A History of the Theories of Aether and Electricity (1910) — part 14 of 29

The conception which he at this time entertained of the polarization may be reconstructed from what he had already written concerning electrolytes. He supposedf that in the ordinary or unpolarized condition of a body, the molecules con- sist of atoms which are bound to each other by the forces of chemical affinity, these forces being really electrical in their nature ; and that the same forces are exerted, though to a less

• Exp.Res., § 1338.

t This must not be taken to be more than an idea which Faraday mentioned as present to his mind. He declined as yet to formulate a definite hypothesis.

Faraday. 209

degree, between atoms which belong to different molecules, thus producing the phenomena of cohesion. When an electric field is set up, a change takes place in the distribution of these forces ; some are strengthened and some are weakened, the effect being symmetrical about the direction of the applied electric force.

Such a polarized condition acquired by a dielectric when placed in an electric field presents an evident analogy to the condition of magnetic polarization which is acquired by a mass of soft iron when placed in a magnetic field ; and it was there- fore natural that in discussing the matter Faraday should introduce lines of electric force, similar to the lines of magnetic force which he had employed so successfully in his previous researches. A line of electric force he defined to be a curve whose tangent at every point has the same direction as the electric intensity.

The changes which take place in an electric field when the dielectric is varied may be very simply described in terms of lines of force. Thus if a mass of sulphur, or other substance of high specific inductive capacity, is introduced into the field, the effect is as if the lines of force tend to crowd into it : as W. Thomson (Kelvin) showed later, they are altered in the same way as the lines of flow of heat, in a case of steady con- duction of heat, would be altered by introducing a body of greater conducting power for heat. By studying the figures of the lines of force in a great number of individual cases, Faraday was led to notice that they always dispose themselves as if they were subject to a mutual repulsion, or as if the tubes of force had an inherent tendency to dilate.*

It is interesting to interpret by aid of these conceptions the law of Priestley and Coulomb regarding the attraction between two oppositely-charged spheres. In Faraday's view, the medium intervening between the spheres is the seat of a system of stresses, which may be represented by an attraction or tension along the lines of electric force at every point, together with a

• Exp. Res., §§ 1224, 1297 (1837). P

210 Faraday.

mutual repulsion of these lines, or pressure laterally. Where a line of force ends on one of the spheres, its tension is exercised on the sphere: in this way, every surface-element of each sphere is pulled outwards. If the spheres were entirely removed from each other's influence, the state of stress would be uniform round each sphere, and the pulls on its surf ace -elements would balance, giving no resultant force on the sphere. But when the two spheres are brought into each other's presence, the unit lines of force become somewhat more crowded together on the sides of the spheres which face than on the remote sides, and thus the resultant pull on either sphere tends to draw it toward the other. When the spheres are at distances great compared with their radii, the attraction is nearly proportional to the inverse square of the distance, which is Priestley's law.

In the following year (1838) Faraday amplified* his theory of electrostatic induction, by making further use of the analogy with the induction of magnetism. Fourteen years previously Poisson had imaginedf an admirable model of the molecular processes which accompany magnetization; and this was now applied with very little change by Faraday to the case of induc- tion in dielectrics. " The particles of an insulating dielectric," he suggested, J " whilst under induction may be compared to a series of small magnetic needles, or, more correctly still, to a series of small insulated conductors. If the space round a charged globe were filled with a mixture of an insulating dielectric, as oil of turpentine or air, and small globular conductors, as shot, the latter being at a little distance from each other so as to be insulated, then these would in their condition and action exactly resemble what I consider to be the condition and action of the particles of the insulating dielectric itself. If the globe were charged, these little con- ductors would all be polar ; if the globe were discharged, they would all return to their normal state, to be polarized again upon the recharging of the globe/'

That this explanation accounts for the phenomena of specific

• Exp. Res., Series xiv. t Cf. p. 65. J Exp. Res., § 1679.

Faraday. 211

inductive capacity may be seen by what follows, which is substantially a translation into electrostatical language of Poisson's theory of induced magnetism.*

Let p denote volume-density of electric charge. For each of Faraday's " small shot " the integral

JJJ pdx dy dz,

integrated throughout the shot, will vanish, since the total charge of the shot is zero : but if r denote the vector (x, y, z),

the integral

J/J p r dx dy dz

will not be zero, since it represents the electric polarization of the shot : if there are N shot per unit volume, the quantity

P = &!!! P r dx dy dz

will represent the total polarization per unit volume. If d denote the electric force, and E the average value of d, P will be proportional to E, say

P - (€ - 1) E.

By integration by parts, assuming all the quantities concerned to vary continuously and to vanish at infinity, we have

• p* D * (x> y> z} ** dy ds = "Iff* ^ p ** dy dz>

where ^ denotes an arbitrary function, and the volume-integrals are taken throughout infinite space. This equation shows that the polar-distribution of electric charge on the shot is equivalent to a volume- distribution throughout space, of density

P = - div P.

Now the fundamental equation of electrostatics may in suitable units be written,

div d = p ;

• W. Thomson (Kelvin), Camb. and Dub. Matb. Journal, November, 1845 ; "W. Thomson's Papers on Electrostatics and Magnetism, § 43 sqq. ; F. 0. Mossotti, Arcb. des sc. phys. (Geneva) vi (1847), p. 193 ; Mem. della Soc. Ital. Modena, (2)xiv(1850), p. 49.

P 2

212 Faraday.

and this gives on averaging

div E = pi + jo,

where pi denotes the volume-density of free electric charge, i.e. excluding that in the doublets ; or

div (E + P) = Plt or div (* E) = plt

This is the fundamental equation of electrostatics, as modified in order to take into account the effect of the specific inductive capacity «.

The conception of action propagated step by step through a medium by the influence of contiguous particles had a firm hold on Faraday's mind, and was applied by him in almost every part of physics. " It appears to me possible," he wrote in 1838,* " and even probable, that magnetic action may be communicated to a distance by the action of the intervening particles, in a manner having a relation to the way in which the inductive forces of static electricity are transferred to a distance ; the intervening particles assuming for the time more or less of a peculiar condition, which (though with a very imperfect idea) I have several times expressed by the term electro-tonic state."^

The same set of ideas sufficed to explain electric currents. Conduction, Faraday suggested,* might be " an action of contiguous particles, dependent on the forces developed in electrical excitement ; these forces bring the particles into a state of tension or polarity ;§ and being in this state the contiguous particles have a power or capability of communicating these forces, one to the other, by which they are lowered and discharge occurs."

• Exp Res., § 1729.

f This name had been devised in 1831 to express the state of matter subject to magneto-electric induction ; cf. Exp.^Res., § 60. J Exp. Res. iii, p. 513. § As in electrostatic induction in dielectrics.

Faraday. 213

After working strenuously for the ten years which followed the discovery of induced currents, Faraday found in 1841 that his health was affected ; and for four years he rested. A second period of brilliant discoveries began in 1845.

Many experiments had been made at different times by various investigators* with the purpose of discovering a connexion between magnetism and light. These had generally taken the form of attempts to magnetize bodies by exposure in particular ways to particular kinds of radiation ; and a successful issue had been more than once reported, only to be negatived on re-examination.

The true path was first indicated by Sir John Herschel. After his discovery of the connexion between the outward form of quartz crystals and their property of rotating the plane of polarization of light, Herschel remarked that a rectilinear electric current, deflecting a needle to right and left all round it, possesses a helicoidal dissymmetry similar to that displayed by the crystals. " Therefore," he wrote,f " induction led me to conclude that a similar connexion exists, and must turn up somehow or other, between the electric current and polarized light, and that the plane of polarization would be deflected by magneto-electricity."

The nature of this connexion was discovered by Faraday, who so far back as 1834J had transmitted polarized light through an electrolytic solution during the passage of the current, in the hope of observing a change of polarization. This early attempt failed ; but in September, 1845, he varied the experiment by placing a piece of heavy glass between the poles of an excited electro-magnet ; and found that the plane of polarization of a beam of light was rotated when the beam travelled through the glass parallel to the lines of force of the magnetic field. §

*e.g. by Morichini, of Rome, in 1813, Quart. Journ. Sci. xix, p. 338; by Samuel Hunter Christie, of Cambridge, in 1825, Phil. Trans., 1826, p. 219 ; and by Mary Somerville in the same year, Phil. Trans., 1826, p. 132.

t Sir. J. Herschel in Bence Jones's Life of Faraday, p. 205.

lExp. Res., § 951. \Ib., § 2152.

214 Faraday.

In the year following Faraday's discovery, Airy* suggested a way of representing the effect analytically; as might have been expected, this was by modifying the equations which had been already introduced by MacCullagh for the case of naturally active bodies. In Mac Cullagh's equations |8^F=c2822r+ VZ

d^Y

the terms 83^/8#3 and 83F/8#3 change sign with x, so that the rotation of the plane of polarization is always right-handed or always left-handed with respect to the direction of the beam. This is the case in naturally-active bodies ; but the rotation due to a magnetic field is in the same absolute direction whichever way the light is travelling, so that the derivations with respect to x must be of even order. Airy proposed the equations /82F 2 82F a£

I *-, * o 1 <*N 9 r^ *"\ /

dx* dt'

where p denotes a constant, proportional to the strength of the magnetic field which is used to produce the effect. He remarked, however, that instead of taking p dZ/dt and fj. 8 Y/dt as the additional terms, it would be possible to take « 83^/8£3 and /u 83 F/8^3, or im()3Z/dz?dt> and ju83F/8^28^, or any other derivates in which the number of differentiations is odd with respect to t and even with respect to x. It may, in fact, be shown by the method pre- viously applied to Mac Cullagh's formulae that, if the equations

are

, 82F 8r+s^

8r+sF

where (r + s) is an odd number, the angle through which the

• Phil. Mag. xxviii (1846) p. 469.

Faraday. 215

plane of polarization rotates in unit length of path is a numerical multiple of

where T denotes the period of the light. Now it was shown by Verdet* that the magnetic rotation is approximately proportional to the inverse square of the wave-length ; and hence we must

have

r + s= 3;

so that the only equations capable of correctly representing Faraday's effect are either

w ^dx-dt

or

dt* W * dt

d'Z _ 2 d'Z W '" °l W^'W

The former pair arise, as will appear later, in Maxwell's theory of rotatory polarization : the latter pair, which were suggested in 1868 by Boussinesq,f follow from that physical theory of the phenomenon which is generally accepted at the present time.*

Airy's work on the magnetic rotation of light was limited in the same way as MacCullagh's work on the rotatory power of quartz ; it furnished only an analytical representation of the effect, without attempting to justify the equations. The earliest endeavour to provide a physical theory seems to have been made in 1858, in the inaugural dissertation of Carl Neumann,

• Comptes Rendus, Ivi (1863), p. 630.

t Journal de Math., xiii (1868), p. 430.

J Fand Z being interpreted as components of electric force.

216 Faraday.

of Halle.* Neumann assumed that every element of an electric current exerts force on the particles of the aether ; and in parti- cular that this is true of the molecular currents which constitute magnetization, although in this case the force vanishes except when the aethereal particle is already in motion. If e denote the displacement of the aethereal particle ra, the force in question may be represented by the term

km [ e. K]

where K denotes the imposed magnetic field, and k denotes a magneto-optic constant characteristic of the body. When this term is introduced into the equations of motion of the aether, they take the form which had been suggested by Airy ; whence Neumann's hypothesis is seen to lead to the incorrect conclusion that the rotation is independent of the wave-length.

The rotation of plane-polarized light depends, as Fresnel had shown,f on a difference between the velocities of propagation of the right-handed and left-handed circularly polarized waves into which plane-polarized light may be resolved. In the case of magnetic rotation, this difference was shown by Verdet to be proportional to the component of the magnetic force in the direction of propagation of the light; and CornuJ showed further that the mean of the velocities of the right-handed and left- handed waves is equal to the velocity of light in the medium when there is no magnetic field. From these data, by Fresnel's geometrical method, the wave- surf ace in the medium may be obtained; it is found to consist of two spheres (one relating to the right-handed and one to the left-handed light), each identical with the spherical wave-surface of the unmagnetized medium, displaced from each other along the lines of magnetic force.§

The discovery of the connexion between magnetism and

• Explicare tentatur, quomodojiat, ut lucis planum polarisationis per vires el. vel mag. declinetur. Halis Saxonum, 1858. The results were republished in a tract Die magnetische Drehung der Polarisationsebene des Lichtes. Halle, 1863.

t Cf. p. 174. J Comptes Rendus, xcii (1881), p. 1368.

§ Cornu, Comptes Rendus, xcix (1884), p. 1045.

Faraday. 217

light gave interest to a short paper of a speculative character which Faraday published* in 1846, under the title " Thoughts on Kay- Vibrations." In this it is possible to trace the progress of Faraday's thought towards something like an electro-magnetic theory of light.

Considering first the nature of ponderable matter, he suggests that an ultimate atom may be nothing else than a field of force — electric, magnetic, and gravitational — surrounding a point- centre ; on this view, which is substantially that of Michell and Boscovich, an atom would have no definite size, but ought rather to be conceived of as completely penetrable, and extend- ing throughout all space ; and the molecule of a chemical compound would consist not of atoms side by side, but of " spheres of power mutually penetrated, and the centres even coinciding."t

All space being thus permeated by lines of force, Faraday suggested that light and radiant heat might be transverse vibrations propagated along these lines of force. In this way he proposed to " dismiss the aether," or rather to replace it by lines of force between centres, the centres together with their lines of force constituting the particles of material substances.

If the existence of a luminiferous aether were to be admitted, Faraday suggested that it might be the vehicle of magnetic force ; " for," he wrote in 1851,{ "it is not at all unlikely that if there be an aether, it should have other uses than simply the conveyance of radiations." This sentence may be regarded as the origin of the electro-magnetic theory of light.

At the time when the " Thoughts on Eay -Vibrations " were published, Faraday was evidently trying to comprehend every- thing in terms of lines of force ; his confidence in which had been recently justified by another discovery. A few weeks after the first observation of the magnetic rotation of light, he noticed § that a bar of the heavy glass which had been used in

• Phil. Mag. (3), xxviii (1846) : Exp. Res., iii, p. 447.

t Cf. Bence Jones's Life of Faraday, ii, p. 178.

J Exp. Res., $ 3075. § Phil. Trans., 1846, p. 21 : Exp. Res., § 2253.

218 Faraday.

this investigation, when suspended between the poles of an electro-magnet, set itself across the line joining the poles : thus behaving in the contrary way to a bar of an ordinary magnetic substance, which would tend to set itself along this line. A simpler manifestation of the effect was obtained when a cube or sphere of the substance was used ; in such forms it showed a disposition to move from the stronger to the weaker places of the magnetic field. The pointing of the bar was then seen to be merely the resultant of the tendencies of each of its particles to move outwards into the positions of weakest magnetic action.

Many other bodies besides heavy glass were found to display the same property ; in particular, bismuth.* The name diamagnetic was given to them.

" Theoretically," remarked Faraday, " an explanation of the movements of the diamagnetic bodies might be offered in the supposition that magnetic induction caused in them a contrary state to that which it produced in magnetic matter ; i.e. that if a particle of each kind of matter were placed in the magnetic field, both would become magnetic, and each would have its axis parallel to the resultant of magnetic force passing through it ; but the particle of magnetic matter would have its north and south poles opposite, or facing toward the contrary poles of the inducing magnet, whereas with the diamagnetic particles the reverse would be the case ; and hence would result approxi- mation in the one substance, recession in the other. Upon Ampere's theory, this view would be equivalent to the sup- position that, as currents are induced in iron and magnetics parallel to those existing in the inducing magnet or battery wire, so in bismuth, heavy glass, and diamagnetic bodies, the currents induced are in the contrary direction."

This explanation became generally known as the " hypothesis of diamagnetic polarity " ; it represents diamagnetism as similar

• The repulsion of bismuth in the magnetic field had been previously observed by A. Brugmans in 1778; Antonii Brugmans Magnetismus, Lugd. Bat., 1778. t Exp. Res., § 2429.

Faraday. 219

to ordinary induced magnetism in all respects, except that the direction of the induced polarity is reversed. It was accepted by other investigators, notably by W. Weber, Pliicker, Eeich, and Tyndall ; but was afterwards displaced from the favour of its inventor by another conception, more agreeable to his peculiar views on the nature of the magnetic field. In this second hypothesis, Faraday supposed an ordinary magnetic or para- magnetic* body to be one which offers a specially easy passage to lines of magnetic force, so that they tend to crowd into it in preference to other bodies ; while he supposed a dia- magnetic body to have a low degree of conducting power for the lines of force, so that they tend to avoid it. " If, then," he reasoned,f "a medium having a certain conducting power occupy the magnetic field, and then a portion of another medium or substance be placed in the field having a greater conducting power, the latter will tend to draw up towards the place of greatest force, displacing the former." There is an electrostatic effect to which this is quite analogous ; a charged body attracts a body whose specific inductive capacity is greater than that of the surrounding medium, and repels a body whose specific inductive capacity is less; in either case the tendency is to afford the path of best conductance to the lines of force .J

For some time the advocates of the "polarity" and " conduction " theories of diarnagnetism carried on a contro- versy which, indeed, like the controversy between the adherents of the one-fluid and two-fluid theories of electricity, persisted after it had been shown that the rival hypotheses were mathe- matically equivalent, and that no experiment could be suggested which would distinguish between them.

Meanwhile new properties of magnetizable bodies were being discovered. In 1847 Julius Pliicker (b. 1801, d. 1868), Professor of Natural Philosophy in the University of Bonn, while repeating and extending Faraday's magnetic experiments,

• This term was introduced by Faraday, Exp. Res., § 2790. t Exp. £es., § 2798.

J The mathematical theory of the motion of a magnetizable body in a non- uniform field of force was discussed by "W. Thomson (Kelvin) in 1847.

220 Faraday.

observed* that certain uniaxal crystals, when placed between the two poles of a magnet, tend to set themselves so that the optic axis has the equatorial position. At this time Faraday was continuing his researches ; and, while investigating the diamagnetic properties of bismuth, was frequently embarrassed by the occurrence of anomalous results. In 1848 he ascertained that these were in some way connected with the crystalline form of the substance, and showedf that when a crystal of bismuth is placed in a field of uniform magnetic force (so that no tendency to motion arises from its diamagnetism) it sets itself so as to have one of its crystalline axes directed along the lines of force.

At first he supposed this effect to be distinct from that which had been discovered shortly before by Pliicker. " The results," he wrote,J " are altogether very different from those produced by diamagnetic action. They are equally distinct from those discovered and described by Pliicker, in his beautiful researches into the relation of the optic axis to magnetic action ; for there the force is equatorial, whereas here it is axial. So they appear to present to us a new force, or a new form of force, in the molecules of matter, which, for convenience sake, I will conventionally designate by a new word, as the magne- crystallic force." Later in the same year, however, he recognized^ that " the phaenomena discovered by Pliicker and those of which I have given an account have one common origin and cause."

The idea of the " conduction " of lines of magnetic force by different substances, by which Faraday had so successfully explained the phenomena of diamagnetism, he now applied to the study of the magnetic behaviour of crystals. " If," he wrote,|| "the idea of conduction be applied to these magnecrystallic bodies, it would seem to satisfy all that requires explanation in their special results. A magnecrystallic substance would then be one which in the crystallized state could conduct onwards, or

• Ann. d. Phys. Ixxii (1847), p. 315; Taylor's Scientific Memoirs, v, p. 353.

t Phil. Trans., 1849, p. 1 ; Exp. Res., § 2454.

i Exp. Res., § 2469. § Ibid., § 2605. || Ibid., § 2837.

Faraday. 221

permit the exertion of the magnetic force with more facility in one direction than another ; and that direction would be the magnecrystallic axis. Hence, when in the magnetic field, the magnecrystallic axis would be urged into a position coincident with the magnetic axis, by a force correspondent to that difference, just as if two different bodies were taken, when the one with the greater conducting power displaces that which is weaker."

This hypothesis led Faraday to predict the existence of another type of magnecrystallic effect, as yet unobserved. " If such a view were correct/' he wrote,* " it would appear to follow that a diamagnetic body like bismuth ought to be less diamagnetic when its magnecrystallic axis is parallel to the magnetic axis than when it is perpendicular to it. In the two positions it should be equivalent to two substances having different conducting powers for magnetism, and therefore if submitted to the differential balance ought to present differential phaenomena." This expectation was realized when the matter was subjected to the test of experiment. f

The series of Faraday's " Experimental Researches in Electricity " end in the year 1855. The closing period of his life was quietly spent at Hampton Court, in a house placed at his disposal by the kindness of the Queen ; and here on August 25th, 1867, he passed away.

Among experimental philosophers Faraday holds by uni- versal consent the foremost place. The memoirs in which his discoveries are enshrined will never cease to be read with admiration and delight; and future generations will preserve with an affection not less enduring the personal records and familiar letters, which recall the memory of his humble and unselfish spirit.

*Exp. Res., § 2839. ^ Ibid., § 2841.

222 The Mathematical Electricians of the

CHAPTEK VII.

THE MATHEMATICAL ELECTRICIANS OF THE MIDDLE OF THE NINETEENTH CENTURY.

WHILE Faraday was engaged in discovering the laws of induced currents in his own way, by use of the conception of lines of force, his contemporary Franz Neumann was attacking the same problem from a different point of view. Xeumann preferred to take Ampere as his model ; and in 1845 published a memoir,* in which the laws of induction of currents were deduced by the help of Ampere's analysis.

Among the assumptions on which Neumann based his work was a rule which had been formulated, not long after Faraday's original discovery, by Emil Lenz,f and which may be enunciated as follows : when a conducting circuit is moved in a magnetic field, the induced current flows in such a direction that the ponderomotive forces on it tend to oppose the motion.

Let ds denote an element of the circuit which is in motion, and let C ds denote the component, taken in the direction of motion, of the ponderomotive force exerted by the inducing current on d$, when the latter is carrying unit current ; so that the value of C is known from Ampere's theory. Then Lenz's rule requires that the product of C into the strength of the induced current should be negative. Xeumann assumed that this is because it consists of a negative coefficient multiplying the square of C\ that is, he assumed the induced electro- motive force to be proportional to C. He further assumed it to be proportional to the velocity v of the motion; and thus obtained for the electromotive force induced in ds the expression

• ei-Cds, where e denotes a constant coefficient. By aid of this formula,

•Berlin Abhandlungen, 1845, p. 1 ; 1848, p. 1 ; reprinted as Xo. 10 and No. 36 of Ostwald's Klassiker-, translated Journal de Math, xiii (1848), p. 113. t Ann. d. Phys. xxxi (1834), p. 483.

Middle of the Nineteenth Century. 223

in the earlier part* of the memoir, he calculated the induced currents in various particular cases.

But having arrived at the formulae in this way, Neumann noticedf a peculiarity in them which suggested a totally different method of treating the subject. In fact, on examining the expression for the current induced in a circuit which is in motion in the field due to a magnet, it appeared that this induced current depends only on the alteration caused by the motion in the value of a certain function ; and, moreover, that this function is no other than the potential of the ponderomotive forces which, according to Ampere's theory, act between the circuit, supposed traversed by unit current, and the magnet.

Accordingly, Neumann now proposed to reconstruct his theory by taking this potential function as the foundation.

The nature of Neumann's potential, and its connexion with Faraday's theory, will be understood from the following considerations : —

The potential energy of a magnetic molecule M in a field of magnetic intensity B is (B . M) ; and therefore the potential energy of a current i flowing in a circuit s in this field is

where S denotes a diaphragm bounded by the circuit s ; as is seen at once on replacing the circuit by its equivalent magnetic shell S. If the field B be produced by a current i' flowing in a circuit s', we have, by the formula of Biot and Savart,

1 *•'

curl —

• §§ 1-8. It may be remarked that Neumann, in making use of Ohm's law, was (like everyone else at this time) unaware of the identity of electroscopic force with electrostatic potential. t § 9.

224 The Mathematical Electricians of the

Hence, the mutual potential energy of the two currents is

• . dS

which hy Stokes's transformation may be written in the form

(ds.ds')

This expression represents the amount of mechanical work which must be performed against the electro-dynamic pondero- motive forces, in order to separate the two circuits to an infinite distance apart, when the current-strengths are maintained unaltered.

The above potential function has been obtained by con- sidering the ponderomotive forces ; but it can now be connected with Faraday's theory of induction of currents. For by interpreting the expression

(B . dS)

If'

in terms of lines of force, we see that the potential function represents the product of i into the number of unit-lines of magnetic force due to s't which pass through the gap formed by the circuit s ; and since by Faraday's law the currents induced in s depend entirely on the variation in the number of these lines, it is evident that the potential function supplies all that is needed for the analytical treatment of the induced currents. This was Neumann's discovery.

The electromotive force induced in a circuit s by the motion of other circuits s', carrying currents i't is thus proportional to the time-rate of variation of the potential

(ds.ds').

so that if we denote by a the vector

Middle of the Nineteenth Century. 225

which, of course, is a function of the position of the element ds from which r is measured, then the electromotive force induced in any circuit-element ds by any alteration in the currents

which give rise to a is

(a. ds).

The induction of currents is therefore governed by the vector a ; this, which is generally known as the vector-potential, has from Neumann's time onwards played a great part in electrical theory. It may be readily interpreted in terms of Faraday's conceptions ; for (a . ds) represents the total number of unit lines of magnetic force which have passed across the line-element ds prior to the instant t. The vector-potential may in fact be regarded as the analytical measure of Faraday's electrotonic state*

While Neumann was endeavouring to comprehend the laws of induced currents in an extended form of Ampere's theory, another investigator was attempting a still more ambitious project : no less than that of uniting electrodynamics into a coherent whole with electrostatics.

Wilhelm Weber (6. 1804, d. 1890) was in the earlier part of his scientific career a friend and colleague of Gauss at Gottingen. In 1837, however, he became involved in political trouble. The union of Hanover with the British Empire, which had subsisted since the accession of the Hanoverian dynasty to the British throne, was in that year dissolved by the operation of the Salic law ; the Princess Victoria succeeded to the crown of England, and her uncle Ernest- Augustus to that of Hanover. The new king, who was a pronounced reactionary, revoked the free constitution which the Hanoverians had for some time enjoyed ; and Weber, who took a prominent part in opposing this action, was deprived of his professorship. From 1843 to 1849, when his principal theoretical researches in electricity were made, he occupied a chair in the University of Leipzig.

The theory of Weber was in its origin closely connected with the work of another Leipzig professor, Fechner, who in 1845f introduced certain assumptions regarding the nature of

• Cf. pp. 212, 272. t Ann. d. Phys. Lxiv (1845), p. 337.

Q

226 The Mathematical Electricians of the

electric currents. Fechner supposed every current to consist in a streaming of electric charges, the vitreous charges travelling in one direction, and the resinous charges, equal to them in magnitude and number, travelling in the opposite direction with equal velocity. He further supposed that like charges attract each other when they are moving parallel to the same direction, while unlike charges attract when they are moving in opposite directions. On these assumptions he succeeded in bringing Faraday's induction effects into connexion with Ampere's laws of electrodynamics.

In 1846 Weber,* adopting the same assumptions as Fechner, analysed the phenomena in the following way : —

The formula of Ampere for the ponderomotive force between two elements ds, ds' of currents it i ', may be written

r ds ds r2 ds ds'

Suppose now that X units of vitreous electricity are contained in unit length of the wire s, and are moving with velocity u ; and that an equal quantity of resinous electricity is moving with velocity u in the opposite direction ; so that