# A History of the Theories of Aether and Electricity/Chapter 12

A History of the Theories of Aether and Electricity  (1910)  by Edmund Taylor Whittaker
Chapter XII: The Theory of Aether and Electrons in the Closing Years of the Nineteenth Century

Chapter XII.

The Theory of Aether and Electrons in the Closing Years of the Nineteenth Century.

The attempts of Maxwell[1] and of Hertz[2] to extend the theory of the electromagnetic field to the case in which ponderable bodies are in motion had not been altogether successful. Neither writer had taken account of any motion of the material particles relative to the aether entangled with them, so that in both investigations the moving bodies were regarded simply as homogeneous portions of the medium which fills all space, distinguished only by special values of the electric and magnetic constants. Such an assumption is evidently inconsistent with the admirable theory by which Fresnel[3] had explained the optical behaviour of moving transparent bodies; it was therefore not surprising that writers subsequent to Hertz should have proposed to replace his equations by others designed to agree with Fresnel's formulae. Before discussing these, however, it may be well to review briefly the evidence for and against the motion of the aether in and adjacent to moving ponderable bodies, as it appeared in the last decade of the nineteenth century.

The phenomena of aberration had been explained by Young[4] on the assumption that the aether around bodies is unaffected by their motion. But it was shown by Stokes[5] in 1845 that this is not the only possible explanation. For suppose that the motion of the earth communicates motion to the neighbouring portions of the aether; this may be regarded as superposed on the vibratory motion which the aethereal particles have when transmitting light: the orientation of the wave-fronts of the light will consequently in general be altered; and the direction in which a heavenly body is seen, being normal to the wavefronts will thereby be allected. But if the aethereal motion is irrotational, so that the elements of the aether do not rotate, it is easily seen that the direction of propagation of the light in space is unaffected; the luminous disturbance is still propagated in straight lines from the star, while the normal to the wave-front at any point deviates from this line of propagation by the small angle u/c, where u denotes the component of the aethereal velocity at the point, resolved at right angles to the line of propagation, and c denotes the velocity of light. If it be supposed that the aether near the earth is at rest relatively to the earth's surface, the star will appear to be displaced towards the direction in which the earth is moving, through an angle measured by the ratio of the velocity of the earth to the velocity of light, multiplied by the sine of the angle between the direction of the earth's motion and the line joining the earth and star. This is precisely the law of aberration.

An objection to Stokes's theory has been pointed out by several writers, amongst others by H. A. Lorentz.[6] This is, that the irrotational motion of an incompressible fluid is completely determinate when the normal component of the velocity at its boundary is given: so that if the aether were supposed to have the same normal component of velocity as the earth, it would not have the same tangential component of velocity. It follows that no motion will in general exist which satisfies Stokes's conditions; and the difficulty is not solved in any very satisfactory fashion by either of the suggestions which, have been proposed to meet it. One of those is to suppose that the moving earth does generate a rotational disturbance, which, however, being radiated away with the velocity of light, does not affect the steadier irrotational motion; the other, which was advanced by Planck,[7] is that the two conditions of Stokes's theory—namely, that the motion of the aether is to be irrotational and that at the earth's surface its velocity is to be the same as that of the earth—may both be satisfied if the aether is supposed to be compressible in accordance with Boyle's law, and subject to gravity, so that round the earth it is compressed like the atmosphere; the velocity of light being supposed independent of the condensation of the aether.

Lorentz,[8] in calling attention to the defects of Stokes's theory, proposed to combine the ideas of Stokes and Fresnel, by assuming that the aether near the earth is moving irrotationally (as in Stokes's theory), but that at the surface of the earth the aethereal velocity is not necessarily the same as that of ponderable matter, and that (as in Fresnel's theory) a material body imparts the fraction (μ2 - 1)/μ2 of its own motion to the aether within it. Fresnel's theory is a particular case of this new theory, being derived from it by supposing the velocity-potential to be zero.

Aberration is by no means the only astronomical phenomenon which depends on the velocity of propagation of light; we have indeed seen[9] that this velocity was originally determined by observing the retardation of the eclipses of Jupiter's satellites. It was remarked by Maxwell[10] in 1879 that these eclipses. furnish, theoretically at least, a means of determining the velocity of the solar system relative to the aether. For if the distance from the eclipsed satellite to the earth be divided by the observed retardation in time of the eclipse, the quotient. represents the velocity of propagation of light in this direction, relative to the solar system; and this will differ from the velocity of propagation of light relative to the aether by the component, in this direction, of the sun's velocity relative to the aether. By taking observations when Jupiter is in different signs of the zodiac, it should therefore be possible to determine the sun's velocity relative to the aether, or at least that component of it which lies in the ecliptic.

The same principles may be applied to the discussion of other astronomical phenomena. Thus the minimum of a variable star of the Algol type will be retarded or accelerated by an interval of time which is found dividing the projection of the radius from the sun to the earth on the direction from the sun to the Algol variable by the velocity, relative to the solar system, of propagation of light from the variable; and thus the latter quantity may be deduced from observations of the retardation.[11]

Another instance in which the time taken by light to cross an orbit influences an observable quantity is afforded by the astronomy of double stars. Savary[12] long ago remarked that when the plane of the orbit of a double star is not at right angles to the line of sight, an inequality in the apparent motion must be caused by the circumstance that the light from the remoter star has the longer journey to make. Yvon Villarceau[13] showed that the effect might be represented by a constant alteration of the elliptic elements of the orbit (which alteration is of course beyond detection), together with a periodic inequality, which may be completely specified by the following statement: the apparent coordinates of one star relative to the other have the values which in the absence of this effect they would have at an earlier or later instant, differing from the actual time by the amount

${\displaystyle {\frac {m_{1}-m_{2}}{m_{1}+m_{2}}}\cdot {\frac {z}{c}}}$,

where m1, and m2 denote the masses of the stars, c the velocity of light, and z the actual distance of the two stars from each other at the time when the light was emitted, resolved along the line of sight. In the existing state of double-star astronomy, this effect would be masked by errors of observation.

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

Among the consequences of the finite velocity of propagation of light which are of importance in astronomy, a leading place must be assigned to the principle enunciated in 1842 by Christian Doppler,[14] that the motion of a source of light relative to an observer modifies the period of the disturbance which is received by him. The phenomenon resembles the depression of the pitch of a note when the source of sound is receding from the observer. In either case, the period of the vibrations perceived by the observer is (c + v)/c × the natural period, where v denotes the velocity of separation of the source and observer, and c denotes the velocity of propagation of the disturbance. If, e.g., the velocity of separation is equal to the orbital velocity of the earth, the D lines of sodium in the spectrum of the source will be displaced towards the red, as compared with lines derived from a terrestrial sodium flame, bs about one-tenth of the distance between them. The application of this principle to the determination of the relative velocity of stars in the line of sight, which has proved of great service in astrophysical research, was suggested by Fizeau in 1848.[15]

Passing now from the astronomical observatory, we must examine the information which has been gained in the physical laboratory regarding the effect of the earth's motion on optical phenomena. We have already[16] referred to the investigations by which the truth of Fresnel's formula was tested. An experiment of a different type was suggested in 1852 by Fizeau,[17] who remarked that, unless the aether is carried along by the earth, the radiation emitted by a terrestrial source should have different intensities in different directions. It was, however, shown long afterwards by Lorentz[18] that such an experiment would not be expected on theoretical grounds to yield a positive result; the amount of radiant energy imparted to an absorbing body is independent of the earth's motion. A few years later Fizeau investigated[19] another possible effect. If a beam of polarized light is sent obliquely through a glass plate, the azimuth of polarization is altered to an extent which depends, amongst other things, on the refractive index of the glass. Fizeau performed this experiment with sunlight, the light being sent through the glass in the direction of the terrestrial motion, and in the opposite direction; the readings seemed to differ in the two cases, but on account of experimental difficulties the result was indecisive.

Some years later, the effect of the earth's motion on the rotation of the plane of polarization of light propagated along the axis of a quartz crystal was investigated by Mascart.[20] The result was negative, Mascart stating that the rotation could not have been altered by more than the (1/40,000)th part when the orientation of the apparatus was reversed from that of the terrestrial motion to the opposite direction. This was afterwards confirmed by Lord Rayleigh,[21] who found that the alteration, if it existed, could not amount to (1/100,000)th part.

In terrestrial methods of determining the velocity of light the ray is made to retrace its path, so that any velocity which the earth might possess with respect to the luminiferous medium would affect the time of the double passage only by an amount proportional to the square of the constant of aberration.[22] In 1881, however, A. A. Michelson[23] remarked that the effect, though of the second order, should be manifested by a measurable difference between the times for rays describing equal paths parallel and perpendicular respectively to the direction of the earth's motion. Ho produced interference-fringes between two pencils of light which had traversed paths perpendicular to each other; but when the apparatus was rotated through a right angle, so that the difference would be reversed, the expected displacement of the fringes could not be perceived. This result was regarded by Michelson himself as a vindication of Stokes's theory,[24] in which the aether in the neighbourhood of the earth is supposed to be set in motion. Lorentz[25], however, showed that the quantity to be measured had only half the value supposed by Michelson, and suggested that the negative result of the experiment might be explained by that combination of Fresnel's and Stokes's theories which was developed in his own memoir[26]; since, if the velocity of the aether near the earth were (say) half the earth's velocity, the displacement of Michelson's fringes would be insensible.

A sequel to the experiment of Michelson and Morley was performed in 1897, when Michelson[27] attempted to determine by experiment whether the relative motion of earth and aether varies with the vertical height above the terrestrial surface. No result, however, could be obtained to indicate that the velocity of light depends on the distance from the centre of the earth; and Michelson concluded that if there were no choice but between the theories of Fresnel and Stokes, it would be necessary to adopt the latter, and to suppose that the earth's influence on the aether extends[errata 1] to many thousand kilometres above its surface. By this time, however, as will subsequently appear, a different explanation was at hand.

Meanwhile the perplexity of the subject was increased by experimental results which pointed in the opposite direction to that of Michelson. In 1892 Sir Oliver Lodge[28] observed the interference between the two portions of a bifurcated beam of light, which were made to travel in opposite directions round a closed path in the space between two rapidly rotating steel disks. The observations showed that the velocity of light is not affected by the motion of adjacent matter to the extent of (1/200)th part of the velocity of the matter. Continuing his investigations, Lodge[29] strongly magnetized the moving matter (iron in this experiment), so that the light was propagated across a moving magnetic field; and electrified it so that the path of the beams lay in a moving electrostatic field; but in no case was the velocity of the light appreciably affected.

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

The first memoir in which the new conceptions were unfolded) was published by H. A. Lorentz[30] in 1892. The theory of Lorentz was, like those of Weber, Riemann, and Clausius,[31] a theory of electrons; that is to say, all electrodynamical phenomena were ascribed to the agency of moving electric charges, which were supposed in a magnetic field to experience forces proportional to their velocities, and to communicate these forces to the ponderable matter with which they might be associated.[32]

In spite of the fact that the earlier theories of electrons had failed to fulfil the expectations of their authors, the assumption that all electric and magnetic phenomena are due to the presence or motion of individual electric charges was one to which physicists were at this time disposed to give a favourable consideration, for, as we have seen,[33] evidence of the atomic nature of electricity was now contributed by the study of the conduction of electricity through liquids and gases. Moreover, the discoveries of Hertz[34] had shown that a molecule which is emitting light must contain some system resembling a Hertzian vibrator; and the essential process in a Hertzian vibrator is the oscillation of electricity to and fro. Lorentz himself from the outset of his career[35] had supposed the interaction of ponderable matter with the electric field to be effected by the agency of electric charges associated with the material atoms.

The principal difference by which the theory now advanced by Lorentz is distinguished from the theories of Weber, Riemann, and Clausius, and from Lorentz' own earlier work, lies in the conception which is entertained of the propagation of influence from one electron to another. In the older writings, the electrons were assumed to be capable of acting on each other at a distance, with forces depending on their charges, mutual distances, and velocities; in the present memoir, on the other hand, the electrons were supposed to interact not directly with each other, but with the medium in which they were embedded. To this medium were ascribed the properties characteristic of the aether in Maxwell's theory.

The only respect in which Lorentz' medium differed from Maxwell's was in regard to the effects of the motion of bodies. Impressed by the success of Fresnel's beautiful theory of the propagation of light in moving transparent substances,[36] Lorentz designed his equations so as to accord with that theory, and showed that this might be done by drawing a distinction between matter and aether, and assuming that a moving ponderable body cannot communicate its motion to the aether which surrounds it, or even to the aether which is entangled in its own particles; so that no part of the aether can be in motion relative to any other part. Such an aether is simply space endowed with certain dynamical properties.

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

Suppose that e(x, y, z) and e′(x′, y′, z′) are two electrons. In the theory of Clausius,[37] the kinetic potential of their mutual action is

${\displaystyle {\frac {ee^{\prime }}{r}}({\dot {x}}{\dot {x}}^{\prime }+{\dot {y}}{\dot {y}}^{\prime }+{\dot {z}}{\dot {z}}^{\prime }-c^{2})}$;

so when any number of electrons are present, the part of the kinetic potential which concerns any one of them—say, e—may be written

${\displaystyle L_{e}=e(a_{x}{\dot {x}}+a_{y}{\dot {y}}+a_{z}{\dot {z}}-c^{2}\phi )}$,

where a and φ denote potential functions, defined by the

${\displaystyle \mathbf {a} =\iiint {\frac {\rho ^{\prime }\mathbf {v} ^{\prime }}{r}}\ dx^{\prime }\ dy^{\prime }\ dz^{\prime },\qquad \phi =\iiint {\frac {\rho ^{\prime }}{r}}\ dx^{\prime }\ dy^{\prime }\ dz^{\prime }}$;

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

We shall now reject Clausius' assumption that electrons act instantaneously at a distance, and replace it by the assumption that they act on each other only through the mediation of an aether which fills all space, and satisfies Maxwell's equations, This modification may be effected in Clausius' theory without difficulty; for, as we have seen,[38] if the state of Maxwell's aether at any point is defined by the electric vector d and magnetic vector h,[39] these vectors may be expressed in terms of potentials a and φ by the equations

${\displaystyle \mathbf {d} =c^{2}{\text{grad }}\phi -\mathbf {\dot {a}} ,\qquad \mathbf {h} ={\text{curl }}\mathbf {a} }$;

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

${\displaystyle \mathbf {a} =\textstyle \iiint \{({\bar {\rho \mathbf {v} _{x}}})^{\prime }/r\}\ dx^{\prime }\ dy^{\prime }\ dz^{\prime },\qquad \phi =\iiint \{({\bar {\rho }})^{\prime }/r\}\ dx^{\prime }\ dy^{\prime }\ dz^{\prime }}$,

where the bars indicate that the values of (ρvx)′ and (ρ)′ refer to the instant (t - r/c). Comparing these formulae with those given above for Clausius' potentials, we see that the only change which it is necessary to make in Clausius' theory is that of retarding the potentials in the way indicated by L. Lorenz.[40] The electric and magnetic forces, thus defined in terms of the position and motion of the charges, satisfy the Maxwellian equations

${\displaystyle {\begin{cases}{\text{div }}\mathbf {d} &=&4\pi c^{2}\rho ,\\{\text{div }}\mathbf {h} &=&0,\\{\text{curl }}\mathbf {d} &=&-\mathbf {\dot {h}} ,\\{\text{curl }}\mathbf {h} &=&4\pi \rho \mathbf {v} .\end{cases}}}$

The theory of Lorentz is based on these four aethereal equations of Maxwell, together with the equation which determines the ponderomotive force on a charged particle; this, which we shall now derive, is the contribution furnished by Clausius' theory. The Lagrangian equations of motion of the electron e are

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)-{\frac {\partial L}{\partial x}}=0}$,

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

${\displaystyle {\frac {\partial L_{e}}{\partial x}}-{\frac {d}{dt}}\left({\frac {\partial L_{e}}{\partial {\dot {x}}}}\right)}$,

or

${\displaystyle e\left({\frac {\partial a_{x}}{\partial x}}{\dot {x}}+{\frac {\partial a_{y}}{\partial x}}{\dot {y}}+{\frac {\partial a_{z}}{\partial x}}{\dot {z}}-c^{2}{\frac {\partial \phi }{\partial x}}\right)-e{\frac {da_{x}}{dt}}}$;

which, since

${\displaystyle {\frac {da_{x}}{dt}}={\frac {\partial a_{x}}{\partial t}}+{\frac {\partial a_{x}}{\partial x}}{\dot {x}}+{\frac {\partial a_{x}}{\partial y}}{\dot {y}}+{\frac {\partial a_{x}}{\partial z}}{\dot {z}}}$,

reduces to

${\displaystyle -e\left(c^{2}{\frac {\partial \phi }{\partial x}}+{\frac {\partial a_{x}}{\partial t}}\right)+e{\dot {z}}\left({\frac {\partial a_{z}}{\partial x}}-{\frac {\partial a_{x}}{\partial z}}\right)+e{\dot {y}}\left({\frac {\partial a_{y}}{\partial x}}-{\frac {\partial a_{x}}{\partial y}}\right)}$,

or ${\displaystyle ed_{x}+e({\dot {y}}h_{z}-{\dot {z}}h_{y})}$, so that the force in question is

${\displaystyle e\mathbf {d} +e[\mathbf {v.h} ]}$.

This was Lorentz' expression for the ponderomotive force on an electrified corpuscle of charge e moving with velocity v in a field defined by the electric force d and magnetic force h.

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

The electromagnetic field in ponderable bodies, which to our senses appears in general to vary continuously, would present a different aspect if we were able to discern molecular structure; we should then perceive the individual electrons by which the field is produced, and the rapid fluctuations of electric and magnetie* force between them. As it is, the values furnished by our instruments represent averages taken over volumes which, though they appear small to us, are large compared with molecular dimensions.[41] We shall denote an average value of this kind by a bar placed over the corresponding symbol.

Lorentz supposed that the phenomena of electrostatic charge and of conduction-currents are due to the presence or motion of simple electrons such as have been considered above. The part of ${\displaystyle {\bar {\rho }}}$ arising from these is the measurable density of electrostatic charge; this we shall denote by ρ1. If w denote the velocity of the ponderable matter, and if the velocity v of the electrons be written w + u, then the quantity ${\displaystyle {\bar {\rho \mathbf {v} }}}$, so far as it arises from electrons of this type, may be written ${\displaystyle \rho _{1}\mathbf {w} +{\bar {\rho \mathbf {u} }}}$. The former of these terms represents the convection-current, and the latter the conduction-current.

Consider next the phenomena of dielectrics. Following Faraday, Thomson, and Mossotti,[42] Lorentz supposed that each dielectric molecule contains corpuscles charged vitreously and also corpuscles charged resinously. These in the absence of an external field are so arranged as to neutralize each other's electric fields outside the molecule. For simplicity we may suppose that in each molecule only one corpuscle, of charge e, is capable of being displaced from its position; it follows from what has been assumed that the other corpuscles in the molecule exert the same electrostatic action as a charge e situated at the original position of this corpuscle. Thus if e is displaced to an adjacent position, the entire molecule becomes equivalent to an electric doublet, whose moment is measured by the product of e and the displacement of e. The molecules in unit volume, taken together, will in this way give rise to a (vector) electric moment per unit volume, P, which may be compared to the (vector) intensity of magnetization in Poisson's theory of magnetism.[43] As in that theory, we may replace the doublet-distribution P of the scalar quantity ρ by a volume-distribution of ρ, determined by the equation[44]

${\displaystyle {\bar {\rho }}=-{\text{div }}\mathbf {P} }$.

This represents the part of ${\displaystyle {\bar {\rho }}}$ due to the dielectric molecules.

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

${\displaystyle {\bar {\rho w_{x}}}=-{\text{div }}(w_{x}\mathbf {P} )=-w_{x}{\text{div }}\mathbf {P} -(\mathbf {P.} \nabla )w_{x}}$,

or

${\displaystyle {\bar {\rho \mathbf {w} }}=-{\text{div }}\mathbf {P.w} -(\mathbf {P.} \nabla )\mathbf {w} }$.

We have now to find that part of ${\displaystyle {\bar {\rho \mathbf {u} }}}$ which is due to dielectric molecules. For a single doublet of moment p we have, by differentiation,

${\displaystyle \textstyle \iiint \rho \mathbf {u} \ dx\ dy\ dz=d\mathbf {p} /dt}$,

where the integration is taken throughout the molecule; so that

${\displaystyle \textstyle \iiint \rho \mathbf {u} \ dx\ dy\ dz=(d/dt)(V\mathbf {P} )}$,

where the integration is taken throughout a volume V, which encloses a large number of molecules, but which is small compared with measurable quantities; and this equation may be written

${\displaystyle {\bar {\rho \mathbf {u} }}={\frac {1}{V}}{\frac {d}{dt}}(VP)}$.

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

${\displaystyle (d/dt)\mathbf {P} =\mathbf {\dot {P}} +(\mathbf {w.} \nabla )\mathbf {P} }$,

and ${\displaystyle (d/dt)V=V{\text{ div }}\mathbf {w} }$; so that

${\displaystyle {\begin{matrix}{\bar {\rho \mathbf {u} }}&=&{\bar {\mathbf {P} }}+(\mathbf {w.} \nabla )\mathbf {P} +{\text{div }}\mathbf {w.P} \\\ &=&{\bar {\mathbf {P} }}+{\text{curl }}[\mathbf {P.w} ]+{\text{div }}\mathbf {P.w} +(\mathbf {P.} \nabla )\mathbf {w} ,\end{matrix}}}$

and therefore

${\displaystyle {\bar {\rho \mathbf {u} }}+{\bar {\rho \mathbf {w} }}=\mathbf {\dot {P}} +{\text{curl }}[\mathbf {P.w} ]}$.

This equation determines the part of ${\displaystyle {\bar {\rho \mathbf {v} }}}$ which arises from the dielectric molecules.

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

${\displaystyle {\begin{matrix}{\text{div }}\mathbf {\bar {d}} &=&4\pi c^{2}\rho _{1}-4\pi c^{2}{\text{div }}\mathbf {P} ,\qquad {\text{div }}\mathbf {\bar {h}} =0,\\{\text{curl }}\mathbf {\bar {d}} &=&-\mathbf {\bar {\dot {h}}} ,\\{\text{curl }}\mathbf {\bar {h}} &=&(1/c^{2})\mathbf {\bar {\dot {d}}} +4\pi \left\{{\begin{matrix}{\text{convection-current}}+{\text{conduction-current}}\\+\mathbf {\dot {P}} +{\text{curl }}[\mathbf {P.w} ]\end{matrix}}\right\}.\end{matrix}}}$.

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

${\displaystyle {\begin{matrix}\mathbf {\bar {d}} =\mathbf {E} ,&{\text{the electric force;}}\\(1/4\pi c^{2})\mathbf {E} +\mathbf {P} =\mathbf {D} ,&{\text{the electric induction;}}\\\mathbf {\bar {h}} =\mathbf {H} ,&{\text{the electric vector.}}\end{matrix}}}$

The equations then become (writing ρ for ρ1, as there is no longer any need to use the subscript),

${\displaystyle {\begin{matrix}{\text{div }}\mathbf {D} &=&\rho ,\qquad \qquad -&{\text{curl }}\mathbf {E} &=&\mathbf {\dot {H}} ,\\{\text{div }}\mathbf {H} &=&0,\qquad \qquad &{\text{curl }}\mathbf {H} &=&4\pi \mathbf {S} ,\end{matrix}}}$

where

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


The term in S evidently represents the displacementcurrent of Maxwell; and the term curl [P.w] will be recognized as a modified form of the term curl [D.w], which was first introduced into the equations by Hertz.[45] It will be remembered that Hertz supposed this term to represent the generation of a magnetic force within a dielectric which is in motion in an electric field, and that Heaviside[46] by adducing considerations relative to the energy, showed that the term ought to be regarded as part of the total current, and inferred from its existence that a dielectric which moves in an electric field is the seat of an electric current, which produces a magnetic field in the surrounding space. The modification introduced by Lorentz consisted in replacing D by P in the vector-product; this implied that the moving dielectric does not carry along the aethereal displacement, which is represented by the term E/4πc2 in D, but only carries along the charges which exist at opposite ends of the molecules of the ponderable dielectric, and which are represented by the term P. The part of the total current represented by the term curl [P.w] is generally called the current of dielectric convection.

That a magnetic field is produced when an uncharged dielectric is in motion at right angles to the lines of force of a constant electrostatic field had been shown experimentally in 1888 by Röntgen.[47] His experiment consisted in rotating a dielectric disk between the plates of a condenser; a magnetic field was produced, equivalent to that which would be produced by the rotation of the "fictitious charges" on the two faces of the dielectric, i.e., charges which bear the same relation to the dielectric polarization that Poisson's equivalent surfacedensity of magnetism[48] bears to magnetic polarization. If U denote the difference of potential between the opposite coatings of the condenser, and ε the specific inductive capacity of the dielectric, the surface-density of electric charge on the coatings is proportional to ±εU, and the fictitious charge on the surfaces of the dielectric is proportional to ${\displaystyle \mp (\epsilon -1)U}$. It is evident from this that if a plane condenser is charged to a given difference of potential, and is rotated in its own plane, the magnetic field produced is proportional to ε if (as in Rowland's experiment[49]) the coatings are rotated while the dielectric remains at rest, but is in the opposite direction, and is proportional to (ε - 1) if (as in Röntgen's experiment) the dielectric is rotated while the coatings remain at rest. If the coatings and dielectric are rotated together, the magnetic action (being the sum of these) should be independent of ε—a conclusion which was verified later by Eichenwald.[50]

Hitherto we have taken no account of the possible magnetization of the ponderable body. This would modify the equations in the usual manner,[51] so that they finally take the form

${\displaystyle {\begin{matrix}{\text{div }}&\mathbf {D} &=&\rho ,\qquad \qquad (I)\\{\text{div }}&\mathbf {B} &=&0,\qquad \qquad (II)\\{\text{curl }}&\mathbf {H} &=&4\pi \mathbf {S} ,\qquad \qquad (III)\\-{\text{curl }}&\mathbf {E} &=&\mathbf {\dot {B}} ,\qquad \qquad (IV),\end{matrix}}}$

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

${\displaystyle \mathbf {S} ={\bar {\rho \mathbf {v} }}+\mathbf {\dot {d}} /4\pi c^{2}}$,

we have

${\displaystyle {\begin{matrix}{\text{div }}\mathbf {S} &=&{\text{div }}{\bar {\rho \mathbf {v} }}&+&(1/4\pi c^{2}){\text{div }}(\partial \mathbf {\bar {d}} /\partial t)\\{\text{div }}\ &=&{\text{div }}{\bar {\rho \mathbf {v} }}&+&\partial {\bar {\rho }}/\partial t\end{matrix}}}$,

which vanishes by virtue of the principle of conservation of electricity. Thus

${\displaystyle {\text{div }}\mathbf {S} =0,\qquad \qquad (V)}$

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

We have now to consider the relation by which the polarization P of dielectrics is determined. If the dielectric is moving with velocity w, the ponderomotive force on unit electric charge moving with it is (as in all theories)[52]

${\displaystyle \mathbf {E} ^{\prime }=\mathbf {E} +[\mathbf {w.B} ].\qquad \qquad (1)}$

In order to connect P with E′, it is necessary to consider the motion of the corpuscles. Let e denote tho charge and m the mass of a corpuscle, (ξ, η, ζ, ) its displacement from its position of equilibrium, k2(ξ, η, ζ, ) the restitutive force which retains it in the vicinity of this point; then the equations of motion of the corpuscle are

${\displaystyle m{\ddot {\xi }}+k^{2}\xi =eE_{x}^{\prime }}$

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

${\displaystyle \xi =Ae^{nt{\sqrt {-1}}},\qquad E_{x}^{\prime }=E_{0}e^{nt{\sqrt {-1}}}}$

Substituting these values in the equations of motion, we obtain

${\displaystyle A(k^{2}-mn^{2})=eE_{0},\qquad {\text{and therefore}}\qquad \xi (k^{2}-mn^{2})=eE_{x}^{\prime }}$.

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

${\displaystyle \mathbf {P} =Ne(\xi ,\eta ,\zeta )=Ne^{2}\mathbf {E} ^{\prime }/(k^{2}-mn^{2})}$.

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

${\displaystyle =(1/4\pi c^{2})\mathbf {E} +\mathbf {P} =(1/4\pi c^{2})\mathbf {E} +Ne^{2}\mathbf {E} /(k^{2}-mn^{2})}$.

But, as we have seen[53] D bears to E the ratio μ2/4πc2, where μ denotes the refractive index of the dielectric; and therefore the refractive index is determined in terms of the frequency by the equation

${\displaystyle \mu ^{2}=1+4\pi e^{2}c^{2}N/(k^{2}-mn^{2})}$.

This formula is equivalent to that which Maxwell and Sellmeier[54] had derived from the elastic-solid theory. Though superficially different, the derivations are alike in their essential feature, which is the assumption that the molecules of the dielectric contain systems which possess free periods of vibration, and which respond to the oscillations of the incident light. The formula may be derived on electromagnetic principles without any explicit reference to electrons; all that is necessary is to assume that the dielectric polarization has a free period of vibration.[55]

When the luminous vibrations are very slow, so that n is small, μ2 reduces to the dielectric constant ε[56]; so that the theory of Lorentz leads to the expression

${\displaystyle \epsilon =1+4\pi Ne^{2}c^{2}/k^{2}}$

for the specific inductive capacity in terms of the number and circumstances of the electrons.[57]

Returning now to the case in which the dielectric is supposed to be in motion, the equation for the polarization may be written

${\displaystyle 4\pi c^{2}\mathbf {P} =(\mu ^{2}-1)\mathbf {E} ^{\prime };\qquad \qquad (2)}$

from this equation, Fresnel's formula for the velocity of light in a moving dielectric may be deduced. For, let the axis of z be taken parallel to the direction of motion of the dielectric, which is supposed to be also the direction of propagation of the light; and, considering a plane-polarized wave, take the axis of x parallel to the electric vector, so that the magnetic vector must be parallel to the axis of y. Then equation (III) above becomes

${\displaystyle -\partial H_{y}/\partial z=4\pi \mathbf {\dot {D}} _{x}+4\pi w\partial P_{x}/\partial z}$;

equation (IV) becomes (assuming B equal to II, as is always the case in optics),

${\displaystyle -\partial E_{x}/\partial z={\dot {H}}_{y}}$.

The equation which defines the electric induction gives

${\displaystyle D_{x}=(1/4\pi c^{2})E_{x}+P_{x}}$;

and equations (1) and (2) give

${\displaystyle 4\pi c^{2}P_{x}=(\mu ^{2}-1)(E_{x}-wH_{y})}$.

Eliminating D., Px, and Ily, we have

${\displaystyle c^{2}{\frac {\partial ^{2}E_{x}}{\partial z_{2}}}={\frac {\partial ^{2}E_{x}}{\partial t^{2}}}+\left(\mu ^{2}-1\right)\left({\frac {\partial }{\partial t}}+w{\frac {\partial }{\partial z}}\right)^{2}E_{x}}$;

or, neglecting w/cº,

${\displaystyle {\frac {\partial ^{2}E_{x}}{\partial z^{2}}}={\frac {\mu ^{2}}{c^{2}}}{\frac {\partial ^{2}E_{x}}{\partial t^{2}}}+{\frac {}{c^{2}}}{\frac {\partial ^{2}E_{x}}{\partial t\partial z}}}$.[58]

Substituting ${\displaystyle E_{x}=e^{n(t-z/V){\sqrt {-1}}}}$, so that V denotes the velocity of light in the moving dielectric with respect to the fixed aether we have

${\displaystyle c^{2}=\mu ^{2}V^{2}-2w(\mu ^{2}-1)V}$,

or (neglecting w2/c2)

${\displaystyle V={\frac {c}{\mu }}+{\frac {\mu ^{2}-1}{\mu ^{2}}}w}$,

which is the formula of Fresnel.[59] The hypothesis of Fresnel, that a ponderable body in motion carries with it the excess of aether which it contains as compared with space free from matter, is thus seen to be transformed in Lorentz' theory into the supposition that the polarized molecules of the dielectric, like so many small condensers, increase the dielectric constant, and that it is (so to speak) this augmentation of the dielectric constant which travels with the moving matter. One evident objection to Fresnel's theory, namely, that it required the relative velocity of aether and matter to be different for light of different colours, is thus removed; for the theory of Lorentz only requires that the dielectric constant should have different values for light of different colours, and of this a satisfactory explanation is provided by the theory of dispersion.

The correctness of Lorentz' hypothesis, as opposed to that of Hertz (in which the whole of the contained aether was supposed to be transported with the moving body), was afterwards confirmed by various experiments. In 1901 R. Blondlot[60] drove a current of air through a magnetic field, at right angles to the lines of magnetic force. The air-current was made to pass between the faces of a condenser, which were connected by a wire, so as to be at the same potential. An electromotive force E′ would be produced in the air by its motion in the magnetic field; and, according to the theory of Hertz, this should produce an electric induction D of amount (ε/4πc2)E′ (where ε denotes the specific inductive capacity of the air, which is practically unity); so that, according to Hertz, the faces of the condenser should become charged. According to Lorentz theory, on the other hand, the electric induction D is determined by the equation

${\displaystyle 4\pi c^{2}\mathbf {D} =\mathbf {E} +(\epsilon -1)\mathbf {E} ^{\prime }}$

where E denotes the electric force on a charge at rest, which is zero in the present case. Thus, according to Lorentz' theory, the charges on the faces would have only (ε - 1)/ε of the values which they would have in Hertz' theory; that is, they would be practically zero. The result of Blondlot's experiment was in favour of the theory of Lorentz.

An experiment of a similar character was performed in 1905 by H. A. Wilson.[61] In this, the space between the inner and outer coatings of a cylindrical condenser was filled with the dielectric ebonite. When the coatings of such a condenser are maintained at a definite difference of potential, charges are induced on thein; and if the condenser be rotated on its axis in a magnetic field whose lines of force are parallel to the axis, these charges will be altered, owing to the additional polarization which is produced in the dielectric molecules by their motion in the magnetic field. As before, the value of the additional charge according to the theory of Lorentz is (ε - 1)/ε times its value as calculated by the theory of Hertz. The result of Wilson's experiments was, like that of Blondlot's, in favour of Lorentz.

The reconciliation of the electromagnetic theory with Fresnel's law of the propagation of light in moving bodies was a distinct advance. But the theory of the motionless aether was hampered by one difficulty: it was, in its original form, incompetent to explain the negative result of the experiment of Michelson and Morley.[62] The adjustment of theory to observation in this particular was achieved by means of a remarkable hypothesis which must now be introduced.

In the issue of "Nature" for June 16th, 1892,[63] Lodge mentioned that Fitz Gerald had communicated to him a new suggestion for overcoming the difficulty. This was, to suppose that the dimensions of material bodies are slightly altered when they are in motion relative to the aether. Five months afterwards, this hypothesis of Fitz Gerald's was adopted by Lorentz, in a communication to the Amsterdam Academy;[64]after which it won favour in a gradually widening circle, until eventually it came to be generally taken as the basis of all theoretical investigations on the notion of ponderable bodies through the aether.

Let us first see how it explains Michelson's result. On the supposition that the aether is motionless, one of the two portions into which the original beam of light is divided should accomplish its journey in a time less than the other by w2l/c2, where w denotes the velocity of the earth, c the velocity of light, and l the length of each arm. This would be exactly compensated if the arm which is pointed in the direction of the terrestrial motion were shorter than the other by an amount w2l/2c2; as would be the case if the linear dimensions of moving bodies were always contracted in the direction of their motion in the ratio of (1 - w2/2c2) to unity. This is FitzGerald's hypothesis of contraction. Since for the earth the ratio w/c is only

${\displaystyle {\frac {30{\text{km./sec.}}}{300,000{\text{km./sec.}}}},}$

the fraction w2/c2 is only one hundred-millionth.

Several further contributions to the theory of electrons in motionless aether were made in a short treatise[65] which was published by Lorentz in 1895. One of these related to the explanation of an experimental result obtained some years previously by Th, des Coudres,[66] of Leipzig. Des Coudres had observed the mutual inductance of coils in different circumstances of inclination of their common axis to the direction of the earth's motion, but had been unable to detect any effect depending on the orientation. Lorentz now showed that this could be explained by considerations similar to those which Budde and FitzGerald[67] had advanced in a similar case; a conductor carrying a constant electric current and moving with the earth would exert a force on electric charges at relative rest in its vicinity, were it not that this force induces on the surface of the conductor itself a compensating electrostatic charge, whose action annuls the expected effect.

The most satisfactory method of discussing the influence of the terrestrial motion on electrical phenomena is to transform the fundamental equations of the aether and electrons to axes moving with the earth. Taking the axis of x parallel to the direction of the earth's motion, and denoting the velocity of the earth by w, we write

${\displaystyle x=x_{1}+wt,\qquad y=y_{1},\qquad z=z_{1}}$,

so that (x1, y1, z1) denote coordinates referred to axes moving with the earth. Lorentz completed the change of coordinates by introducing in place of the variable t a "local time" t1 defined by the equation

${\displaystyle t=t_{1}+wx_{1}/c^{2}}$.

It is also necessary to introduce, in place of d and h, the electric and magnetic forces relative to the moving axes: these are[68]

${\displaystyle \mathbf {d} _{1}=\mathbf {d} +[\mathbf {w.h} ]}$

${\displaystyle \mathbf {h} _{1}=\mathbf {h} +(1/c^{2})[\mathbf {d.w} ]}$;

and in place of the velocity v of an electron referred to the original fixed axes, we must introduce its velocity v1, relative to the moving axes, which is given by the equation

${\displaystyle \mathbf {v} _{1}=\mathbf {v} -\mathbf {w} }$.

The fundamental equations of the aether and electrons, referred to the original axes, are

${\displaystyle {\begin{matrix}{\text{div }}\mathbf {d} &=&4\pi c^{2}\rho ,\qquad \qquad &{\text{curl }}\mathbf {d} &=&-\mathbf {\dot {h}} ,\\{\text{div }}\mathbf {h} &=&0,\qquad \qquad &{\text{curl }}\mathbf {h} &=&(1/c^{2})\mathbf {\dot {d}} +4\pi \rho \mathbf {v} ,\end{matrix}}}$

${\displaystyle \mathbf {F} =\mathbf {d} +[\mathbf {v.h} ]}$,

where F denotes the ponderomotive force on a particle carrying a unit charge.

By direct transformation from the original to the new variables it is found that, when quantities of order w2/c2 and wv/c2 are neglected, these equations take the form

${\displaystyle {\begin{matrix}{\text{div}}_{1}\ \mathbf {d} _{1}&=&4\pi c^{2}\rho ,\qquad \qquad &{\text{curl}}_{1}\ \mathbf {d} _{1}&=&-\partial \mathbf {h} _{1}/\partial t_{1},&\ \\{\text{div}}_{1}\ \mathbf {h} _{1}&=&0,\qquad \qquad &{\text{curl}}_{1}\ \mathbf {h} _{1}&=&(1/c^{2})\partial \mathbf {d} _{1}/\partial t_{1}&+\ 4\pi \rho \mathbf {v} _{1},\end{matrix}}}$

${\displaystyle \mathbf {F} =\mathbf {d} _{1}+[\mathbf {v_{1}.h_{1}} ]}$,

where div1 d1 stands for

${\displaystyle {\frac {\partial d_{x_{1}}}{\partial x_{1}}}+{\frac {\partial d_{y_{1}}}{\partial y_{1}}}+{\frac {\partial d_{z_{1}}}{\partial z_{1}}}}$.

Since these have the same form as the original equations, it follows that when terms depending on the square of the constant of aberration are neglected, all electrical phenomena may be expressed with reference to axes moving with the earth by the same equations as if the axes were at rest relative to the aether.

In the last chapter of the Versuch Lorentz discussed those experimental results which were as yet unexplained by the theory of the motionless aether. That the terrestrial motion exerts no influence on the rotation of the plane of polarization in quartz[69] might be explained by supposing that two independent effects, which are both due to the earth's motion, cancel each other; but Lorentz left the question undecided. Five years later Larmor[70] criticized this investigation, and arrived at the conclusion that there should be no first-order effect; but Lorentz[71] afterwards maintained his position against Larmor's criticism.

Although the physical conceptions of Lorentz had from the beginning included that of atomic electric charges, the analytical equations had hitherto involved ρ, the volume-density of electric charge; that is, they had been conformed to the hypothesis of a continuous distribution of electricity in space. It might hastily be supposed that in order to obtain an analytical theory of electrons, nothing more would be required than to modify the formulae by writing e (the charge of an electron) in place of ρdxdydz. That this is not the case was shown[72] a few years after the publication of the Versuch.

Consider, for example, the formula for the scalar potential at any point in the aether,

${\displaystyle \phi =\textstyle \iiint ({\bar {\rho }}^{\prime }/r)\ dx^{\prime }\ dy^{\prime }\ dz^{\prime }}$,

where the bar indicates that the quantity underneath it is to have its retarded value.[73]

This integral, in which the integration is extended over all elements of space, must be transformed before the integration can be taken to extend over moving elements of charge. Let de′ denote the sum of the electric charges which are accounted for under the heading of the volume-element dx′dy′dz′ in the above integral. This quantity de′ is not identical with ${\displaystyle {\bar {\rho }}^{\prime }dx^{\prime }dy^{\prime }dz^{\prime }}$. For, to take the simplest case, suppose that it is required to compute the value of the potential-function for the origin at the time t, and that the charge is receding from the origin along the axis of x with velocity u. The charge which is to be ascribed to any position x is the charge which occupies that position at the instant t - x/c; so that when the reckoning is made according to intervals of space, it is necessary to reckon within a segment (x2x1) not the electricity which at any one instant occupies that segment, but the electricity which at the instant (t - x3/c) occupies a segment (x2x′1), where x′1 denotes the point from which the electricity streams to x1, in the interval between the instants (t - x2/c) and (t - x1/c). We have evidently

${\displaystyle x_{1}-x_{1}^{\prime }=u(x_{2}-x_{1})/c,\qquad {\text{ or }}\qquad x_{2}-x_{1}^{\prime }=(x_{2}-x_{1})(1+u/c)}$.

For this case we should therefore have

${\displaystyle de^{\prime }={\frac {x_{2}-x_{1}^{\prime }}{x_{2}-x_{1}}}{\bar {\rho }}^{\prime }\ dx^{\prime }dy^{\prime }dz^{\prime }=\left(1+{\frac {u}{c}}\right){\bar {\rho }}^{\prime }\ dx^{\prime }dy^{\prime }dz^{\prime }}$.

In the general case, it is only necessary to replace u by the component of velocity of the electric charge in the direction of the radius vector from the point at which the potential is to be computed. This component may be written ${\displaystyle v\cos({\hat {v.r}})}$, where r is measured positively from the point in question to the charge, and v denotes the velocity of the charge. Thus

${\displaystyle cde^{\prime }=\{c+v\cos({\hat {v.r}})\}{\bar {\rho }}^{\prime }\ dx^{\prime }\ dy^{\prime }\ dz^{\prime }}$,

and therefore

${\displaystyle \phi =c\int {\frac {de^{\prime }}{c{\bar {r}}+(\mathbf {{\bar {r}}.{\bar {v}}} )}}}$,

where the integration is extended over all the charges in the field, and the bars over the letters imply that the position of the charge considered is that which it occupied at the instant ${\displaystyle t-{\bar {r}}/c}$. In the same way the vector-potential may be shown to have the value

${\displaystyle \mathbf {a} =c\int {\frac {\mathbf {\bar {v}} de^{\prime }}{c{\bar {r}}+(\mathbf {{\bar {r}}.{\bar {v}}} )}}}$.

Meanwhile the unsettled problem of the relative motion of earth and aether was provoking a fresh series of experimental investigations. The most interesting of these was due to FitzGerald,[74] who shortly before his death in February, 1901, commenced to examine the phenomena manifested by a charged electrical condenser, as it is carried through space in consequence of the terrestrial motion. On the assumption that a moving charge develops a magnetic field, there will be associated with the condenser a magnetic force at right angles to the lines of electric force and to the direction of the motion: magnetic energy must therefore be stored in the medium, when the plane of the condenser includes the direction of the drift; but when the plane of the condenser is at right angles to the terrestrial motion, the effects of the opposite charges neutralize each other. FitzGerald's original idea was that, in order to supply the magnetic energy, there must be a mechanical drag on the condenser at the moment of charging, similar to that which would be produced if the mass of a body at the surface of the earth were suddenly to become greater. Moreover, it was conjectured that the condenser, when freely suspended, would tend to move so as to assume the longitudinal orientation, which is that of maximum kinetic energy[75]: the transverse position would therefore be one of unstable equilibrium.

For both effects a search was made by FitzGerald's pupil Trouton:[76] in the experiments designed to observe the turning couple, a condenser was suspended in a vertical plane by a fine wire, and charged. If the plane of the condenser were that of the meridian, about noon there should be no couple tending to alter the orientation, because the drift of aether due to the earth's motion would be at right angles to this plane; at any other hour, a couple should act. The effect to be detected was extremely small; for the magnetic force due to the motion of the charges would be of order w/c, where w denotes the velocity of the earth; so the magnetic energy of the system, which depends on the square of the force, would be of order (w/c)2; and the couple, which depends on the derivate of this with respect to the azimuth, would therefore be likewise of the second order in (w/c).

No couple could be detected. As the energy of the magnetic field must be derived from some source, there seems to be no escape from the conclusion that the electrostatic energy of a charged condenser is diminished by the fraction (w/c)2 of its amount when the condenser is moving with velocity w at right angles to its lines of electrostatic force. To explain this diminution, it is necessary to admit FitzGerald's hypothesis of contraction. The negative result of the experiment may be taken to indicate[77] that the kinetic potential of the system, when the FitzGerald contraction is taken into account as a constraint, is independent of the orientation of the plates with respect to the direction of the terrestrial motion.

It may be remarked that the existence of the couple, had it been observed, would have demonstrated the possibility of drawing on the energy of the earth's motion for purposes of terrestrial utility.

The FitzGerald contraction of matter as it moves through the aether might conceivably be supposed to affect in some way the optical properties of the moving natter; for instance, transparent substances might become doubly refracting. Experiments designed to test this supposition were performed by Lord Rayleigh in 1902,[78] and by D. B. Brace in 1904[79]; but no double refraction comparable with the proportion (w/c)2 of the single refraction could be detected. The FitzGerald contraction of a material body cannot therefore be of the same nature as the contraction which would be produced in the body by pressure, but must be accompanied by such concomitant changes in the relations of the molecules to the aether that an isotropic substance does not lose its simply refracting character.

By this time, indeed, the hypothesis of contraction, which originally had no direct connexion with electric theory, had assumed a new aspect. Lorentz, as we have seen,[80] had obtained the equations of a moving electric system by applying a transformation to the fundamental equations of the aether. In the original form of this transformation, quantities of higher order than the first in w/c were neglected. But in 1900 Larmor[81] extended the analysis so as to include small quantities of the second order, and thereby discovered a remarkable connexion between the equations of transformation and the equations which represent FitzGerald's contraction. After this Lorentz[82] went further still, and obtained the transformation in a form which is exact to all orders of the small quantity w/c. In this form we shall now consider it.

The fundamental equations of the aether are

${\displaystyle {\begin{matrix}{\text{div }}\mathbf {d} &=&4\pi c^{2}\rho ,\qquad \qquad &{\text{curl }}\mathbf {d} &=&-\mathbf {\dot {h}} ,\\{\text{div }}\mathbf {h} &=&0,\qquad \qquad &{\text{curl }}\mathbf {h} &=&(1/c^{2})\mathbf {\dot {d}} +4\pi \rho \mathbf {v} ,\end{matrix}}}$

It is desired to find a transformation from the variables x, y, z, t, ρ, d, h, v, to new variables x1, y1, z1, t1, ρ1, d1, h1, v1, such that the equations in terms of these new variables may take the same form as the original equations, namely:

${\displaystyle {\begin{matrix}{\text{div}}_{1}\ \mathbf {d} _{1}&=&4\pi c^{2}\rho _{1},\qquad \qquad &{\text{curl}}_{1}\ \mathbf {d} _{1}&=&-\partial \mathbf {h} _{1}/\partial t_{1},&\ \\{\text{div}}_{1}\ \mathbf {h} _{1}&=&0,\qquad \qquad &{\text{curl}}_{1}\ \mathbf {h} _{1}&=&(1/c^{2})\partial \mathbf {d} _{1}/\partial t_{1}&+\ 4\pi \rho _{1}\mathbf {v} _{1},\end{matrix}}}$

Evidently one particular class of such transformations is that which corresponds to rotations of the axes of coordinates about the origin: these may be described as the linear homogeneous transformations of determinant unity which transform the expression (x2 + y2 + z2) into itself.

These particular transformations are, however, of little interest, since they do not change the variable t. But in place of them consider the more general class formed of all those linear homogeneous transformations of determinant unity in the variables x, y, z, ct, which transform the expression (x2 + y2 + z2 - c2t2)[errata 2] into itself: we shall show that these transformations have the property of transforming the differential equations into themselves.

All transformations of this class may be obtained by the combination and repetition (with interchange of letters) of one of them, in which two of the variables—say, y and z—are unchanged. The equations of this typical transformation may easily be derived by considering that the equation of the rectangular hyperbola

${\displaystyle x^{2}-(ct)^{2}=1}$

(in the plane of the variables x, ct) is unaltered when any pair of conjugate diameters are taken as new axes, and a new unit of length is taken proportional to the length of either of these diameters. The equations of transformation are thus found to be

${\displaystyle {\begin{matrix}x&=&x_{1}\cosh a&+&ct_{1}\sinh a,&\qquad &y&=&y_{1},\\t&=&t_{1}\cosh a&+&(x_{1}/c)\sinh a,&\qquad &z&=&z_{1},\end{matrix}}}$

where a denotes a constant. The simpler equations previously given by Lorentz[83] may evidently be derived from these by writing w/c for tanh a, and neglecting powers of w/c above the first. By an obvious extension of the equations given by Lorentz for the electric and magnetic forces, it is seen that the corresponding equations in the present transformation are

 ${\displaystyle {\begin{cases}d_{x}&=&d_{x_{1}},\\d_{y}&=&d_{y_{1}}\cosh a+ch_{z_{1}}\sinh a,\\d_{z}&=&d_{z_{1}}\cosh a-ch_{y_{1}}\sinh a,\end{cases}}}$ ${\displaystyle {\begin{cases}h_{x}&=&h_{x_{1}},\\h_{y}&=&h_{y_{1}}\cosh a-(1/c)d_{z_{1}}\sinh a,\\h_{z}&=&h_{z_{1}}\cosh a+(1/c)d_{y_{1}}\sinh a,\end{cases}}}$

The connexion between ρ and ρ1 may be obtained in the following way. It is assumed that if a charge e is attached to a particle which occupies the position (ξ, η, ζ) at the instant t, an equal charge will be attached to the corresponding point (ξ1, η1, ζ1) at the corresponding instant t1, in the transformed system; so that a charge e′ attached to an adjacent particle (ξ + Δξ, η + Δη, ζ + Δζ) at the instant t will give rise in the derived system to a charge e′ at the place

 ${\displaystyle \left(\xi _{1}+{\frac {\partial \xi _{1}}{\partial \xi }}\Delta \xi +{\frac {\partial \xi _{1}}{\partial \eta }}\Delta \eta +{\frac {\partial \xi _{1}}{\partial \zeta }}\Delta \zeta ,\qquad \eta _{1}+{\frac {\partial \eta _{1}}{\partial \xi }}\Delta \xi +{\frac {\partial \eta _{1}}{\partial \eta }}\Delta \eta +{\frac {\partial \eta _{1}}{\partial \zeta }}\Delta \zeta ,\right.}$ ${\displaystyle \left.\zeta _{1}+{\frac {\partial \zeta _{1}}{\partial \xi }}\Delta \xi +{\frac {\partial \zeta _{1}}{\partial \eta }}\Delta \eta +{\frac {\partial \zeta _{1}}{\partial \zeta }}\Delta \zeta \right)}$

at the instant

${\displaystyle \left(t_{1}+{\frac {\partial t_{1}}{\partial \xi }}\Delta \xi +{\frac {\partial t_{1}}{\partial \eta }}\Delta \eta +{\frac {\partial t_{1}}{\partial \zeta }}\Delta \zeta \right)}$;

that is to say, at the place

${\displaystyle (\xi _{1}+\Delta \xi \cosh a,\qquad \eta _{1}+\Delta \eta ,\qquad \zeta _{1}+\Delta \zeta )}$

at the instant {{Wikimath|(t1 - sinh a. Δξ/c). Thus at the instant t1, this charge will occupy the position

 ${\displaystyle (\xi _{1}+\Delta \xi \cosh a+\sinh a.\Delta \xi .v_{x_{1}}/c,\qquad \eta _{1}+\Delta \eta +\sinh a.\Delta \xi .v_{y_{1}}/c\qquad }$ ${\displaystyle \zeta _{1}+\Delta \zeta +\sinh a.\Delta \xi .v_{a_{1}}/c)}$

The charges corresponding to those in the original system which were at the instant t contained in a volume ΔξΔηΔζ will therefore in the derived system at the instant t1, occupy a volume

${\displaystyle \left|{\begin{matrix}\cosh a&+&\sinh a.v_{x_{1}}/c.&0&0\\\ &\ &\sinh a.v_{y_{1}}/c&1&0\\\ &\ &\sinh a.v_{z_{1}}/c&0&1\\\end{matrix}}\right|.\Delta \xi \Delta \eta \Delta \zeta }$

or,

${\displaystyle (\cosh a+\sinh a.v_{x_{1}}/c)\Delta \xi \Delta \eta \Delta \zeta }$.

Thus if ρ1 denote the volume-density of electric charge in the transformed system, we shall have

${\displaystyle \rho _{1}(\cosh a+\sinh a.v_{x_{1}}/c)=\rho }$;

this equation expresses the connexion between ρ1 and ρ. We have moreover

${\displaystyle {\begin{matrix}v_{x}&=&{\frac {{\frac {\partial x}{\partial x_{1}}}v_{x_{1}}+{\frac {\partial x}{\partial y_{1}}}v_{y_{1}}+{\frac {\partial x}{\partial z_{1}}}v_{z_{1}}+{\frac {\partial x}{\partial t_{1}}}}{{\frac {\partial t}{\partial x_{1}}}v_{x_{1}}+{\frac {\partial t}{\partial y_{1}}}v_{y_{1}}+{\frac {\partial t}{\partial z_{1}}}v_{z_{1}}+{\frac {\partial t}{\partial t_{1}}}}}\\\ &=&e\tanh a+{\frac {v_{x_{1}}{\text{sech }}a}{\cosh a+v_{x_{1}}e^{-1}\sinh a}}\end{matrix}}}$

and similarly

${\displaystyle v_{y}={\frac {v_{y_{1}}}{\cosh a+v_{x_{1}}e^{-1}\sinh a}}}$,

and

${\displaystyle v_{x}={\frac {v_{x_{1}}}{\cosh a+v_{x_{1}}e^{-1}\sinh a}}}$.

When the original variables are by direct substitution replaced by the new variables in the differential equations, the latter take the form

${\displaystyle {\begin{matrix}{\text{div}}_{1}\ \mathbf {d} _{1}&=&4\pi c^{2}\rho _{1},\qquad \qquad &{\text{curl}}_{1}\ \mathbf {d} _{1}&=&-\partial \mathbf {h} _{1}/\partial t_{1},&\ \\{\text{div}}_{1}\ \mathbf {h} _{1}&=&0,\qquad \qquad &{\text{curl}}_{1}\ \mathbf {h} _{1}&=&(1/c^{2})\partial \mathbf {d} _{1}/\partial t_{1}&+\ 4\pi \rho _{1}\mathbf {v} _{1},\end{matrix}}}$

that is to say, the fundamental equations of the aether retain their form unaltered, when the variables are subjected to the transformation which has been specified.

We are now in a position to show the connexion of this transformation with FitzGerald's hypothesis of contraction. Suppose that two material particles are moving along the axis of x with velocity w = c tanh α. From the relation

${\displaystyle v_{x}=c\tanh \alpha +{\frac {v_{x_{1}}{\text{sech }}\alpha }{\cosh \alpha +v_{x_{1}}c^{-1}\sinh \alpha }}}$,

it follows that ${\displaystyle v_{x_{1}}}$, is zero for each of the particles, which implies that they are at rest relative to the new axes. Let x1, and x′1 denote their coordinates with respect to this latter system; then the coordinates of one particle at the instant t1, referred to the original axes, will be given by the equations

${\displaystyle x=x_{1}\cosh \alpha +ct_{1}\sinh \alpha ,\qquad t=t_{1}\cosh \alpha +x_{1}c^{-1}\sinh \alpha }$;

and the coordinates of the other particle will be given by

${\displaystyle x^{\prime }=x_{1}^{\prime }\cosh \alpha +ct_{1}\sinh \alpha ,\qquad t^{\prime }=t_{1}\cosh \alpha +x_{1}^{\prime }c^{-1}\sinh \alpha }$;

so that at time t the latter particle will have the coordinate x′′, where

${\displaystyle {\begin{matrix}x^{\prime \prime }&=&x^{\prime }+w(t-t^{\prime })\\\ &=&x_{1}^{\prime }\cosh \alpha +ct_{1}\sinh \alpha +(x-x_{1}^{\prime })\sinh ^{2}\alpha {\text{ sech }}\alpha ,\end{matrix}}}$

which gives

${\displaystyle x^{\prime \prime }-x=(x_{1}^{\prime }-x_{1})(1-w^{2}/c^{2})^{\frac {1}{2}}}$.

This equation shows that the distance between the particles in the system of measurement furnished by the original axes, with reference to which the particles were moving with velocity w, bears the ratio (1 - w2/c2)12:1 to their distance in the system of measurement furnished by the transformed axes, with reference to which the particles are at rest. But according to FitzGerald's hypothesis of contraction, when a material body is in motion relative to the aether, in a direction parallel to the axis of x, its dimensions parallel to this direction contract in precisely this ratio; so that the equation of the body, in terms of the coordinates x1, y1, z1, which move with it, is unaltered. Thus the hypothesis of FitzGerald may be expressed by the statement that the equations of the figures of ponderable bodies are covariant with respect to those transformations for which the fundamental equations of the aether are covariant.

The covariance holds with respect to all linear homogeneous transformations in the variables (x, y, z, t), of determinant unity, which transform the expression (x2 + y2 + z2 - c2t2) into itself. This group comprises an infinite number of transformations; so that there are an infinite number of sets of variables resembling (x1, y1, z1, t1), of which any one set (xr, yr, zr, tr) can be derived from any other set (xs, ys, zs, ts) by a transformation of the group; among the sets wo must of course include the original set of coordinates (x, y, z, t). But hitherto we have proceeded on the assumption that the original set (x, y, z, t) is entitled to a primacy among all the other sets, since the axes (x, y, z, t) have been supposed to possess the special property of having no motion relative to the aether, and the time represented by the variable t has been understood to be a definite physical quantity. The other sets of variables (xr, yr, zr, tr) have been regarded merely as symbols convenient for use in problems relating to moving bodies, but not as corresponding to physical entities in the same degree as (x, y, z, ct). We must now inquire whether this view is justified.

The question amounts to asking whether absolute position in space, or at any rate absolute fixity relative to the aether, is something which can be brought within the bounds of human knowledge.

It is well known that the science of dynamics, as founded on Newton's laws of motion, does not supply any criterion by which rest may be distinguished from uniform motion; for if the laws of motion are applicable when the position of bodies is referred to any particular set of axes, they will be equally applicable when position is referred to any other set of axes which have a uniform motion of translation relative to these.

The older theories of electrostatics, magnetism, and electrodynamics, which are based on the conception of action at a distance, are concerned only with relative configurations and motions, and are therefore useless in the search for a basis of absolute reckoning.

But the existence of an aether, which is postulated in the undulatory theory of light, seems at first sight to involve the conceptions of rest and motion relative to it, and thus to afford a means of specifying absolute position. Suppose, for instance, that a disturbance is generated at any point in free aether; this disturbance will spread outwards in the form of a sphere; and the centre of this sphere will for all subsequent time occupy an unchanged position relative to the aether. In this way, or in many other ways, we might hope to determine, by electrical or optical experiments, the velocity of the earth relative to the aether.

The failure of such experiments as had been tried led FitzGerald[84] to suggest that the dimensions of material bodies undergo contraction when the bodies are in motion relative to the aether. By the transformation of Lorentz and Larmor, as we have seen, this hypothesis came to be expressed in a new form; namely that the equation of the figure of the body, referred to a frame of reference moving with it, is always the same, but that frames of reference which are in notion relative to each other are based on different standards of length and time. This way of regarding the matter brings into prominence the fundamental questions involved. Before speaking of lengths and velocities, it is necessary to examine the nature of systems of measurement of space and time.

Of the events with which Natural Philosophy is concerned, each is perceived to happen at some definite location at some definite moment. When a material object has been observed to occupy a certain position at a certain instant, the same object may again be observed at a subsequent instant; but it is impossible to determine whether the object is or is not in the same position, since there is no obvious means of preserving the identity of any location from one moment to another. The physicist, however, finds it convenient to construct a framework of axes in space and time for the purpose of fitting his experiences into an orderly arrangement; and the question at issue is whether experience furnishes the means of determining a framework completely and uniquely by absolute properties, or whether the selection inevitably rests on arbitrary choice and accidental circumstance.

In attempting to answer this question, it may first be observed that the choice is always made so as to simplify the description of natural phenomena as much as possible; thus, the variable which is to measure time is so chosen that its increment in the interval between any two consecutive beats of a pendulum is the same as its increment in the interval between any other two consecutive beats. If the selection of the four variables (x, y, z, t) is well made, it should be possible to express the laws of nature by statements of a simple character, e.g., that a body isolated from the influence of external agents moves through equal intervals of space in equal intervals of time.

Accepting, then, the principle that the framework of axes is to be chosen so as to furnish the simplest possible expression of the natural laws, it becomes of importance to determine which of the natural laws are entitled, by reason of their primary importance, to receive the greatest consideration.

Now many indications point to the probability that the various types of forces which are observed in ponderable bodies—forces of cohesion, of chemical union, and so forth—are ultimately electric in their nature. Such an assumption would have the great advantage of explaining the contraction postulated by Fitz Gerald, since it would represent the contraction as actually produced by the notion. But if this assumption be correct, the theory of electricity and aether is without doubt the fundamental theory of Natural Philosophy, and the framework of space and time should be chosen with a view chiefly to the expression of electrical phenomena. This may most naturally be done by stipulating that the wave-fronts of disturbances generated in free aether shall, in the system of length and time adopted, be accounted spheres whose centres are at the origins of disturbance and whose radii are proportional to the times elapsed since their initiation. Referred to axes of (x, y, z, t) which satisfy these conditions, the fundamental equations of the electric field assume the form which has been taken as the basis of all our theoretical investigations.

Imagine now a distant star which is moving with a uniform velocity w or o tanh a relative to this framework (x, y, z, t). The theorem of transformation shows that there exists another framework (x1, y1, z1, t1), with respect to which the star is at rest, and in which moreover the condition laid down regarding the wave-surface is satisfied. This framework is peculiarly fitted for the representation of the phenomena which happen on the star; whose inhabitants would therefore naturally adopt it as their system of space and time. Beings, on the other hand, who dwell on a body which is at rest with respect to the axes (x, y, z, t) would prefer to use the latter system, and from the point of view of the universe at large, either of these systems is as good as the other. The equations of motion of the aether are the same with respect to both sets of coordinates, and therefore neither can claim to possess the only property which could confer a primacy—namely, an absolute relation to the aether.[85]

To sum up, we may say that the phenomena whose study is the object of Natural Philosophy take place each at a definite location at a definite moment; the whole constituting a four-dimensional world of space and time. To construct a set of axes of space and time is equivalent to projecting this four-dimensional world into a three-dimensional world of space and a one-dimensional world of time, and this projection may be performed in an infinite number of ways, each of which is distinguished from the others only by characteristics merely arbitrary and accidental.[86]

In order to represent natural phenomena without introducing this contingent element, it would be necessary to abandon the customary three-dimensional system of coordinates, and to operate in four dimensions. Analysis of this kind has been devised, and has been applied to the theory of the aether; but its development belongs to the twentieth century, and consequently falls outside the scope of the present work.

From what has been said, it will be evident that, in the closing years of the nineteenth century, electrical investigation was chiefly concerned with systems in motion. The theory of electrons was, however, applied with success in other directions, and notably to the explanation of a new experimental discovery.

The last recorded observation of Faraday[87] was an attempt to detect changes in the period, or in the state of polarization, of the light emitted by a sodium flame, when the flame was placed in a strong magnetic field. No result was obtained; but the conviction that an effect of this nature remained to be discovered was felt by many of his successors. Tait[88] examined the influence of a magnetic field on the selective absorption of light; impelled thereto, as he explained, by theoretical considerations. For from the phenomenon of magnetic rotation it may be inferred[89] that rays circularly polarized in opposite senses are propagated with different velocities in the magnetized medium; and therefore if only those rays are absorbed which have a certain definite wave-length in the medium, the period of the ray absorbed from a beam of circularly polarized white light will not be the same when the polarization is right-handed as when it is left-handed. "Thus," wrote Tait, "what was originally a single dark absorption-line might become a double line."

The effect anticipated under different forms by Faraday and Tait was discovered, towards the end of 1896, by I. Zeeman.[90] Repeating Faraday's procedure, he placed a sodium fame between the poles of an electromagnet, and observed a widening of the D-lines in the spectrum when the magnetizing current was applied.

A theoretical explanation of the phenomenon was immediately furnished to Zeeman by Lorentz.[91] The radiation is supposed to be emitted by electrons which describe orbits. within the sodium atoms. If e denote the charge of an electron of mass m, the ponderomotive force which acts on it by virtue of the external magnetic field is e ['i.K], where K denotes the magnetic force and r denotes the displacement of the electron from its position of equilibrium; and therefore, if the force which restrains the electron in its orbit be κ2r, the equation of motion of the electron is

${\displaystyle m{\ddot {\mathbf {r} }}+\kappa ^{2}\mathbf {r} =e[\mathbf {{\dot {r}}.K} ]}$.

The motion of the electron may (as is shown in treatises on dynamics) be represented by the superposition of certain particular solutions called principal oscillations, whose distinguishing property is that they are periodic in the time. In order to determine the principal oscillations, we write ${\displaystyle r_{0}e^{nt{\sqrt {-1}}}}$ for r, where r0, denotes a vector which is independent of the time, and n denotes the frequency of the principal oscillation: substituting in the equation, we have

${\displaystyle (k^{2}-mn^{2})\mathbf {r} _{0}=en{\sqrt {-1}}[\mathbf {r_{0}.K} ]}$.

This equation may be satisfied either (1) if r0, is parallel to K, in which case it reduces to

${\displaystyle \kappa ^{2}-mn^{2}=0}$,

so that n has the value κm-12, or (2) if r0 is at right angles to K, in which case by squaring both sides of the equation we obtain the result

${\displaystyle (k^{2}-mn^{2})^{2}=e^{2}n^{2}K^{2}}$,

which gives for n the approximate values κm-12 ± eK/2m.

When there is no external magnetic field, so that K is zero, the three values of n which have been obtained all reduce to κm-12, which represents the frequency of vibration of the emitted light before the magnetic field is applied. When the field is applied, this single frequency is replaced by the three frequencies. κm-12, κm-12 + eK/2m, κm-12 - eK/2m; that is to say, the single line in the spectrum is replaced by three lines close together. The apparatus used by Zeeman in his earliest experiments was not of sufficient power to exhibit this triplication distinctly, and the effect was therefore described at first as a widening of the spectral lines.[92]

We have seen above that the principal oscillation of the electron corresponding to the frequency κm-12 is performed in a direction parallel to the magnetic force K. It will therefore give rise to radiation resembling that of a Hertzian vibrator, and the electric vector of the radiation will be parallel to the lines of force of the external magnetic field. It follows that when the light received in the spectroscope is that which has been emitted in a direction at right angles to the magnetic field, this constituent (which is represented by the middle line of the triplet in the spectrum) will appear polarized in a plane at right angles to the field; but when the light received in the spectroscope is that which has been emitted in the direction of the magnetic force, this constituent will be absent.

We have also seen that the principal oscillations of the electron corresponding to the frequencies κm-12 ± eK/2m are performed in a plane at right angles to the magnetic field K. In order to determine the nature of these two principal oscillations, we observe that it is possible for the electron to describe a circular orbit in this plane, if the radius of the orbit be suitably chosen; for in a circular motion the forces κ2r and ${\displaystyle e[\mathbf {{\dot {r}}.K} ]}$ would be directed towards the centre of the circle; and it would therefore be necessary only to adjust the radius so that these furnish the exact amount of centripetal force required. Such a motion, being periodic, would be a principal oscillation. Moreover, since the force ${\displaystyle e[\mathbf {{\dot {r}}.K} ]}$ changes sign when the sense of the movement in the circle is reversed, it is evident that there are two such circular orbits, corresponding to the two senses in which the electron may circulate; these must, therefore, be no other than the two principal oscillations of frequencies κm-12 ± eK/2m. When the light received in the spectroscope is that which has been emitted in a direction at right angles to the external magnetic field, the circles are seen edgewise, and the light appears polarized in a plane parallel to the field; but when the light examined is that which has been emitted in a direction parallel to the external magnetic force, the radiations of frequencies κm-12 ± eK/2m are seen to be circularly polarized in opposite senses. All these theoretical .conclusions have been verified by observation.

It was found by Cornu[93] and by C. G. W. König[94] that the more refrangible component (i.e., the one whose period is shorter than that of the original radiation) has its circular vibration in the same sense as the current in the electromagnet. From this. it may be inferred that the vibration must be due to a resinously charged electron; for let the magnetizing current and the electron be supposed to circulate round the axis of z in the direction in which a right-handed screw must turn in order to progress along the positive direction of the axis of z; then the magnetic force is directed positively along the axis of z, and, in order that the force on the electron may be directed inward to the axis of z (so as to shorten the period), the charge on the electron must be negative.

The value of e/m for this negative electron may be determined by measurement of the separation between the components of the triplet in a magnetic field of known strength; for, as we have seen, the difference of the frequencies of the outer components is eK/m. The values of e/m thus determined agree well with the estimations[95] of e/m for the corpuscles of cathode rays.

The phenomenon discovered by Zeeman is closely related to the magnetic rotation of the plane of polarization of light.[96] Both effects may be explained by supposing that the molecules of material bodies contain electric systems which possess natural periods of vibration, the simplest example of such a system being an electron which is attracted to a fixed centre with a force proportional to the distance. Zeeman's effect represents the influence of al external magnetic field on the free oscillations of these electric systems, while Faraday's effect represents the influence of the external magnetic field on the forced oscillations which the systems perform under the stimulus of incident light. The latter phenomenon may be analysed without difficulty on these principles, the equation of motion of one of the electrons being taken in the form

${\displaystyle m\mathbf {\ddot {r}} +\kappa ^{2}\mathbf {r} =e\mathbf {E} +e[\mathbf {{\dot {r}}.H} ]}$

where m denotes the mass and e the charge of the electron, r its distance from the centre of force, κ2r the restitutive force, E and H the electric and magnetic forces. When the electron performs forced oscillations under the influence of light of frequency n, this equation becomes

${\displaystyle (\kappa ^{2}-mn^{2})\mathbf {r} =e\mathbf {E} +e[\mathbf {{\dot {r}}.H} ]}$

The influence of the magnetic force on the motion of the electron is small compared with the influence of the electric force, i.e. the second term on the right is small compared with the first term; so in the second term we may replace r by its value as found from the first term, namely, eE/(κ2mn2). The equation thus becomes

${\displaystyle \mathbf {r} ={\frac {e\mathbf {E} }{\kappa ^{2}-mn^{2}}}+{\frac {e^{2}}{(\kappa ^{2}-mn^{2})^{2}}}[\mathbf {{\dot {E}}.H} ]}$.

If P denote[97] the electric moment per unit volume, we have

P = er × the number of such systems in unit volume of the medium;

so P must be of the form

${\displaystyle {\frac {\epsilon -1}{4\pi c^{2}}}\mathbf {E} +\sigma [\mathbf {{\dot {E}}.H} ]}$,

where ε evidently represents the dielectric constant of the medium, and σ is the coefficient which measures the magnetic rotatory power. In the magneto-optic term we may replace H by K, the external magnetic force, since this is large compared with the magnetic force of the luminous vibrations. Thus if D denote the electric induction, we have

${\displaystyle \mathbf {D} =\epsilon \mathbf {E} /4\pi c^{2}+[\mathbf {{\dot {E}}.K} ]}$.

Combining this with the usual electromagnetic equations,

${\displaystyle {\begin{matrix}{\text{curl }}\mathbf {H} &=&4\pi \mathbf {\dot {D}} ,\\{\text{curl }}\mathbf {E} &=&-\mathbf {\dot {H}} ,\end{matrix}}}$

we have

${\displaystyle -{\text{curl curl }}\mathbf {E} =\epsilon \mathbf {\ddot {E}} /c^{2}+4\pi \sigma [\mathbf {{\overset {...}{E}}.K} ]}$.

When a plane wave of light is propagated through the medium in the direction of the lines of magnetic force, and the axis of x is taken parallel to this direction, the equation gives

${\displaystyle {\begin{cases}{\frac {\partial ^{2}E_{y}}{\partial x^{2}}}&=&{\frac {\epsilon }{c^{2}}}{\frac {\partial ^{2}E_{y}}{\partial t^{2}}}+4\pi \sigma K{\frac {\partial ^{3}E_{z}}{\partial t^{3}}}\\{\frac {\partial ^{2}E_{z}}{\partial x^{2}}}&=&{\frac {\epsilon }{c^{2}}}{\frac {\partial ^{2}E_{z}}{\partial t^{2}}}-4\pi \sigma K{\frac {\partial ^{3}E_{y}}{\partial t^{3}}}\end{cases}}}$

and these equations, as we have seen,[98] are competent to explain the rotation of the plane of polarization.

From the occurrence of the factor (κ2mn2) in the denomi. nator of the expression for the magneto-optic constant σ, it may be inferred that the magnetic rotation will be very large for light whose period is nearly the same as a free period of vibration of the electrons. A large rotation is in fact observed[99] when plane-polarized light, whose frequency differs but little from the frequencies of the D-lines, is passed through sodium vapour in a direction parallel to the lines of magnetic force.

The optical properties of metals may be explained, according to the theory of electrons, by a slight extension of the analysis which applies to the propagation of light in transparent substances, It is, in fact, only necessary to suppose that some of the electrons in metals are free instead of being bound to the molecules: a supposition which may be embodied in the equations by assuming that an electric force E gives rise to a polarization P, where

${\displaystyle \mathbf {E} =\alpha \mathbf {\ddot {P}} +\beta \mathbf {\dot {P}} +\gamma \mathbf {P} }$;

the term in α represents the effect of the inertia of the electrons; the term in β represents their ohmic drift; and the term in γ represents the effect of the restitutive forces where these exist. This equation is to be combined with the customary electromagnetic equations

${\displaystyle {\text{curl }}\mathbf {H} =\mathbf {\dot {E}} /c^{2}+4\pi \mathbf {\dot {P}} ,\qquad -{\text{curl }}\mathbf {E} =\mathbf {\dot {H}} }$.

In discussing the propagation of light through the metal, we may for convenience suppose that the beam is plane-polarized and propagated parallel to the axis of z, the electric vector being parallel to the axis of x. Thus the equations of motion reduce to

${\displaystyle {\begin{cases}{\frac {\partial ^{2}E_{x}}{\partial z^{2}}}&=&{\frac {1}{c^{2}}}{\frac {\partial ^{2}E_{x}}{\partial t^{2}}}+4\pi {\frac {\partial ^{2}P_{x}}{\partial t^{2}}}\\E_{x}&=&\alpha {\frac {\partial ^{2}P_{x}}{\partial t^{2}}}+\beta {\frac {\partial ^{2}P}{\partial t}}+\gamma P_{x}.\end{cases}}}$

For Ex, and Px we may substitute exponential functions of

${\displaystyle n{\sqrt {-1}}(t-z\mu /c)}$,

where n denotes the frequency of the light, and μ the quasi-index of refraction of the metal: the equations then give at once

${\displaystyle (\mu ^{2}-1)(-\alpha n^{2}+\beta n{\sqrt {-1}}+\gamma )=4\pi c^{2}}$.

Writing ${\displaystyle \nu (1-\kappa {\sqrt {-1}})}$ for μ, so that ν is inversely proportional to the velocity of light in the medium, and κ denotes the coefficient of absorption, and equating separately the real and imaginary parts of the equation, we obtain

${\displaystyle {\begin{cases}\nu ^{2}(1-\kappa ^{2})&=&1+{\frac {4\pi c^{2}(\gamma -\alpha n^{2})}{\beta ^{2}n^{2}+(\gamma -\alpha n^{2})^{2}}};\\\nu ^{2}\kappa &=&{\frac {2\pi c^{2}\beta n}{\beta ^{2}n^{2}+(\gamma -\alpha n^{2})^{2}}}\end{cases}}}$

When the wave-length of the light is very large, the inertia represented by the constant α has but little influence, and the equations reduce to those of Maxwell's original theory[100] of the propagation of light in metals. The formulae were experimentally confirmed for this case by the researches of E. Hagen and H, Rubens[101] with infra-red light; a relation being thus established between the ohmic conductivity of a metal and its optical properties with respect to light of great wavelength.

When, however, the luminous vibrations are performed more rapidly, the effect of the inertia becomes predominant; and if the constants of the metal are such that, for a certain range of values of n, ν2κ is small, while ν2(1 - κ2) is negative, it is evident that, for this range of values of n, ν will be small and κ large, i.e., the properties of the metal will approach those of ideal silver.[102] Finally, for indefinitely great values of n, ν2κ is small and ν2(1 - κ2) is nearly unity, so that ν tends to unity and κ to zero: an approximation to these conditions is realized in the X-rays.[103]

In the last years of the nineteenth century, attempts were made to form more definite conceptions regarding the behaviour of electrons within metals. It will be remembered that the original theory of electrons had been proposed by Weber[104] for the purpose of explaining the phenomena of electric currents in metallic wires. Weber, however, made but little progress towards an electric theory of metals; for being concerned chiefly with magneto-electric induction and electromagnetic ponderomotive force, be scarcely brought the metal into the discussion at all, except in the assumption that electrons of opposite signs travel with equal and opposite velocities relative to its substance. The more comprehensive scheme of his successors half a century afterwards aimed at connecting in a unified theory all the known electrical properties of metals, such as the conduction of currents according to Ohm's law, the thermo-electric effects of Seebeck, Peltier, and W. Thomson, the galvano-magnetic effect of Hall, and other phenomena which will be mentioned subsequently.

The later investigators, indeed, ranged beyond the group of purely electrical properties, and sought by aid of the theory of electrons to explain the conduction of heat. The principal ground on which this extension was justified was an experimental result obtained in 1853 by G. Wiedemann and R. Franz,[105] who found that at any temperature the ratio of the thermal conductivity of a body to its ohmic conductivity is approximately the same for all metals, and that the value of this ratio is proportional to the absolute temperature. In fact, the conductivity of a pure metal for heat is almost independent of the temperature; while the electric conductivity varies in inverse proportion to the absolute temperature, so that a pure metal as it approaches the absolute zero of temperature tends to assume the character of a perfect conductor. That the two conductivities are closely related was shown to be highly probable by the experiments of Tait, in which pieces of the same metal were found to exhibit variations in ohmic conductivity exactly parallel to variations in their thermal conductivity.

The attempt to explain the electrical and thermal properties of metals by aid of the theory of electrons rests on the assumption that conduction in metals is more or less similar to conduction in electrolytes; at any rate, that positive and negative charges drift in opposite directions through the substance of the conductor under the influence of an electric field. It was remarked in 1888 by J. J. Thomson,[106] who must be regarded as the founder of the modern theory, that the differences which are perceived between metallic and electrolytic conduction may be referred to special features in the two cases, which do not affect their general resemblance. electrolytes the carriers are provided only by the salt, which is dispersed throughout a large inert mass of solvent; whereas in metals it may be supposed that every molecule is capable of furnishing carriers. Thomson, therefore, proposed to regard the current in metals as a series of intermittent discharges, caused by the rearrangement of the constituents of molecular systems—a conception similar to that by which Grothuss[107] had pictured conduction in electrolytes. This view would, as he showed, lead to a general explanation of the connexion between thermal and electrical conductivities,

Most of the later writers on metallic conduction have preferred to take the hypothesis of Arrhenius[108] rather than that of Grothuss as a pattern; and have therefore supposed the interstices between the molecules of the metal to be at all times swarming with electric charges in rapid motion. 11: 1898 E. Riecke[109] effected an important advance by examining the consequences of the assumption that the average velocity of this random motion of the charges is nearly proportional to the square root of the absolute temperature T. P. Drude[110] in 1900 replaced this by the more definite assumption that the kinetic energy of each moving charge is equal to the average kinetic energy of a molecule of a perfect gas at the same temperature, and may therefore be expressed in the form qT, where q denotes a universal constant.

In the same year J. J. Thomson[111] remarked that it would accord with the conclusions drawn from the study of ionization in gases to suppose that the vitreous and resinous charges play different parts in the process of conduction: the resinous charges may be conceived of as carried by simple negative corpuscles or electrons, such as constitute the cathode rays: they may be supposed to move about freely in the interstices between the atoms of the metal. The vitreous charges, on the other hand, may be regarded as more or less fixed in attachment to the metallic atoms. According to this view the transport of electricity is due almost entirely to the motion of the negative charges.

An experiment which was performed at this time by Riecke[112] lent some support to Thomson's hypothesis. A cylinder of aluminium was inserted between two cylinders of copper in a circuit, and a current was passed for such a time that the amount of copper deposited in an electrolytic arrangement would have amounted to over a kilogramme. The weight of each of the three cylinders, however, showed no measurable change; from which it appeared unlikely that metallic conduction is accompanied by the transport of metallic ions.

The ideas of Thomson, Riecke, and Drude were combined by Lorentz[113] in an investigation which, as it is the most complete, will here be given as the representative of all of them.

It is supposed that the atoms of the metal are fixed, and that in the interstices between them a large number of resinous electrons are in rapid motion. The mutual collisions of the electrons are disregarded, so that their collisions with the fixed atoms alone come under consideration; these are regarded as analogous to collisions between moving and fixed elastic spheres.

The flow of heat and electricity in the metal is supposed to take place in a direction parallel to the axis of x, so that the metal is in the same condition at all points of any plane perpendicular to this direction; and the flow is supposed to be steady, so that the state of the system is independent of the time.

Consider a slab of thickness dx and of unit area; and suppose that the number of electrons in this slab whose x-components of velocity lie between u and u + du, whose y-components of velocity lie between v and v + du, and whose z-components of velocity lie between w and w + du, is

f(u, v, w, x) dx du dv dw.

One of these electrons, supposing it to escape collision, will in the interval of time dt travel from (x, y, z) to (x + u dt, y + vdt, z + wdt): and its x-component of velocity will at the end of the interval be increased by an amount eEdt/m, if m and e denote its mass and charge, and E denotes the electric force. Suppose that the number of electrons lost to this group by collisions in the interval dt is a dx du dv dw dt, and that the number added to the group by collisions in the same interval is b dx du dv dw dt. Then we have

f (u, v, w, x) + (b - a)dt = f(u + eEdt/m, v, w, x + udt),

and therefore

${\displaystyle b-a={\frac {eE}{m}}{\frac {\partial f}{\partial u}}+u{\frac {\partial f}{\partial x}}}$.

Now, the law of distribution of velocities which Maxwell postulated for the molecules of a perfect gas at rest is expressed by the equation

${\displaystyle f=\pi ^{-{\frac {3}{2}}}a^{-3}Ne^{-{\frac {r^{2}}{a^{2}}}}}$,

where N denotes the number of moving corpuscles in unit volume, r denotes the resultant velocity of a corpuscle (so that r2 = u2 + v2 + w2), and a denotes a constant which specifies the average intensity of agitation, and consequently the temperature. It is assumed that the law of distribution of velocities among the electrons in a metal is nearly of this form; but a term must be added in order to represent the general drifting of the electrons parallel to the axis of x. The simplest assumption that can be made regarding this term is that it is of the form

u × a function of r only;


we shall, therefore, write

${\displaystyle f=N\pi ^{-{\frac {3}{2}}}a^{-3}e^{-{\frac {r^{2}}{a^{2}}}}+u\chi (r)}$.

The value of χ(r) may now be determined from the equation

${\displaystyle b-a={\frac {eE}{m}}{\frac {\partial f}{\partial u}}+u{\frac {\partial f}{\partial x}}}$;

for on the left-hand side, the Maxwellian term

${\displaystyle \pi ^{-{\frac {3}{2}}}a^{-3}Ne^{-{\frac {r^{2}}{a^{2}}}}}$

would give a zero result, since b is equal to a in Maxwell's system; thus b-a must depend solely on the term (r); and an examination of the circumstances of a collision, in the manner of the kinetic theory of gases, shows that (ba) must have the form -urχ(r)/l, where l denotes a constant which is closely related to the mean free path of the electrons. In the terms on the right-hand side of the equation, on the other hand, Maxwell's term gives a result different from zero; and in comparison with this we may neglect the terms which arisefrom -(r). Thus we have

${\displaystyle -{\frac {ur\chi (r)}{l}}=\left({\frac {eE}{m}}{\frac {\partial }{\partial u}}+u{\frac {\partial }{\partial x}}\right){\frac {N}{\pi ^{\frac {3}{2}}a^{3}}}e^{-{\frac {r^{2}}{a^{2}}}}}$,

or

${\displaystyle u\chi (r)={\frac {lu}{\pi ^{\frac {3}{2}}r}}.e^{-{\frac {r^{2}}{a^{2}}}}.\left\{{\frac {2eNE}{ma^{5}}}-{\frac {d}{dx}}\left({\frac {N}{a^{3}}}\right)-{\frac {2Nr^{2}}{a^{6}}}{\frac {da}{dx}}\right\}}$;

and thus the law of distribution of velocities is determined.

The electric current i is determined by the equation

${\displaystyle i=e\textstyle \iiint uf(u,v,w)\ du\ dv\ dw}$,

where the integration is extended over all possible values of the components of velocity of the electrons. The Maxwellian term in f(u, v, w) furnishes no contribution to this integral, so we have

${\displaystyle i=e\textstyle \iiint u^{2}\chi (r)\ du\ dv\ dw}$.

When the integration is performed, this formula becomes

${\displaystyle i={\frac {2le}{3\pi ^{\frac {1}{2}}}}\left({\frac {2eNE}{ma}}-a{\frac {dN}{dx}}-N{\frac {da}{dx}}\right)}$,

or

${\displaystyle E={\frac {3\pi ^{\frac {1}{2}}m}{4le^{2}}}{\frac {a}{N}}i+{\frac {m}{2e}}\left({\frac {a^{2}}{N}}{\frac {dN}{dx}}+a{\frac {da}{dx}}\right)}$.

The coefficient of i in this equation must evidently represent the ohmic specific resistance of the metal; so if γ denote the specific conductivity, we have

${\displaystyle \gamma ={\frac {4le^{2}}{3\pi ^{\frac {1}{2}}m}}{\frac {N}{a}}}$.

Let the equation be next applied to the case of two metals. A and B in contact at the same temperature T, forming an open circuit in which there is no conduction of heat or electricity (so that i and da/dx are zero). Integrating the equation

${\displaystyle E={\frac {m}{2e}}{\frac {a^{2}}{N}}{\frac {dN}{dx}}}$

across the junction of the metals, we have

Discontinuity of potential at junction ${\displaystyle ={\frac {ma^{2}}{2e}}\log {\frac {N_{b}}{N_{a}}}}$;


or since 34ma2, which represents the average kinetic energy of an electron, is by Drude's assumption equal to qT, where q denotes a universal constant, we have

Discontinuity of potential at junction ${\displaystyle ={\frac {2}{3}}{\frac {q}{e}}T\log {\frac {N_{b}}{N_{a}}}}$;


This may be interpreted as the difference of potential connected with the Peltier[114] effect at the junction of two metals; the product of the difference of potential and the current measures the evolution of heat at the junction. The Peltier discontinuity of potential is of the order of a thousandth of a volt, and must be distinguished from Volta's contact-difference of potential, which is generally much larger, and which, as it presumably depends on the relation of the metals to the medium in which they are immersed, is beyond the scope of the present investigation.

Returning to the general equations, we observe that the flux of energy W is parallel to the axis of x, and is given by the equation

${\displaystyle W={\tfrac {1}{2}}m\textstyle \iiint ur^{2}f(u,v,w)du\ dv\ dw}$,

where the integration is again extended over all possible values of the components of velocity; performing the integration, we have

${\displaystyle W={\frac {2ml}{3\pi ^{\frac {1}{2}}}}\left({\frac {2ea}{m}}NE-a^{2}{\frac {dN}{dx}}-3Na^{2}{\frac {da}{dx}}\right)}$;

or, substituting for E from the equation already found,

${\displaystyle W={\frac {ma^{2}}{e}}i-{\frac {4ml}{3\pi ^{\frac {1}{2}}}}Na^{2}{\frac {da}{dx}}}$.

Consider now the case in which there is conduction of heat without conduction of electricity. The flux of energy will in this case be given by the equation

${\displaystyle W=-\kappa {\frac {dT}{dx}}}$,

where κ a denotes the thermal conductivity of the metal expressed in suitable units; or

${\displaystyle W=-\kappa .{\frac {3ma}{2q}}{\frac {da}{dx}}}$.

If it be assumed that the conduction of heat in metals is effected by motion of the electrons, this expression may be compared with the preceding; thus we have

${\displaystyle \kappa {\tfrac {8}{9}}\pi ^{-{\frac {1}{2}}}\log N}$;

and comparing this with the formula already found for the electric conductivity, we have

${\displaystyle {\frac {\kappa }{\gamma }}={\tfrac {8}{9}}T\left({\frac {q}{e}}\right)^{2}}$,

an equation which shows that the ratio of the thermal to the electric conductivity is of the form T × a constant which is the same for all metals. This result, accords with the law of Wiedemann and Franz,

Moreover, the value of q is known from the kinetic theory of gases; and the value of e has been determined by J.J. Thomson[115] and his followers; substituting these values in the formula for κ/γ, a fair agreement is obtained with the values of κ/γ determined experimentally.

It was remarked by J. J. Thomson that if, as is postulated in the above theory, a metal contains a great number of free electrons in temperature equilibrium with the atoms, the specific heat of the metal must depend largely on the energy required in order to raise the temperature of the electrons. Thomson considered that the observed specific heats of metals are smaller than is compatible with the theory, and was thus led to investigate[116] the consequences of his original hypothesis[117] regarding the motion of the electrons, which differs from the one just described in much the same way as Grothuss' theory of electrolysis differs from Arrhenius'. Each electron was now supposed to be free only for a very short time, from the moment when it is liberated by the dissociation of an atom to the moment when it collides with, and is absorbed by, a different atom. The atoms were conceived to be paired in doublets, one pole of each doublet being negatively, and the other positively, electrified. Under the influence of an external electric field the doublets orient themselves parallel to the electric force, and the electrons which are ejected from their negative poles give rise to a current predominantly in this direction. The electric conductivity of the metal may thus be calculated. In order to comprise the conduction of heat in his theory, Thomson assumed that the kinetic energy with which an electron leaves an atom is proportional to the absolute temperature; so that if one part of the metal is hotter than another, the temperature will be equalized by the interchange of corpuscles. This theory, like the other, leads to a rational explanation of the law of Wiedemann and Franz.

The theory of electrons in metals has received support from the study of another phenomenon. It was known to the philosophers of the eighteenth century that the air near an incandescent metal acquires the power of conducting electricity. "Let the end of a poker," wrote Canton,[118] "when red-hot, be brought but for a moment within three or four inches of a small electrified body, and its electrical power will be almost, if not entirely, destroyed."

The subject continued to attract attention at intervals[119]; and as the process of conduction in gases came to be better understood, the conductivity produced in the neighbourhood of incandescent metals was attributed to the emission of electrically charged particles by the metals. But it was not until the develop- mout of J.J. Thomson's theory of ionization in gages that notable advances were made. In 1899, Thomson[120] determined the ratio of the charge to the mass of the resinously charged ions emitted by a hot filament of carbon in rarefied hydrogen, by observing their deflexion in a magnetic field. The value obtained for the ratio was nearly the same as that which he had found for the corpuscles of cathode rays; whence he concluded that the negative ions emitted by the hot carbon were negative electrons.

The corresponding investigation[121] for the positive leak from hot bodies yielded the information that the mass of the positive ions is of the same order of magnitude as the mass of material atoms. There are reasons for believing that these ions are produced from gas which has been absorbed by the superficial layer of the metal.[122]

If, when a hot metal is emitting ions in a rarefied gas, an electromotive force be established between the metal and a neighbouring electrode, either the positive or the negative ions are urged towards the electrode by the electric field, and a current is thus transmitted through the intervening space. When the metal is at a higher potential than the electrode, the current is carried by the vitreously charged ions: when the electrode is at the higher potential, by those with resinous charges. In either case, it is found that when the electromotive force is increased indefinitely, the current does not increase indefinitely likewise, but acquires a certain "saturation" value. The obvious explanation of this is that the supply of ions available for carrying the current is limited.

When the temperature of the metal is high, the ions emitted are mainly negative; and it is found[123] that in these circumstances, when the surrounding gas is rarefied, the saturation-current is almost independent of the nature of the gas or of its pressure. The leak of resinous electricity from a metallic surface in a rarefied gas must therefore depend only on the temperature and on the nature of the metal; and it was shown by 0. W. Richardson[124] that the dependence on the temperature may be expressed by an equation of the form

${\displaystyle i=AT^{\frac {1}{2}}e^{-{\frac {b}{T}}}}$,

where i denotes the saturation-current per unit area of surface (which is proportional to the number of ions emitted in unit time), T denotes the absolute temperature, and A and b are constants.[125]

In order to account for these phenomena, Richardson[126]adopted the hypothesis which had previously been proposed[127] for the explanation of metallic conductivity; namely, that a metal is to be regarded as a sponge-like structure of comparatively large fixed positive ions and molecules, in the interstices of which negative electrons are in rapid motion. Since the electrons do not all escape freely at the surface, he postulated a superficial discontinuity of potential, sufficient to restrain most of them. Thus, let N denote the number of free electrons in unit volume of the metal; then in a parallelepiped whose height measured at right angles to the surface is dx, and whose base is of unit area, the number of electrons whose x-components of velocity are comprised between u and u + du is

${\displaystyle \pi ^{-{\frac {1}{2}}}\alpha ^{-1}Nc^{-{\frac {u^{2}}{\alpha ^{2}}}}\ du\ dx}$, where ${\displaystyle {\tfrac {3}{4}}m\alpha ^{2}=qT}$,

m denoting the mass of an electron, T the absolute temperature, and q, the universal constant previously introduced.

Now, an electron whose x-component of velocity is u will arrive at the interface within an interval dt of time, provided that at the beginning of this interval it is within a distance udt of the interface. So the number of electrons whose x-components of velocity are comprised between u and u + du which arrive at unit area of the interface in the interval dt is

${\displaystyle \pi ^{-{\frac {1}{2}}}\alpha ^{-1}Nc^{-{\frac {u^{2}}{\alpha ^{2}}}}u\ du\ dt}$.

If the work which an electron must perform in order to escape through the surface layer be denoted by φ, the number of electrons emitted by unit area of metal in unit time is therefore

${\displaystyle \int _{{\tfrac {1}{2}}mu^{2}=\phi }^{\inf }\pi ^{-{\tfrac {1}{2}}}\alpha ^{-1}Ne^{-{\frac {u^{2}}{\alpha ^{2}}}}u\ du}$, or ${\displaystyle {\tfrac {1}{2}}\pi ^{-{\tfrac {1}{2}}}N\alpha e^{-{\frac {2\phi }{m\alpha ^{2}}}}}$.

The current issuing from unit area of the hot metal is thus

${\displaystyle {\tfrac {1}{2}}\pi ^{-{\tfrac {1}{2}}}N\epsilon \alpha e^{-{\frac {2\phi }{m\alpha ^{2}}}}}$, or ${\displaystyle N\epsilon .(qT/3\pi m)^{\tfrac {1}{2}}e^{-{\frac {3\phi }{2qT}}}}$,

where ε denotes the charge on an electron. This expression, being of the form

${\displaystyle AT^{b}c^{-{\frac {b}{T}}}}$,

agrees with the experimental measures; and the comparison furnishes the value of the superficial discontinuity of potential which is implied in the existence of φ.[128]

A few years after the date of this investigation, a plan was devised and successfully carried out[129] for determining experimentally the kinetic energy possessed by the ions after emission. The mean kinetic energy of both negative and positive ions was found to be the same for various metals (platinum, gold, silver, etc.), and to be directly proportional to the absolute temperature, and the distribution of velocities among the ions proved to be that expressed by Maxwell's law. The ions may therefore be regarded as kinetically equivalent to the molecules of & gas whose temperature is the same as that of the metal.

By the investigations which have been recorded, the hypothesis of atomic electric charges has been, to all appearances, decisively established. But all the parts of the theory of electrons do not enjoy an equal degree of security; and in particular, it is possible that the future may bring important changes in the conception of the aether. The hope was formerly entertained of discovering an aether by reference to which motion might be estimated absolutely; but such a hope has been destroyed by the researches which have sprung from FitzGerald's hypothesis of contraction, and in some recent writings it is possible to recognize a tendency to replace the classical aether by other conceptions, which, however, have been as yet but indistinctly outlined.

In any event, the close of the nineteenth century brought to an end a well-marked era in the history of natural philosophy: and this is true not only with respect to the discoveries themselves, but also in regard to the conditions of scientific organization and endeavour, which in the last decades of that period became profoundly changed. The investigators who advanced the theories of aether and electricity, from the time of Descartes to that of Lord Kelvin, were, with very few exceptions, congregated within a narrow territory: from Dublin to the western provinces of Russia, and from Stockholm to the north of Italy, may be circumscribed by a circle of no more than six hundred miles radius. But throughout the whole of Kelvin's long life, the domain of culture was rapidly extending: the learning of the Germanic and Latin peoples was carried to the furthest regions of the earth: new universities were founded, and inquiries into the secrets of nature were instituted in every quarter of the globe. Let this record close with the anticipation that fellowship in the pursuit of knowledge will increase in the nations the spirit of generous emulation and mutual respect.

## Notes

1. Cf p. 288.
2. Cf. p. 365.
3. Cf. p. 116.
4. Cf. p. 115.
5. Phil. Mag. xxvii (1845), p. 9; xxviii (1846), p. 76; xxix (1846), p. 6.
6. Archives Néerl, xxi (1896), p. 103.
7. Cf. Lorentz, Proc. Amsterdam Acad. (English ed.), i (1899), p. 443.
8. Archives Néerl. xxi (1856), p. 103: cf. also Zittinsgavers10. Kon, Ak. Amsterdam, 1897-98, p. 266.
9. Cf. p. 22.
10. Proc. R. S. xxx (1880), p. 108.
11. The velocity of light was found from observations of Algol, by C. Y. L. Charlier, Öfversigt af K. Vet.-Ak. Förhandl, xlvi (1889), p. 523.
12. Conn. des Temps, 1830.
13. Additions à la Connaissance des Temps, 1878: an improved deduction was given by H. Seeliger, Sitzungsberichte d. K. Ak, zu München, xix (1889), p. 19.
14. Abhandl. der K. Böhm, Ges. der Wissensch. (5) ii (1842), p. 465.
15. An apparatus for demonstrating the Doppler-Fizeau effect in the laboratory was constructed by Belopolsky, Astrophys. Journal xiii (1901), p. 15.
16. Cf. pp. 117-120.
17. Ann. d. Phys. xcii (1854), p. 652.
18. Proc. Amsterdam Acad. (English edition), iv (1902) p. 678
19. Annales de Chim. (3) lxviii (1860), p. 129; Ann. d. Phys. cxiv (1861), p. 654.
20. Annales de l'Ec. Norm. (2) i (1872), p. 157.
21. Phil. Mag. iv. (1902), p. 215.
22. The constant of aberration is the ratio of the earth's orbital velocity ω, the velocity of light; cf. supra, p. 100.
23. Amer. Journ. Sci. xxii (1881), p. 20. His method was afterwards improved: cf. Michelson and Morley, Amer. Journ. Sci. xxxiv (1887), . 333; Phil. Mag. xxiv (1887), p. 449.
24. Cf. p. 411.
25. Arch. Néerl. xxi (1886), p. 103. On the Michelson-Morley experiment cf. also Hicks, Phil. Mag. iii (1902), p. 9.
26. Cf. p. 413.
27. Amer. Journ. Sci. (4) iii (1897), p. 475.
28. Phil. Trans. clxxxiv (1893), p. 727.
29. Ibid., clxxxix (1897), p. 149.
30. Archives Néerl. xxv (1892), p. 363: the theory is given in ch. iv, pp. 132 et sqq.
31. Cf. pp. 226, 231, 262.
32. Some writers have inclined to use the term 'electron-theory' as if it were specially connected with Sir Joseph Thomson's justly celebrated discovery (cf. p. 407, supra) that all negative electrons have equal charges. But Thomson's discovery, though undoubtedly of the greatest importance as a guide to the structure of the universe, has hitherto exercised but little influence on general electromagnetic theory. The reason for this is that in theoretical investigations it is customary to denote the changes of electrons by symbols, e1, e2, …; and the equality or non-equality of these makes no difference to the equations. To take an illustration from Celestial Mechanics, it would clearly make no difference in the general equations of the planetary theory if the masses of the planets happened to be all equal.
33. Cf. chapter xi.
34. Cf. pp. 357-363.
35. Verh. d. Ak. v. Wetenschappen, Amsterdam, Deel xviii (1878).
36. Cf. pp. 116 et sqq.
37. Cf. p. 262.
38. Cf. pp. 298, 299.
39. We shall use the small letters d and h in place of E and H, when we are concerned with Lorentz' fundamental case, in which the system consists solely of free aether and isolated electrons.
40. Cf. p. 298.
41. These principles had been enunciated, and to some extent developed, by J. Willard Gibbs in 1882-3: Amer. Journ. Sci. xxiii, pp. 262, 460, xxv, p. 107: Gibbs' Scientific papers, ii, pp. 182, 195, 211.
42. Cf. pp. 210, 211.
43. Cf. p. 64.
44. We assume all transitions gradual, so as to avoid surface-distributions.
45. Cf. p. 386.
46. Cf. p. 367.
47. Ann. d. Phys. XXXV (1888), p. 264: xl (1890), p. 93.
48. Cf. p. 64.
49. Cf. p. 339.
50. Ann. d. Phys. xi (1903), p. 421; xiii (1904), p. 919. Eichenwald performed other experiments of a similar character, e.g. he observed the magnetic field due to the changes of polarization in a dielectric which was moved in a non-homogeneous electric field.
51. It is possible to construct a purely electronic theory of magnetization, a magnetic molecule being supposed to contain electrons in orbital revolution. It then appears that the vector which represents the average value of h is not H, but B.
52. Cf. p. 365.
53. Cf. p. 281.
54. Ct. p. 293.
55. A theory of dispersion, which, so far as its physical assumptions and results. are concerned, resembles that described above, was published in the same year (1892) by Helmholtz, Berl. Ber., 1892, p. 1093, Ann d. Phys. xlviii (1893), pp. 389, 723. Io this, as in Lorentz' theory, the incident light is supposed to excite sympathetic vibrations in the electric doublets which exist in the molecules of transparent bodies. Helmholtz' equations were, however, derived in a different way from those of Lorentz, being deduced from the Principle of Least Action. The final result is, as in Lorentz' theory, represented (when the effect of damping is neglected) by the Maxwell-Sellmeier formula. Helmholtz' theory was developed further by Reiff, Ann. d. Phys. lv (1895), p. 82.
In a theory of dispersion given by Planck, Berl. Ber., 1902, p. 470, the damping of the oscillations is assumed to be due to the loss of energy by radiation: so that no new constant is required in order to express it.
Lorentz, in his lectures on the Theory of Electrons (Leipzig, 1909), p. 141, suggested that the dissipative term in the equations of motion of dielectric electrons might be ascribed to the destruction of the regular vibrations of the electrons within a molecule by the collisions of the molecule with other molecules.
Some interesting references to the ideas of Hertz on the electromagnetic explanation of dispersion will be found in a memoir by Drude, Ann. d. Phys. (6) i (1900), p. 437.
56. Cf. p. 283.
57. Cf. p. 211.
58. This equation was first given as a result of the theory of electrons by Lorentz in the last chapter of his memoir of 1892, Arch. Néerl. xxv, p. 525. It was also given by Larmor, Phil. 'Trans., clxxxv (1894), p. 891.
59. Cf. p. 117.
60. Comptes Rendus cxxxiii (1901), p. 778.
61. Phil. Trans, cciv (1905), p. 121.
62. Cf. p 417.
63. Nature, xlvi (1892), p. 165.
64. Verslagen d. Kon. Ak, van Wetenschappen, 1892-3, p. 74 (November 26th 1892).
65. Versuch einer Theorie der electrischen und optischen Erscheinungen in beweglen Körpern, von H. A. Lorentz; Leiden, E. J. Brill. It was reprinted by Teubner, of Leipzig, in 1906.
66. Ann. d. Phys. xxxviii (1889), p. 73.
67. Cf. p. 263.
68. Cf. pp. 365, 366.
69. Cf. p. 416.
70. Larmor, Aether and Matter, 1900.
71. Proc. Amsterdam Acad. (English ed.), iv (1902), p. 669.
72. E. Wiechert, Arch. Néerl. (2) v (1900), p. 549. Cf. also A. Liénard, L' Éclairage élect. xvi (1898), pp. 5, 53, 106.
73. Cf. p. 298.
74. FitzGerald's Scientific Writings, p. 557.
75. Larmor, in FitzGerald's Scientific papers, p 566.
76. F. T. Trouton. Trans. Roy. Dub. Soc., April, 1902; F. T. Trouton and H. R. Noble, Phil. Trans, ccii (1903), p. 165.
77. Cf. P. Langevin, Comptes Rendus, cxl (1905), p. 1171.
78. Phil. Mag. iv (1902), p. 678.
79. Phil. Mag. vii (1904), p. 317.
80. Cf. p. 434. Cf. also Lorentz, Proc. Amsterdam Acad. (English ed.), i (1899), p. 427.
81. Larmor, Aether and Matter, p. 173.
82. Proc. Amsterdam Acad. (English ed.), vi, p. 809. Lorentz' work was completed in respect to the formulae which connect ρ1, v1 with ρ, v, by Einstein, Ann. d. Phys., xvii (1905), p. 891. It should be added that the transformation in question had been applied to the equation of vibratory motions many years before by Voigt, Gött. Nach, 1887, p. 41.
83. Cf. p. 434.
84. Cf. p. 432.
85. This was first clearly expressed by Einstein, Ann. d. Phys. xvii (1905), P. 891.
86. Cf. H. Minkowski, Raum und Zeit.: Leipzig, 1909.
87. Bence Jones' Life of Faraday, ii, p. 449.
88. Proc. R.S. Edinb. ix (1875), p. 118.
89. Cf. pp. 174, 216.
90. Zittingsverslagen der Akad. v. Wet. te Amsterdam v (1896), pp. 181, 242; vi (1897), pp. 13, 99; Phil. Mag. (5) xliii (1897), p. 226.
91. Phil. Mag. xliii (1897), p. 232.
92. Later observations, with more powerful apparatus, have shown that the primitive spectral line is frequently replaced by more than three components.
93. Comptes Rendus, cxxv (1897), p. 555.
94. Ann. d. Phys. lxii (1897), p. 240.
95. Cf. p. 405.
96. Cf. pp. 213-216, 307-309, 367-370.
97. Cf. p. 428.
98. Cf. p 215.
99. The phenomenon was first observed by D. Macaluso and O. M. Corbino, Comptes Rendus, cxxvii (1898), p. 548, Rend. Lincei (5) vii (2) (1898), p. 293. The theoretical explanation was supplied by W. Voigt, Gött. Nach., 1898, p. 349, Ann. d. Phys. lxvii (1899), p. 345. Cf. also P. Zeeman, Proc. Amst, Acad. v (1902), p. 41, and J. J. Hallo, Arch. Néerl. (2) x (1905), p. 148.
Voigt also predicted that if plane-polarized light, of period nearly the same as that of the D radiation, were passed through sodium vapour in a magnetic field, in a direction perpendicular to the lines of magnetic force, the velocity of propagution would be found to depend on the orientation of the plane of polarization, so that the sodium vapour would behave as a uniaxal crystal. This prediction was confirmed experimentally by Voigt and Wiechert: cf. Voigt, Gött. Nach., 1898, p. 355: Ann. d. Phys. lxvii. (1899), p. 345. Cf. also A. Cotton, Comptes Rendus, cxxviii (1899), p. 294, and J. Goest, Arch. Néerl. (2), x (1905), p. 291.
100. Cf. p. 290.
101. Berlin Sitzungsber., 1903, pp. 269, 410; Ann. d. Phys. xi (1903), p. 873; Phil. Mag. vii (1904), p. 157.
102. Cf. p. 179.
103. Models illustrating the selective reflexion and absorption of light by metallic bodies and by gases were discussed by H. Lamb, Mem. and Proc. Manchester Lit. and Phil. Soc. xlii (1898), p. I; Proc. Lond. Math, Soc. xxxii (1900), p. 11; Trans. Camb. Phil. Soc. xviii (1900), p. 348.
104. Cf. p. 226.
105. Ann. d. Phys. lxxxix (1853), p. 497.
106. J. J. Thomson, Applications of Dynamics to Physics and Chemistry, 1888, P. 296. Cf. also Giese, Ann. d. Phys. xxxvii (1889), p. 576.
107. Cf. p. 78.
108. Cf. p. 384.
109. Gött. Nach., 1898, pp. 48, 137. Ann. d. Phys. lxvi (1898), pp. 353, 545, 1199; ii, (1900), p. 835.
110. Ann. d. Phys. (4) i (1900), p. 566; iii (1900), p. 369; vii (1902), P. 687.
111. Rapports prés. au Congrès de Physique, Paris, 1900, iii, p. 138.
112. Phys. Zeitsch. iii (1901), p. 639.
113. Amsterdam Proceedings (English edition) vii (1904-1905), pp. 438, 585, 684
114. Cf. p. 264.
115. Cf. p. 407.
116. J.J. Thomson, The Corpuscular Theory of Matter; London, 1907.
117. Cf. p. 457.
118. Phil. Trans. lii (1762), p. 457.
119. Cf. E. Becquerel, Annules de Chimie xxxix (1853), p. 355; Guthrie, Phil. Mag. xlvi (1873), p. 264; also various memoirs by Elster and Geitel in the Annalen d. Phys. from 1882 onwards. The phenomenon is very noticeable, as Edison showed (Engineering, December 12, 1884, p. 553), when a filament of carbon is heated to incandescence in a rarefied gas. In recent years it has been found that ions are emitted when magnesia, or any of the oxides of the alkaline earth metals, is heated to a dull red heat.
120. Phil. Mag. xlviii (1899), p. 517.
121. J.J. Thomson, Proc. Camb, Phil. Soc. xv (1909), p. 61: O. W. Richardson, Phil. Mag. xvi. (1908), p. 740.
122. Cf. Richardson, Phil. Trans, ccvii (1906), p. 1.
123. Cf. J. A. McClelland, Proc, Cumb. Phil. Soc. < (1899), p. 241; xi (1901), p. 296. On the results obtained when the gas is hydrogen, cf. H. A. Wilson, Phil. Trans. ccii (1903), p. 243; ccviii (1908), p. 217; and O. W. Richardson, Phil. Trans. ccvii (1906), p. 1.
124. Proc. Camb. Phil. Soc. xi (1902), p. 286; Phil. Trans. cci (1903), p. 497. Cf. also H. A. Wilson, Phil. Trans. ccii (1903), p. 243.
125. The same law applies to the emission from other bodies, e.g. heated alkaline earths, and to the emission of positive ions—at any rate when a steady state of emission has been reached in a gas which is at a definite pressure.
126. Phil. Trans. cci (1903), p. 197.
127. Cf. pp. 467 et sqq.
128. This discontinuity of potential was found to be 2·45 volts for sodium, 4·1 volts for platinum, and 6·1 volts for carbon.
129. O. W. Richardson and F. C. Brown, Phil. Mag. xvi (1908), pp. 353, 890; F. O. Brown, Phil. Mag. xvii (1909), p. 375; xviii (1909), p. 649

## Errata

1. Original: exends was amended to extends
2. Original: (x2 + y2 + z - c2t2) was amended to (x2 + y2 + z2 - c2t2)