity is direct action at a distance, and depends not only on the distance between the charges but upon their relative motion.

While at the Cavendish Laboratory Maxwell constructed a mechanical model which illustrates in a very beautiful manner the principal phenomena of induced currents. As a piece of mechanism it is simply a differential train, such as is often employed as a dynamometer for measuring the power absorbed by a machine. The apparatus is sketched in Fig. 13. The grooved wheel P is keyed to the same shaft as the bevel wheel A, which therefore turns with it, and the rotation of this piece represents the primary current. A second bevel wheel D turns loosely on the arm C D, which is one of four arms (of which only two are shown in the figure) forming a cross, which can turn freely on the central shaft at C. Sliding weights normal upper M normal upper M prime, etc., can be fixed in any desired position on these arms so as to alter the moment of inertia of the cross, which is the differential piece in the mechanism. A third bevel wheel B is keyed to the same hollow shaft with the wheel S, which is similar to P, and the rotation of the piece B S represents the current in the secondary circuit. As the shaft B S is hollow, and rides loosely on the shaft A C, the wheels A and B can turn quite, independently of one another, except in so far as they are connected by the wheel D. normal upper P prime is an index attached to the interior shaft and turning with P. A loop of string is hung over each of the wheels P and S, and carries a small weight. These strings act as friction brakes to the wheels, and the friction represents the resistance of the primary and secondary circuits respectively. The moment of inertia of the loaded cross, or differential piece, represents the moments of inertia of the cells which constitute the molecular vortices in the dielectric. Its kinetic energy when rotating represents the energy of the vortices, and its angular momentum is proportional to the electro-magnetic momentum of the system. The moments of inertia of the other portions of the mechanism are very small compared with that of the loaded cross. The motion of the cross and the wheel D is impeded by as little friction as possible.

Suppose that the wheel P is made to revolve, representing a current in the primary wire; the heavy cross will not at first move, but the wheel D will revolve and communicate the motion to B, which, with S, will rotate in the direction opposite to that of P, representing a current in the secondary circuit opposite in direction to that in the primary. But the motion of S is resisted by the friction brake, and a finite force must therefore be exerted by D on B to drive it. The reaction of B, together with the force exerted by A, will constantly tend to make the cross revolve in the same direction as P, and the velocity of the cross being constantly accelerated, it will presently revolve with a velocity half that of P, and then D will roll round B, which, with S, will remain at rest. The piece B S will then continue at rest as long as the rotation of P remains constant corresponding to the cessation of the current in the secondary circuit, while that in the primary remains unchanged, but if P be accelerated, S will revolve in the direction opposite to the motion of P. Now suppose P to be suddenly stopped. The kinetic energy of the cross will cause it to continue to revolve until it has done a corresponding amount of work against resistances, and A being at rest, D will roll upon it and compel B, with S, to revolve in the same direction as the cross, that is, in the same direction in which P formerly revolved, and whatever be the resistance to the motion of S, it will be overcome, and S will revolve till the work done against resistance is equal to the kinetic energy originally possessed by the cross. This corresponds to a current induced in the secondary coil on stopping the current in the primary, which current is in the same direction as the primary current, and continues until the energy of rotation of the molecular vortices has been used up in work done against electrical resistance.

If one operator lay hold of the wheel S, and endeavour to keep it at rest while another applies a steady force to P, the motion of P will be accelerated much less rapidly than if the same force had been applied to it, and S had been free, because P can only move by setting in motion the cross with its great moment of inertia. If the operator who is turning P now suddenly stops it, a great shock will be experienced by the machinery, and the wheel S will slip from the grip of the other operator however firmly he may hold it. The force applied to S may correspond to an air-break in the secondary coil, and this is sufficient to prevent a spark when the battery current is started in the primary, but by suddenly stopping the primary current, as in Ruhmkorff’s coil, a disruptive discharge or spark passes through the air between the terminations of the secondary wire. (If the operator who endeavours to keep the wheel S at rest is inexperienced the effect upon him is very striking).

If a pin be placed in the face of the wheel S, and one end of a spring press against the pin, while the other end is fixed to the frame of the apparatus, we have a representation of a secondary coil in which the circuit is broken, and a Leyden jar inserted with its coatings in connection with the ends of the wire. When the motion of P is changed, S will begin to move, and will deflect the spring, corresponding to a current in the secondary coil charging the Leyden jar. If the spring admit of very great deflection, so that a great amount of work must be done upon it before it slips from the pin, the primary current may have attained its full strength before the slip takes place. This corresponds to the capacity of the Leyden jar being too great to allow of its being charged to a sufficient potential to produce a spark. In this case no spark takes place, but when the force between the wheels D and B diminishes on account of the diminution of the acceleration of P, the spring relieves its strain by forcing the wheel S backwards, and the Leyden jar under corresponding circumstances quietly discharges itself through the secondary coil, reversing the operation by which it was charged. But if the pin slips from the spring, the wheel S will revolve, and the spring will fly back corresponding to a disruptive discharge through the air, and if the acceleration of P continue long enough several such disruptive discharges may take place.

We must, of course, be careful not to endeavour to learn from such a model lessons which it was not designed to teach, and we must remember that the behaviour of the mechanism does not represent the electrical action in all respects.

For many years Maxwell rendered valuable service to the British Association, especially in connection with electrical science. Some account of the meetings which he attended will be found in the letters printed in another part of this work, and though during the last few years of his life other engagements prevented his attendance at the annual gatherings, he always showed signs of keen enjoyment when discussing the “British Asses.” In 1862 he was appointed a member of “The Committee on Standards of Electrical Resistance.” In the report issued in 1863, the Appendix, “On the Elementary Relations between Electrical Measurements,” bears the name of Professor Maxwell in conjunction with that of Professor Fleeming Jenkin, while the general description of the method employed in the determination of the Ohm or B. A. unit of resistance, together with the mathematical theory and details of the experiments, are from Maxwell’s pen. In 1863-4 Maxwell was again at work on the same subject in the laboratory of King’s College, and most of the “spins”[259] were conducted under his own supervision. In 1869 the results of Maxwell’s experiments on the relation of the electromagnetic to the electrostatic unit of electricity, described above, were embodied in the Report to the British Association at the meeting at Dundee, and this forms the last of the Reports of the Committee.

In 1874 Professor Maxwell was elected a member of the committee appointed by the British Association for the purpose of investigating Ohm’s law. Most of the work executed by this committee was carried out by Professor Chrystal in the Cavendish Laboratory, under the supervision and at the suggestion of Professor Maxwell. An account of the investigations will be found in the report presented to the Association at the Glasgow meeting in 1876.

Before concluding our notice of Maxwell’s contributions to electrical science we must mention the preparation for the press of The Electrical Researches of the Honourable Henry Cavendish, published in 1879, only a few weeks before the death of its editor. The amount of labour which Professor Maxwell bestowed on this work during the last five years of his life can only be known to those who were constantly in his company. Nearly all the MS. he transcribed with his own hand, the greater part being copied after midnight, while he watched over Mrs. Maxwell during the long illness to which allusion has elsewhere been made. Every obscure passage or allusion was the subject of a long and searching investigation; and many were the letters written to the Librarian of the Royal Society and to scientific and literary friends in different parts of the country, to gain information respecting the meaning of obsolete words and symbols, or the history of individuals. But besides this, and a comparison of Cavendish’s results with those obtained by subsequent investigators, Maxwell repeated many of Cavendish’s experiments almost in their original form, only employing modern instruments for the purposes of measurement. The introduction and the appendices to the work evidence much labour, patient investigation, and very extensive acquaintance with the literature bearing on the subject. Maxwell was by no means one of the class of “thinkers” who only read their own writings; his acquaintance not only with scientific literature, but with nearly every other class of books was astonishing; and if any question of physics was brought before him, he could generally give an account of nearly all that had been done in the subject. In this respect he resembled the late Professor W. H. Miller, whom Cambridge men used to consult about everything.

It would be impossible here to give any adequate account of Maxwell’s work in connection with the Cavendish papers, but we may mention one experiment both on account of its intrinsic importance and of the interest which Maxwell took in it. Cavendish describes an experiment (see p. 104 of The Electrical Researches) in which a sphere of 12·1 inches in diameter was enclosed between two copper hemispheres 13·3 inches in diameter, and the copper hemispheres, which were supported in frames hinged together, and opened or closed by silk cords, were brought together so as to enclose the sphere, contact being made at the same time between it and the hemispheres. The outer sphere was then charged, and the connection between it and the inner sphere removed. On separating the hemispheres, the inner sphere was found to be discharged, or at least the charge, if any, left upon it was too small to affect Cavendish’s pith ball electrometer. Knowing what fraction of the original charge could not fail to be detected by the electrometer, Cavendish deduced from the results of the experiment that the law of variation of electrical action with the distance must lie between the inverse left parenthesis 2 minus one fiftieth right parenthesisth and left parenthesis 2 plus one fiftieth right parenthesisth power of the distance, and inferred that it is the inverse square.

Maxwell placed a copper sphere about 10 inches in diameter within a hollow copper sphere, made of two hemispheres, about 12 inches in diameter, supporting it upon a ring of ebonite, so that the spheres might be concentric. The outer sphere was carefully insulated, and a hole cut in its surface, which could be accurately filled by a little trap door. To this trap door was attached a wire which rested on the inner sphere when the door was shut down. The door was carried on a metal arm which turned about a hinge attached to the outer sphere, and could be raised or lowered by a silk thread which passed over a pulley and hung in front of the operator. When the trap door was raised a wire connection could be allowed to fall by relaxing a second silk cord, so as to make contact with the inner sphere through the hole in the outer. The trap door was first closed and the outer sphere charged by means of a Leyden jar which was immediately removed (the electric machine being placed in a distant apartment). The trap door was then raised, the outer sphere discharged by connecting it to earth, and the inner sphere then put in communication with a quadrant electrometer by means of the wire above mentioned. Not a trace of electricity could be detected on the sphere. To test the accuracy of the method, a small brass sphere was insulated, and supported at a distance of about 60 centimetres from the outer sphere, and the experiment was repeated. When the copper sphere was charged the brass sphere was connected to earth, and thus a small negative charge was induced upon it by the positive electricity on the copper sphere, and the ratio of this charge to that on the copper could be easily calculated. The brass sphere was then insulated, the outer copper sphere discharged as before, and the inner sphere examined with the same result as previously. The brass ball was then discharged, the outer sphere being insulated, when it was found that there was sufficient positive electricity on the outer sphere to deflect the electrometer through more than 300 times the largest deflection which could escape notice. Now when the brass hall was insulated, the negative charge upon it was about one sixty fourthth of the charge originally on the copper sphere, and this induced a positive charge on the sphere when in connection with the earth, which was about StartFraction 1 Over 486 EndFraction of its original charge. Hence StartFraction 1 Over 486 EndFraction of the original charge is more than 300 times the largest charge which could escape observation. From these figures it follows that, in the expression of the law of electrical action, the true index of the inverse power of the distance must be either 2 or differ from it by less than StartFraction 1 Over 21600 EndFraction. These experiments were carried out in the Cavendish Laboratory by Mr. MacAlister of St. John’s College. In note 19 appended to the Cavendish papers, Maxwell describes the experiment, and gives its complete mathematical theory.

The idea that electricity flowing in conductors behaves like an incompressible fluid is at least as old as Cavendish; but before he had the opportunity of reading Cavendish’s papers, Maxwell taught that all electric discharges involve a displacement of electricity in a closed circuit, the electricity behaving both in conductors and in dielectrics like a perfectly incompressible fluid. Thus, when a Leyden jar is discharged, a certain quantity of electricity flows from the inner coating through the knob of the jar, and hence to the outer coating, but an equal quantity also passes across any surface which we may imagine drawn within the glass, so as to include the inner coating and exclude the outer. The glass, therefore, may be regarded as completing a circuit across every section of which the same quantity of electricity flows. Again, if positive electricity be communicated to a conductor by means of a wire which serves as an electrode, an equal quantity passes out from the surface of the conductor and is squeezed into the surrounding dielectric, causing a similar transfer of electricity across every surface drawn in the dielectric so as to surround the conductor, the extent of the displacement of course diminishing as we recede from the conductor on account of the increased area of the surface, but the displacement continuing until some external conductor is reached through which the circuit is completed. Thus, when we say that the charge of a body is increased, we mean that positive electricity is communicated to it through an electrode, but we know that an equal amount passes out of the body through its external surface and is squeezed into the dielectric, and it is this dielectric which is the only portion of the system really affected by the charge, and in which the whole energy of the charged body resides. This view has very recently been put forth under the title of The Conservation of Electricity. According to Maxwell’s view of the constitution of dielectrics, the squeezing of a quantity of electricity into a dielectric does not imply a condensation of the electricity, but a strain in the dielectric on account of the displacement of the electricity, which cannot move without distorting the “cells” of which the dielectric is supposed to be made up.

7. Ever since men began to think about Nature, philosophers have differed in their views respecting the primary constitution of bodies. When, with all the freshness of a first impulse, though unprovided with the barest means of verification, the human mind first went forth in search of physical truth, two paths at once appeared, leading to opposite poles, and both were trodden by pilgrims full of hope. Some found their satisfaction in contemplating the continuous fulness of the universe. They could imagine no gap in Nature, whose endless variety they were contented to refer to a principle of infinite divisibility and to the inter-play of contiguous elements whose changes were ordered by the mind that was interfused with all. To others, of a more analytical turn, it appeared impossible to account for the most obvious phenomena, except by the hypothesis of atoms moving in a void.

The penetrating intellect of Democritus had early given consistency to an atomic theory, which became the foundation of the system of Epicurus. Yet, widely as this philosophy prevailed at certain periods, the doctrine of a plenum was on the whole the more prevalent in the ancient world.

The controversy has been handed on to modern science, or rather it has inevitably reappeared; some, with Descartes and Spinoza, resolving matter into continuous extension; while others, as Bacon would say, cut deeper into Nature, and would divide her if possible into her ultimate constituent parts. Experimental investigations, especially those through which chemistry became an exact science, have greatly favoured atomistic views, yet the tendency to maintain the continuity and plasticity of matter is still apparent, and the bold hypothesis of the vortex atom, due to Sir William Thomson, affords us perhaps the nearest approach to a reconciliation between these two contradictory theories. According to this hypothesis the whole of space is filled with a uniform and perfect fluid, and material atoms are vortices of one form or another which have been created within this fluid.

In order to reach the starting-point of Maxwell’s investigations in molecular physics we must go back to Daniel Bernoulli who, in his hydrodynamics, published in 1738, explained the pressure of the air on the hypothesis that air consists of a number of particles moving about in all directions and impinging on any surface exposed to its action. Le Sage of Geneva in 1818 explained the pressure of a gas in strict accordance with the modern Dynamical Theory. The theory of Le Sage was based upon his doctrine of “ultra-mundane corpuscles,” a doctrine borrowed from the older atomists, and employed by Le Sage to explain gravitation. For a concise account of this and of the older atomic theories the reader is referred to Maxwell’s article “Atom” in the ninth edition of the Encyclopœdia Britannica. In 1847 John Herapath published his Mathematical Physics, in which he supposed gases to consist of perfectly hard molecules impinging against one another, and against any surface exposed to their action; and he points out the relation of the motion to the temperature and pressure of a gas and explains gaseous diffusion.

But we are indebted for all the modern developments of the molecular theory of gases, as well as for its establishment on a sound dynamical basis, mainly to the researches of three men—Professor R. Clausius, Dr. Ludwig Boltzmann of Vienna, and James Clerk Maxwell. Maxwell’s principal contributions to the literature of this subject are his papers on “Illustrations of the Dynamical Theory of Gases,” presented to the British Association at the meeting in Aberdeen in 1859; “On the Viscosity or Internal Friction of Air and other Gases,” which constituted the Bakerian lecture read before the Royal Society on February 8th, 1866, and published in the Phil. Trans. for that year; “On the Dynamical Theory of Gases,” a paper read before the Royal Society on May 31st, 1866, and also published in the Phil. Trans.; the lecture “On Molecules” delivered before the British Association at Bradford in 1873, and which has been referred to in Part I.; a paper “On the Dynamical Evidence of the Molecular Constitution of Bodies” which was read before the Chemical Society in the Spring of 1875, and published in the June number of its Journal; and the article “Atom,” in the ninth edition of the Encyclopœdia Britannica.

According to the molecular theory all bodies are made up of molecules which are more or less free to move relatively to one another. In solids, each molecule can only move through a small distance from its normal position, so that all its movements are essentially of the nature of vibrations. In liquids the molecules are free to move through any distances within the substance of the liquid, but their motion is constantly impeded by neighbouring molecules, from whose interference they are never altogether free, each molecule in its movements resembling an individual endeavouring to work his way through a dense crowd. In a gas each molecule is perfectly free—except on the occasions, comparatively rare, unless the gas be very dense—when it comes into collision with other molecules of gas, or of some other body in contact with the gas. According to this theory, a gas consists of molecules moving in all directions in straight lines, except when they strike one another or some foreign body, when their motion is changed in accordance with the laws of impact of perfectly elastic bodies.

It is probable that the investigations connected with Saturn’s rings, which led Maxwell to contemplate the condition of the air in the midst of “a flight of brickbats,” first led him to take up the subject of the kinetic theory of gases. At any rate, these investigations led him to the adoption of what he aptly termed “the statistical method,” which in his hands became so fruitful in its applications to molecular science. This method consists in the separation of all the things considered into classes, which fulfil certain conditions, and the determination of the number of individuals which at any instant are to be found in each class, without reference to the behaviour of any particular individual. In “the historical method,” on the other hand, the life history of each individual is traced, and if classes are considered at all it is only in their relation to the particular individual we are contemplating. The first method deals with the interests of the community at large and disregards the fate of the individual, except as affecting the average condition of the community; the second deals with the interests of the individual alone. It is obvious that when the individuals are to be counted by millions the only way in which they can be successfully treated by a finite mind is by the statistical method, and this was the method which Maxwell adopted.

In the earliest papers of Clausius he treated all the molecules of a particular gas at the same temperature as moving with the same velocity. Maxwell showed that this could not be the case, and that, even if all the molecules were started with the same velocities, their mutual collisions would increase the velocity of some and diminish that of others, until at length the velocities would be distributed “according to the law of errors.” This law will be best understood from Maxwell’s own illustration. He prepared a diagram illustrating the distribution of bullet marks on a target on the hypothesis that all the shots are aimed at the centre of “the bull” by the same marksman, while the probability of an error of any given magnitude occurring is less the greater the error: diminishing very rapidly as the error increases, according to the ordinary hypothesis respecting “errors of observation” developed by Laplace. If on such a diagram a series of concentric circles be drawn, say an inch apart, about the centre of the target, and the number of shots between each consecutive pair of circles counted, the number will be found to be greatest for a particular pair, but there is nothing to prevent some shots from hitting the target at any distance from the centre. If the length of the line joining the centre of the target with any bullet mark be taken to represent the velocity of a molecule, the diagram will represent the law of distribution of velocities among the molecules of a gas when they have acted on one another for an indefinite time. They will congregate about a particular value, but there is nothing to prevent some individuals possessing a velocity as great or as small as we please.

The momentum of a particle depends on the product of its mass and its velocity; its kinetic energy on the product of its mass and the square of its velocity. One of the first and most important properties of gases which Maxwell showed to be a consequence of the kinetic theory was, that when two groups of molecules are in equilibrium, being separated only by a diaphragm which can transmit the impacts of the particles, the average kinetic energy of the molecules must be the same in each group. Identifying the kinetic energy of the molecules with the heat of the gas, since we know that for equilibrium the gases must be at the same temperature, it follows that the specific heat varies inversely as the mass of a molecule, which is the dynamical expression of the law of Dulong and Petit, a law which states that the products of the specific heat and combining weight is the same for each element. The pressure of a gas is proportional to the kinetic energy of unit volume of a gas, and, since the average kinetic energy of the molecules is the same for each gas at the same temperature, it follows that equal volumes of two gases at the same pressure and temperature contain the same number of molecules, and hence the density of a gas at standard temperature and pressure is proportional to the mass of a molecule, i.e. to its combining weight, which is Gay Lussac’s law of equivalent volumes.

If two gases be in communication with one another, the particles of each are found to penetrate the other until the gases become uniformly mixed, unless some external force act to prevent the uniformity of the mixture. This phenomenon, known as gaseous diffusion, has been studied by Graham, Loschmidt, and others, and is a direct consequence of the dynamical theory. It is plain that the rate of diffusion of one gas into another will be proportional to the average velocity of the particles in the case of two gases being separated by a diaphragm perforated by very fine holes, so that each mixture may be considered homogeneous up to the diaphragm. Maxwell showed that in this case, when the pressures on the two sides of the diaphragm are equal, “the volumes diffused will be as the square roots of the specific gravities inversely, which is the law of diffusion established by Graham.”

The slowness of gaseous diffusion was for some time regarded as an objection to the kinetic theory of gases. Clausius overcame this difficulty by the introduction of the conception of the mean free path of a molecule, and by showing how short this path is. The mean free path is the average distance through which a particle passes between two successive collisions with other particles. The first estimate of the mean free path in air at ordinary pressure and temperature was made by Maxwell from determinations of viscosity or rate of diffusion of momentum in air. Depending on the length of the mean free path are (1) the diffusion of matter; (2) the diffusion of momentum or viscosity; and (3) the diffusion of energy, or the thermal conductivity of the gas. From determinations of the rate at which these three kinds of diffusion proceed, independent determinations of the mean free path in different gases have been made. For hydrogen, at standard pressure and temperature, it is about StartFraction 1 Over 250000 EndFraction of an inch. For other gases and air it is somewhat less. The length of the mean path in air given by Maxwell in 1859 was StartFraction 1 Over 389000 EndFraction of an inch.

Suppose two trains to be moving in opposite directions, side by side on parallel lines, and suppose that as they pass each other a number of passengers from each train jump into the other. It is clear that each passenger will carry with him the momentum he possesses into the other train which is moving in the opposite direction, and will consequently diminish the momentum and therefore the velocity of that train. If a continuous interchange of passengers, backwards and forwards, between the trains were to take place, the trains would ultimately be brought to rest relatively to one another. This is an illustration of the diffusion of momentum, and was originally given by Balfour Stewart. How, suppose two streams of gas to be passing each other tangentially. There will be a continuous interchange of molecules by diffusion between the two streams, and the molecules carrying with them the momentum they possess, the effect of the diffusion will be to tend to bring the two streams to relative rest. Diffusion of momentum consequently introduces a tangential action between layers of gas which are moving relatively to one another, and therefore causes a resistance to any “continuous change of form, depending on the rate at which that change is effected,” that is, it confers upon the gas the property of viscosity. Thus, the so-called viscosity of gases received from Maxwell its complete explanation in accordance with the kinetic theory.

The viscosity of a gas depending on the rate of diffusion of momentum, and therefore on the rate of diffusion of matter, is proportional, like the latter, to the average velocity of the molecules. If the gas be very rare, each particle will meet with fewer collisions, and consequently its course will be less interrupted than when the gas is denser, so that Maxwell found that between two strata of gas at a given distance apart the rate of diffusion was the same whatever the density. He thus “arrived at the startling result that the coefficient of internal friction is independent of the density of any particular kind of gas.” The experimental verification of this result occupied a considerable portion of Maxwell’s leisure time in 1865. The description of the experiments and the statement of the result arrived at form the subject of the Bakerian lecture of 1866. There plates of glass were suspended by a steel piano wire, and caused to oscillate in their own planes between four fixed plates, which were placed at equal distances apart, and so that the moving plates were suspended half way between each pair of fixed plates (see Part I. p. 341). The distance between the plates was varied in different experiments, and the whole system was enclosed in a receiver which could be filled with different gases and the pressure regulated at will. The oscillations of the discs were observed by means of a mirror and scale, and the experiment consisted in determining the rate at which the oscillations died away, from which the viscosity of the gas was calculated. For experiments at high temperatures the receiver was surrounded by a steam jacket. The results arrived at showed that dry air is more viscous than damp air, and that the viscosity of air is greater than that of hydrogen or carbonic acid, nearly in the same ratio as was determined by Graham in his experiments on the transpiration of gases through capillary tubes. But the most remarkable results were (1), that the viscosity of any particular gas is independent of the pressure; and (2) that it is directly proportional to the temperature measured from the absolute zero of the air thermometer.[260] From the last result, Maxwell deduced “that the force between two molecules is proportional inversely to the fifth power of the distance between them.” If the molecules only acted by impact, it would follow that the viscosity must be proportional to the square root of the absolute temperature.

In the paper “On the Dynamical Theory of Gases,” read before the Royal Society on May 31, 1866, the molecules are dealt with in the most general manner, no hypothesis being made respecting their constitution. The paper discusses (1) the phenomena of diffusion, depending on the average value of the velocities; (2) the phenomena depending on the average square of the velocities, which determines pressure and viscosity; and (3) the phenomena depending on the average value of the cubes of the velocities, on which depends the diffusion of energy or the conduction of heat in the gas. In this paper Maxwell showed that if a number of gases be enclosed in any space under the action of any external forces, each gas will arrange itself as if the others were absent, and this result is independent of the law of action of the particles on each other. This is the arrangement which Dalton suggested would probably exist in the atmosphere if the effect of winds could be annulled. He also showed that in a vertical column of air in equilibrium under the action of gravity the temperature must be the same throughout. This was contrary to the then current opinion that the temperature must diminish as we ascend, but Maxwell showed independently that it is a consequence of the principle of the dissipation of energy. Boltzmann was, however, the first to show how, in dealing with a collection of molecules, to take account of external forces acting upon them. In discussing the diffusion of energy Maxwell showed that the thermal conductivity of iron at 25 Superscript ring Baseline normal upper C, as determined by Principal Forbes, is 3525 times that of air at 16 dot 6 Superscript ring Baseline normal upper C.