A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 25 of 27

Calculated by I. 617 780 1000 1210 1603

II. 623 789 1000 1200 1565

III. 976 993 1000 1017 1041 Rotation of the ray E = 21°. 58'.

We are so little acquainted with the details of the molecular

• ' Explicare tentatur quomodo fiat ut lucis planum polarizationis per vires elec- tricas vel magneticas declinetur.' Halis Saxonum, 1858.

•f* These three forms of the equations of motion were first suggested by Sir G. B. Airy (Phil. Mag., June 1846) as a means of analysing the phenomenon then recently discovered by Faraday. Mac Cullagh had previously suggested equations containing

terms of the form — in order to represent mathematically the phenomena of quartz.

These equations were offered by Mac Cullagh and Airy, 'not as giving a mechanical explanation of the phenomena, but as shewing that the phenomena may be explained by equations, which equations appear to be such as might possibly be deduced from some plausible mechanical assumption, although no such assumption lias yet been made.'

831.] ARGUMENT OF THOMSON. 415

constitution of bodies, that it is not probable that any satisfactory theory can be formed relating to a particular phenomenon, such as that of the magnetic action on light, until, by an induction founded on a number of different cases in which visible phenomena are found to depend upon actions in which the molecules are concerned, we learn something more definite about the properties which must be attributed to a molecule in order to satisfy the conditions of ob served facts.

The theory proposed in the preceding pages is evidently of a provisional kind, resting as it does on unproved hypotheses relating to the nature of molecular vortices, and the mode in which they are affected by the displacement of the medium. We must therefore regard any coincidence with observed facts as of much less scientific value in the theory of the magnetic rotation of the plane of polari zation than in the electromagnetic theory of light, which, though it involves hypotheses about the electric properties of media, does not speculate as to the constitution of their molecules.

831.] NOTE. — The whole of this chapter may be regarded as an expansion of the exceedingly important remark of Sir William Thomson in the Proceedings of the Royal Society, June 1856 : — ' The magnetic influence on light discovered by Faraday depends on the direction of motion of moving particles. For instance, in a medium possessing it, particles in a straight line parallel to the lines of magnetic force, displaced to a helix round this line as axis, and then projected tangentially with such velocities as to describe circles, will have different velocities according as their motions are round in one direction (the same as the nominal direction of the galvanic current in the magnetizing coil), or in the contrary direction. But the elastic reaction of the medium must be the same for the same displacements, whatever be the velocities and directions of the par ticles ; that is to say, the forces which are balanced by centrifugal force of the circular motions are equal, while the luminiferous motions are unequal. The absolute circular motions being there fore either equal or such as to transmit equal centrifugal forces to the particles initially considered, it follows that the luminiferous motions are only components of the whole motion ; and that a less luminiferous component in one direction, compounded with a mo tion existing in the medium when transmitting no light, skives an equal resultant to that of a greater luminiferous motion in the con trary direction compounded with the same non -luminous motion. I think it is not only impossible to conceive any other than this

410 MAGNETIC ACTION ON LIGHT.

dynamical explanation of the fact that circularly-polarized light transmitted through magnetized glass parallel to the lines of mag netizing force, with the same quality, right-handed always, or left- handed always, is propagated at different rates according as its course is in the direction or is contrary to the direction in which a north magnetic pole is drawn ; but I believe it can be demonstrated that no other explanation of that fact is possible. Hence it appears that Faraday's optical discovery affords a demonstration of the re ality of Ampere's explanation of the ultimate nature of magnetism ; and gives a definition of magnetization in the dynamical theory of heat. The introduction of the principle of moments of momenta (" the conservation of areas ") into the mechanical treatment of Mr. Rankine's hypothesis of " molecular vortices," appears to indi cate a line perpendicular to the plane of resultant rotatory mo mentum ("the invariable plane") of the thermal motions as the magnetic axis of a magnetized body, and suggests the resultant moment of momenta of these motions as the definite measure of the " magnetic moment." The explanation of all phenomena of electromagnetic attraction or repulsion, and of electromagnetic in duction, is to be looked for simply in the inertia and pressure of the matter of which the motions constitute heat. Whether this matter is or is not electricity, whether it is a continuous fluid inter- permeating the spaces between molecular nuclei, or is itself mole- cularly grouped ; or whether all matter is continuous, and molecular heterogeneousness consists in finite vortical or other relative mo tions of contiguous parts of a body ; it is impossible to decide, and perhaps in vain to speculate, in the present state of science.'

A theory of molecular vortices, which I worked out at consider able length, was published in the Phil. Mag. for March, April, and May, 1861, Jan. and Feb. 1862.

I think we have good evidence for the opinion that some pheno menon of rotation is going on in the magnetic field, that this rota tion is performed by a great number of very small portions of matter, each rotating on its own axis, this axis being parallel to the direction of the magnetic force, and that the rotations of these dif ferent vortices are made to depend on one another by means of some kind of mechanism connecting them.

The attempt which I then made to imagine a working model of this mechanism must be taken for no more than it really is, a de monstration that mechanism may be imagined capable of producing a connexion mechanically equivalent to the actual connexion of the

831.] THEOBY OP MOLECULAK VORTICES. 417

parts of the electromagnetic field. The problem of determining the mechanism required to establish a given species of connexion be tween the motions of the parts of a system always admits of an infinite number of solutions. Of these, some may be more clumsy or more complex than others, but all must satisfy the conditions of mechanism in general.

The following results of the theory, however, are of higher value : —

(1) Magnetic force is the effect of the centrifugal force of the vortices.

(2) Electromagnetic induction of currents is the effect of the forces called into play when the velocity of the vortices is changing.

(3) Electromotive force arises from the stress on the connecting mechanism.

(4) Electric displacement arises from the elastic yielding of the connecting mechanism.

VOL. II.

CHAPTER XXII

FEBROMAQNETISM AND DIAMAGNETISM EXPLAINED BY MOLECULAR CURRENTS.

On Electromagnetic Theories of Magnetism.

832.] WE have seen (Art. 380) that the action of magnets on one another can be accurately represented by the attractions and repulsions of an imaginary substance called * magnetic matter.' We have shewn the reasons why we must not suppose this magnetic matter to move from one part of a magnet to another through a sensible distance, as at first sight it appears to do when we magnetize a bar, and we were led to Poisson's hypothesis that the magnetic matter is strictly confined to single molecules oi" the mag netic substance, so that a magnetized molecule is one in which the opposite kinds of magnetic matter are more or less separated to wards opposite poles of the molecule, but so that no part of either can ever be actually separated from the molecule (Art. 430).

These arguments completely establish the fact, that magnetiza tion is a phenomenon, not of large masses of iron, but of molecules, that is to say, of portions of the substance so small that we cannot by any mechanical method cut one of them in two, so as to obtain a north pole separate from a south pole. But the nature of a mag netic molecule is by no means determined without further investi gation. We have seen (Art. 442) that there are strong reasons for believing that the act of magnetizing iron or steel does not consist in imparting magnetization to the molecules of which it is com posed, but that these molecules are already magnetic, even in un- magnetized iron, but with their axes placed indifferently in all directions, and that the act of magnetization consists in turning the molecules so that their axes are either rendered all parallel to one direction, or at least. are deflected towards that direction.

834-] AMPERE'S THEORY. 419

833.] Still, however, we have arrived at no explanation of the nature of a magnetic molecule, that is, we have not recognized its likeness to any other thing of which we know more. We have therefore to consider the hypothesis of Ampere, that the magnetism of the molecule is due to an electric current constantly circulating in some closed path within it.

It is possible to produce an exact imitation of the action of any magnet on points external to it, by means of a sheet of electric currents properly distributed on its outer surface. But the action of the magnet on points in the interior is quite different from the action of the electric currents on corresponding points. Hence Am pere concluded that if magnetism is to be explained by means of electric currents, these currents must circulate within the molecules of the magnet, and must not flow from one molecule to another. As we cannot experimentally measure the magnetic action at a point in the interior of a molecule, this hypothesis cannot be dis proved in the same way that we can disprove the hypothesis of currents of sensible extent within the magnet.

Besides this, we know that an electric current, in passing from one part of a conductor to another, meets with resistance and gene rates heat ; so that if there were currents of the ordinary kind round portions of the magnet of sensible size, there would be a constant expenditure of energy required to maintain them, and a magnet would be a perpetual source of heat. By confining the circuits to the molecules, within which nothing is known about resistance, we may assert, without fear of contradiction, that the current, in cir culating within the molecule, meets with no resistance.

According to Ampere's theory, therefore, all the phenomena of magnetism are due to electric currents, and if we could make ob servations of the magnetic force in the interior of a magnetic mole cule, we should find that it obeyed exactly the same laws as the force in a region surrounded by any other electric circuit.

834.] In treating of the force in the interior of magnets, we have supposed the measurements to be made in a small crevasse hollowed out of the substance of the magnet, Art. 395. We were thus led to consider two different quantities, the magnetic force and the magnetic induction, both of which are supposed to be observed in a space from which the magnetic matter is removed. We were not supposed to be able to penetrate into the interior of a mag netic molecule and to observe the force within it.

If we adopt Ampere's theory, we consider a magnet, not as a

E e 2

420 ELECTE1C THEORY OF MAGNETISM. [835.

continuous substance, the magnetization of which varies from point to point according to some easily conceived law, but as a multitude of molecules, within each of which circulates a system of electric currents, giving rise to a distribution of magnetic force of extreme complexity, the direction of the force in the interior of a molecule being generally the reverse of that of the average force in its neigh bourhood, and the magnetic potential, where it exists at all, being a function of as many degrees of multiplicity as there are molecules in the magnet.

835.] But we shall find, that, in spite of this apparent complexity, which, however, arises merely from the coexistence of a multitude of simpler parts, the mathematical theory of magnetism is greatly simplified by the adoption of Ampere's theory, and by extending our mathematical vision into the interior of the molecules.

In the first place, the two definitions of magnetic force are re duced to one, both becoming the same as that for the space outside the magnet. In the next place, the components of the magnetic force everywhere satisfy the condition to which those of induction are subject, namely, da dp, dy _ dx dy dz ~

In other words, the distribution of magnetic force is of the same nature as that of the velocity of an incompressible fluid, or, as we have expressed it in Art. 25, the magnetic force has no convergence.

Finally, the three vector functions — the electromagnetic momen tum, the magnetic force, and the electric current — become more simply related to each other. They are all vector functions of no convergence, and they are derived one from the other in order, by the same process of taking the space-variation, which is denoted by Hamilton by the symbol V.

836.] But we are now considering magnetism from a physical point of view, and we must enquire into the physical properties of the molecular currents. We assume that a current is circulating in a molecule, and that it meets with no resistance. If L is the coefficient of self-induction of the molecular circuit, and M the co efficient of mutual induction between this circuit and some other circuit, then if y is the current in the molecule, and y that in the other circuit, the equation of the current y is

=-Sr, (2)

838.] CIRCUITS OF NO RESISTANCE. 421

and since by the hypothesis there is no resistance, R = 0, and we get by integration

Ly + My = constant, = Lyot say. (3)

Let us suppose that the area of the projection of the molecular circuit on a plane perpendicular to the axis of the molecule is A, this axis being defined as the normal to the plane on which the projection is greatest. If the action of other currents produces a magnetic force, X, in a direction whose inclination to the axis of the molecule is 0, the quantity My becomes XA cos0, and we have as the equation of the current

Ly + XAco$e — Ly0, (4)

where y0 is the value of y when X = 0.

It appears, therefore, that the strength of the molecular current depends entirely on its primitive value y0, and on the intensity of the magnetic force due to other currents.

837.] If we suppose that there is no primitive current, but that the current is entirely due to induction, then

• XA

y = j— cos 0. (o)

Jj

The negative sign shews that the direction of the induced cur rent is opposite to that of the inducing current, and its magnetic action is such that in the interior of the circuit it acts in the op posite direction to the magnetic force. In other words, the mole cular current acts like a small magnet whose poles are turned towards the poles of the same name of the inducing magnet.

Now this is an action the reverse of that of the molecules of iron under magnetic action. The molecular currents in iron, therefore, are not excited by induction. But in diamagnetic substances an action of this kind is observed, and in fact this is the explanation of diamagnetic polarity which was first given by Weber.

Weber's Theory of Diamagnetism.

838.] According to Weber's theory, there exist in the molecules of diamagnetic substances certain channels round which an electric current can circulate without resistance. It is manifest that if we suppose these channels to traverse the molecule in every direction, this amounts to making the molecule a perfect conductor.

Beginning with the assumption of a linear circuit within the mo lecule, we have the strength of the current given by equation (5).

422 ELECTRIC THEORY OF MAGNETISM. [8 39.

The magnetic moment of the current is the product of its strength by the area of the circuit, or yA, and the resolved part of this in the direction of the magnetizing force is yAcosO, or, by (5),

Y //2 -^-cos20. (6)

If there are n such molecules in unit of volume, and if their axes are distributed indifferently in all directions, then the average value of cos20 will be J, and the intensity of magnetization of the substance will be ^nXA* ,?.

L Neumann's coefficient of magnetization is therefore

_

The magnetization of the substance is therefore in the opposite direction to the magnetizing force, or, in other words, the substance is diamagnetic. It is also exactly proportional to the magnetizing force, and does not tend to a finite limit, as in the case of ordinary magnetic induction. See Arts. 442, &c.

839.] If the directions of the axes of the molecular channels are arranged, not indifferently in all directions, but with a preponder ating number in certain directions, then the sum

Ju

extended to all the molecules will have different values according to the direction of the line from which 6 is measured, and the dis tribution of these values in different directions will be similar to the distribution of the values of moments of inertia about axes in dif ferent directions through the same point.

Such a distribution will explain the magnetic phenomena related to axes in the body, described by Pliicker, which Faraday has called Magne-crystallic phenomena. See Art. 435.

840.] Let us now consider what would be the effect, if, instead of the electric current being confined to a certain channel within the molecule, the whole molecule were supposed a perfect conductor.

Let us begin with the case of a body the form of which is acyclic, that is to say, which is not in the form of a ring or perforated body, and let us suppose that this body is everywhere surrounded by a thin shell of perfectly conducting matter.

We have proved in Art. 654, that a closed sheet of perfectly conducting matter of any form, originally free from currents, be-

842.] PERFECTLY CONDUCTING MOLECULES. 423

comes, when exposed to external magnetic force, a current-sheet, the action of which on every point of the interior is such as to make the magnetic force zero.

It may assist us in understanding this case if we observe that the distribution of magnetic force in the neighbourhood of such a body is similar to the distribution of velocity in an incompressible fluid in the neighbourhood of an impervious body of the same form.

It is obvious that if other conducting shells are placed within the first, since they are not exposed to magnetic force, no currents will be excited in them. Hence, in a solid of perfectly conducting material, the effect of magnetic force is to generate a system of currents which are entirely confined to the surface of the body.

841.] If the conducting body is in the form of a sphere of radius r, its magnetic moment is

and if a number of such spheres are distributed in a medium, so that in unit of volume the volume of the conducting matter is Xf, then, by putting ^=1, and /x2 = 0 in equation (17), Art. 314, we find the coefficient of magnetic permeability,

f\ n If

(9)

whence we obtain for Poisson's magnetic coefficient

t=-\tf, (10)

and for Neumann's coefficient of magnetization by induction

Since the mathematical conception of perfectly conducting bodies leads to results exceedingly different from any phenomena which we can observe in ordinary conductors, let us pursue the subject somewhat further.

842.] Returning to the case of the conducting channel in the form of a closed curve of area A, as in Art. 836, we have, for the moment of the electromagnetic force tending to increase the angle 0,

n0 m (12)

= — ^-sin0cos0. (13)

This force is positive or negative according as 0 is less or greater than a right angle. Hence the effect of magnetic force on a per fectly conducting channel tends to turn it with its axis at right

424 ELECTRIC THEORY OF MAGNETISM. [843.

angles to the line of magnetic force, that is, so that the plane of the channel becomes parallel to the lines of force.

An effect of a similar kind may be observed by placing a penny or a copper ring between the poles of an electromagnet. At the instant that the magnet is excited the ring turns its plane towards the axial direction, but this force vanishes as soon as the currents are deadened by the resistance of the copper *.

843.] We have hitherto considered only the case in which the molecular currents are entirely excited by the external magnetic force. Let us next examine the bearing of Weber's theory of the magneto-electric induction of molecular currents on Ampere's theory of ordinary magnetism. According to Ampere and Weber, the molecular currents in magnetic substances are not excited by the external magnetic force, but are already there, and the molecule itself is acted on and deflected by the electromagnetic action of the magnetic force on the conducting circuit in which the current flows. When Ampere devised this hypothesis, the induction of electric cur rents was not known, and he made no hypothesis to account for the existence, or to determine the strength, of the molecular currents.

We are now, however, bound to apply to these currents the same laws that Weber applied to his currents in diamagnetic molecules. We have only to suppose that the primitive value of the current y, when no magnetic force acts, is not zero but y0. The strength of the current when a magnetic force, X, acts on a molecular current of area A, whose axis is inclined 6 to the line of magnetic force, is

and the moment of the couple tending to turn the molecule so as

to increase 0 is X2A2

— y0XAsm0 + sin 26. (15)

Hence, putting A

AyQ = m, /- = *, (16)

^7o

in the investigation in Art. 443, the equation of equilibrium becomes Xsin0 — 3X2sin0cos0 = Dsin(a-0). (17)

The resolved part of the magnetic moment of the current in the direction of X is

XA2

y A cosO = y0Acos0 -- ^— cos2 (9, (18)

L

= mcosO(l-3XcoaO). (19)

• See Faraday, Exp. Res., 2310, &c.

845-] MODIFIED THEORY OF INDUCED MAGNETISM. 425

844.] These conditions differ from those in Weber's theory of magnetic induction by the terms involving the coefficient B. If BX is small compared with unity, the results will approximate to those of Weber's theory of magnetism. If BX is large compared with unity, the results will approximate to those of Weber's theory of diamagnetism.

Now the greater y0, the primitive value of the molecular current, the smaller will B become, and if L is also large, this will also diminish B. Now if the current flows in a ring channel, the value

T>

of L depends on log — , where R is the radius of the mean line of

the channel, and r that of its section. The smaller therefore the section of the channel compared with its area, the greater will be L, the coefficient of self-induction, and the more nearly will the phe nomena agree with Weber's original theory. There will be this difference, however, that as X, the magnetizing force, increases, the temporary magnetic moment will not only reach a maximum, but will afterwards diminish as X increases.

If it should ever be experimentally proved that the temporary magnetization of any substance first increases, and then diminishes as the magnetizing force is continually increased, the evidence of the existence of these molecular currents would, I think, be raised almost to the rank of a demonstration.

845.] If the molecular currents in diamagnetic substances are confined to definite channels, and if the molecules are capable of being deflected like those of magnetic substances, then, as the mag netizing force increases, the diamagnetic polarity will always increase, but, when the force is great, not quite so fast as the magnetizing force. The small absolute value of the diamagnetic coefficient shews, however, that the deflecting force on each molecule must be small compared with that exerted on a magnetic molecule, so that any result due to this deflexion is not likely to be perceptible.

If, on the other hand, the molecular currents in diamagnetic bodies are free to flow through the whole substance of the molecules, the diamagnetic polarity will be strictly proportional to the mag netizing force, and its amount will lead to a determination of the whole space occupied by the perfectly conducting masses, and, if we know the number of the molecules, to the determination of the size of each,

CHAPTER XXIII.

THEORIES OF ACTION AT A DISTANCE.

On the Explanation of Ampere's Formula given by Gauss and Weber.

846.] The attraction between the elements ds and da' of two circuits, carrying electric currents of intensity i and i't is, by Ampere's formula,

ii' ds ds' dr dr\ ft\

3--; (1)

zr_ .

r2 v ds ds ds ds '

the currents being estimated in electromagnetic units. See Art. 526. The quantities, whose meaning as they appear in these expres sions we have now to interpret, are

dr dr . d2r

cos e, -jr- -7-7 > and -=— T> ; ds ds dsds

and the most obvious phenomenon in which to seek for an inter pretation founded on a direct relation between the currents is the relative velocity of the electricity in the two elements.

847.] Let us therefore consider the relative motion of two par ticles, moving with constant velocities v and v' along the elements ds and ds' respectively. The square of the relative velocity of these particles is U2 = vz _2vv'cos e + v'2-, (3)

and if we denote by r the distance between the particles,

dr dr ,dr ...

v7 v 7+v -r>> (4)

^ ds ds

.dr dr /9 /dr\2 /ev

v '' 5

848.] FECHNER'S HYPOTHESIS. 427

where the symbol <) indicates that, in the quantity differentiated, the coordinates of the particles are to be expressed in terms of the time.

It appears, therefore, that the terms involving the product vv' in the equations (3), (5), and (6) contain the quantities occurring in (1) and (2) which we have to interpret. We therefore endeavour to

~~

and — 2 • But in order to

express (1) and (2) in terms of ^2, i

do so we must get rid of the first and third terms of each of these expressions, for they involve quantities which do not appear in the formula of Ampere. Hence we cannot explain the electric current as a transfer of electricity in one direction only, but we must com bine two opposite streams in each current, so that the combined effect of the terms involving v2 and v'2 may be zero.

848.] Let us therefore suppose that in the first element, ds, we have one electric particle, £, moving with velocity ?;, and another, elt moving with velocity vl , and in the same way two particles, ef and e, in ds't moving with velocities v' and v'L respectively.

The term involving v2 for the combined action of these particles

Similarly 2 (t/W) = (v'2e' + v\2e) (e + ^) ; (8)

and 2(vtfeS) = (ve + v^^v'e' + vYi). (9)

In order that 2 (o2ee') may be zero, we must have either

/ + e\ = 0, or V2e + v12e1 = 0. (10)

According to Eechner's hypothesis, the electric current consists of a current of positive electricity in the positive direction, com bined with a current of negative electricity in the negative direc tion, the two currents being exactly equal in numerical magnitude, both as respects the quantity of electricity in motion and the velo city with which it is moving. Hence both the conditions of (10) are satisfied by Fechner's hypothesis.

But it is sufficient for our purpose to assume, either —

That the quantity of positive electricity in each element is nu merically equal to the quantity of negative electricity ; or —

That the quantities of the two kinds of electricity are inversely as the squares of their velocities.

Now we know that by charging the second conducting wire as a whole, we can make e' -f e\ either positive or negative. Such a charged wire, even without a current, according to this formula, would act on the first wire carrying a current in which v2e -j- r12el

428 ACTION AT A DISTANCE. [849.

has a value differing from zero. Such an action has never been observed.

Therefore, since the quantity e' + e\ may be shewn experimentally not to be always zero, and since the quantity v2e + v21el is not capable of being experimentally tested, it is better for these specu lations to assume that it is the latter quantity which invariably vanishes.

849.] Whatever hypothesis we adopt, there can be no doubt that the total transfer of electricity, reckoned algebraically, along the first circuit, is represented by

ve--v1ei = dels;

where c is the number of units of statical electricity which are transmitted by the unit electric current in the unit of time, so that we may write equation (9)

2 (vv'ee'} = c2 ii'ds ds'. (11)

Hence the sums of the four values of (3), (5), and (6) become

2 (ee'n2) = -2 c^ii'ds ds' cos e ; (12)

^, (13)

ds ds

and we may write the two expressions (1) and (2) for the attraction between ds and ds'

850.] The ordinary expression, in the theory of statical electri-

PP

city, for the repulsion of two electrical particles e and e' is - , and

which gives the electrostatic repulsion between the two elements if they are charged as wholes.

Hence, if we assume for the repulsion of the two particles either of the modified expressions

we may deduce from them both the ordinary electrostatic forces, and the forces acting between currents as determined by Ampere.

FORMULAE OF GAUSS AND WEBER, 429

851.] The first of these expressions, (18), was discovered by Gauss * in July 1835, and interpreted by him as a fundamental law of electrical action, that ' Two elements of electricity in a state of relative motion attract or repel one another, but not in the same way as if they are in a state of relative rest.' This discovery was not, so far as I know, published in the lifetime of Gauss, so that the second expression, which was discovered independently by W.Weber, and published in the first part of his celebrated Elektrodynamische Maasbe&timmungen^ , was the first result of the kind made known to the scientific world.

852.] The two expressions lead to precisely the same result when they are applied to the determination of the mechanical force be tween two electric currents, and this result is identical with that of Ampere. But when they are considered as expressions of the physical law of the action between two electrical particles, we are led to enquire whether they are consistent with other known facts of nature.

Both of these expressions involve the relative velocity of the particles. Now, in establishing- by mathematical reasoning the well-known principle of the conservation of energy, it is generally assumed that the force acting between two particles is a function of the distance only, and it is commonly stated that if it is a function of anything else, such as the time, or the velocity of the particles, the proof would not hold.

Hence a law of electrical action, involving the velocity of the particles, has sometimes been supposed to be inconsistent with the principle of the conservation of energy.

853.] The formula of Gauss is inconsistent with this principle, and must therefore be abandoned, as it leads to the conclusion that energy might be indefinitely generated in a finite system by physical means. This objection does not apply to the formula of Weber, for he has shewn J that if we assume as the potential energy of a system consisting of two electric particles,

the repulsion between them, which is found by differentiating this quantity with respect to r, and changing the sign, is that given by the formula (19).

• Werke (G-ottingen edition, 1867), \ol.v. p. 616. t Abh. Leibnizens Qes., Leipzig (1846). J Pogg. Ann., Ixxiii. p. 229 (1848).

430 ACTION AT A DISTANCE. [8 54.

Hence the work done on a moving particle by the repulsion of a fixed particle is ^o~"^i' where \ITO and //j are the values of \ff at the beginning and at the end of its path. Now \j/ depends only on the distance, r, and on the velocity resolved in the direction of r. If, therefore, the particle describes any closed path, so that its position, velocity, and direction of motion are the same at the end as at the beginning, ^ will be equal to ^0, and no work will be done on the whole during the cycle of operations.

Hence an indefinite amount of work cannot be generated by a particle moving in a periodic manner under the action of the force assumed by Weber.

854.] But Helmholtz, in his very powerful memoir on the 'Equa tions of Motion of Electricity in Conductors at Rest '*, while he shews that Weber's formula is not inconsistent with the principle of the conservation of energy, as regards only the work done during a complete cyclical operation, points out that it leads to the conclu sion, that two electrified particles, which move according to Weber's law, may have at first finite velocities, and yet, while still at a finite distance from each other, they may acquire an infinite kinetic energy, and may perform an infinite amount of work.

To this Weber f replies, that the initial relative velocity of the particles in Helmholtz's example, though finite, is greater than the velocity of light ; and that the distance at which the kinetic energy becomes infinite, though finite, is smaller than any magnitude which we can perceive, so that it may be physically impossible to bring two molecules so near together. The example, therefore, cannot be tested by any experimental method.