A History of the Theories of Aether and Electricity (1910) — part 15 of 29

Let X', u', denote the corresponding quantities for the other current; and let the suffix ! be taken to refer to the action between the positive charges in the two wires, the suffix 2 to the action between the positive charge in s and the negative charge in s, the suffix 3 to the action between the negative charge in s and the positive charge in s', and the suffix 4 to the action between the negative charges in the two wires. Then we have

'dr\ dr , dr

— = u — + u — ,, dtji ds ds

• Elektrodynamische Maassbestimmungen, Leipzig Abhandl., 1846 : Ann. d. Phys. hcxiii (1848), p. 193: English translation in Taylor's Scientific Memoirs, v (1852), p. 489.

Middle of the Nineteenth Century. 227

and

fdzr\ zdzr , c?r <Fr

__ = u* __ + 2uu --—?-, + u 2 -j-7- • df <fo* dsds ds*

By aid of these and the similar equations with the suffixes 3, 3, 4, the equation for the ponderomotive force may be transformed into the equation

d?r\ f dV

I nt

A A' tl&rtst' \ \ ///z /, \ ///* /„

F =

But this is the equation which we should have obtained had we set out from the following assumptions : that the ponderomotive force between two current-elements is the resultant of the force between the positive charge in ds and the positive charge in ds', of the force between the positive charge in ds and the negative charge in dst etc. ; and that any two electrified particles of charges e and e', whose distance apart is r, repel each other with a force of magnitude

*l&

Two such charges would, of course, also exert on each other an electrostatic repulsion, whose magnitude in these units would be eec'/r2, where c denotes a constant* of the dimensions of a velocity, whose value is approximately 3 x 1010 cm./sec. So that on these assumptions the total repellent force would be

ee'cz f rr r*

«

• The units which have been adopted in the above investigation depend on the electrodynamic actions of currents ; i.e. they are such that two unit currents flowing in parallel circular circuits at a certain distance apart exert unit ponderomotive force on each other. The quantity of electricity conveyed in unit time by such a unit current is adopted as the unit~eharge. This unit charge is not identical with the electrostatic unit charge, which is definedHqbe such that two unit charges at unit distance apart repel each other with unit poniieiQmotive force. Hence the necessity for introducing the factor c.

Q

228 The Mathematical Electricians of the

This expression for the force between two electric charges was taken by Weber as the basis of his theory. Weber's is the first of the electron-theories — a name given to any theory which attributes the phenomena of electrodynamics to the agency of moving electric charges, the forces on which depend not only on the position of the charges (as in electrostatics), but also on their velocity.

The latter feature of Weber's theory led its earliest critics to deny that his law of force could be reconciled with the principle of conservation of energy. They were, however, mistaken on this point, as may be seen from the following considerations. The above expression for the force between two charges may be written in the form

where U denotes the expression

ee'c~

Consider now two material particles at distance r apart, whose mechanical kinetic energy is T, and whose mechanical potential energy is F, and which carry charges e and e'. The equations of motion of these particles will be exactly the same as the equations of motion of a dynamical system for which the kinetic energy is

ee'i*

and the potential energy is

To such a system the principle of conservation of energy may be applied : the equation of energy is, in fact,

m -rr 1 > • 6e ' G"

T + V - — ee r + - = constant. 2r r

Middle of the Nineteenth Century. 229

The first objection made to Weber's theory is thus disposed of ; but another and more serious one now presents itself. The occurrence of the negative sign with the term - ee'r^/Zr implies that a charge behaves somewhat as if its mass were negative, so that in certain circumstances its velocity might increase indefi- nitely under the action of a force opposed to the motion. This is one of the vulnerable points of Weber's theory, and has been the object of much criticism. In fact,* suppose that one charged particle of mass /z. is free to move, and that the other charges are spread uniformly over the surface of a hollow spherical insulator in which the particle is enclosed. The equation of conservation of energy is

^(fi-ep)v*+ V= constant,

where e denotes the charge of the particle, v its velocity, V its potential energy with respect to the mechanical forces which act on it, and p denotes the quantity

• cos-(v.r)dS,

where the integration is taken over the sphere, and where o- denotes the surface-density ; p is independent of the position of the particle p within the sphere. If now the electric charge on the sphere is so great that ep is greate^-tbsciTT^ then v2 and V must increase and diminish together; which is evidently absurd.

Leaving this objection unanswered, we proceed to show how Weber's law of force between electrons leads to the formulae for the induction of currents.

The mutual energy of two moving charges is

~\ ~2cV'

°r ! * " L"v«r'Y' ~1'

r |_ *c r J

where v and v' denote the velocities of the charges ; so that the

• This example was given by Helmholtz, Journal fur Math. Ixxv (1873), p. 35 ; Phil. Mag. xliv (1872), p. 530.

230 The Mathematical Electricians of the

mutual energy of two current-elements containing charges e, e respectively of each kind of electricity, is

r3

If ds, ds' denote the lengths of the elements, and i, if the currents in them, we have

ids = 2ev, i'ds' = 2«V ;

so the mutual energy of two current-elements is

nf

-(r.ds').(r.ds).

The mutual energy of ids with all the other currents is therefore

t(dt.a), where a denotes a vector-potential

By reasoning similar to Neumann's, it may be shown that the electromotive force induced in ds by any alteration in the rest of the field is

-(ds.a);

and thus a complete theory of induced currents may be constructed.

The necessity for induced currents may be inferred by general reasoning from the first principles of Weber's theory. When a circuit s moves in the field due to currents, the velocity of the vitreous charges in s is, owing to the motion of s, not equal and opposite to that of the resinous charges : this gives rise to a difference in the forces acting on the vitreous and resinous charges in s ; and hence the charges of opposite sign separate from each other and move in opposite directions.

The assumption that positive and negative charges move with equal and opposite velocities relative to the matter of

Middle of the Nineteenth Century. 231

the conductor is one to which, for various reasons which will appear later, objection may be taken ; but it is an integral part of Weber's theory, and cannot be excised from it. In fact, if this condition were not satisfied, and if the law of force were Weber's, electric currents would exert forces on electrostatic charges at rest*; as may be seen by the following example. Let a current flow in a closed circuit formed by arcs of two concentric circles and the portions of the radii connecting their extremities; then, if Weber's law were true, and if only one kind of electricity were in motion, the current would evidently exert an electrostatic force on a charge placed at the centre of the circles. It has been shown,f indeed, that the assumption of opposite electricities moving with equal and opposite veloci- ties in a circuit is almost inevitable in any theory of the type of Weber's, so long as the mutual action of two charges is assumed to depend only on their relative (as opposed to their absolute) motion.

The law of Weber is not the only one of its kind; an alterna- tive to it was suggested by Bernhard Eiemann (b. 1826, d. 1866), in a course of lectures which were delivered^ at Gottingen in 1861, and which were published after his death by K. Hattendorff. Kiemann proposed as the electrokinetic energy of two electrons e (x, y, z) and e\xf, y\ z') the expression

this differs from the corresponding expression given by Weber only in that the relative velocity of the two electrons is substituted in place of the component of this velocity along the radius vector. Eventually, as will be seen later, the laws

• This remark was first made by Clausius, Journal fur Math. Ixxxii (1877), p. 86: the simple proof given above is due to Grassmann, Journal fur Math. Ixxxiii (1877), p. 57.

t H. Lorberg, Journal fur Math. Ixxxiv (1878), p. 305.

J Schicere, Elektricitat und Magnetismus, nach den Vorlesungen von B. Riemann : Hannover, 1875, p. 326. Another alternative to Weber's law had been discovered by Gauss so far back as 1835, but was not published until after his death: cf. Gauss' Werke, v, p. 616.

232 The Mathematical Electricians of the

of Riemann and Weber were both abandoned in favour of a third alternative.

At the time, however, Weber's discovery was felt to be a great advance ; and indeed it had, perhaps, the greatest share in awakening mathematical physicists to a sense of the possi- bilities latent in the theory of electricity. Beyond this, its influence was felt in general dynamics ; for Weber's electro- kinetic energy, which resembled kinetic energy in some respects and potential energy in others, could not be precisely classified under either head ; and its introduction, by helping to break down the distinction which had hitherto subsisted between the two parts of the kinetic potential, prepared the way for the modern transformation-theory of dynamics.*

Another subject whose development was stimulated by the work of Weber was the theory of gravitation. That gravitation is propagated by the action of a medium, and consequently is a process requiring time for its accomplishment, had been an article of faith with many generations of physicists. Indeed, the dependence of the force on the distance between the attracting bodies seemed to suggest this idea ; for a propagation which is truly instantaneous would, perhaps, be more naturally conceived to be effected by some kind of rigid connexion between the bodies, which would be more likely to give a force independent of the mutual distance.

It is obvious that, if the simple law of Newton is abandoned, there is a wide field of rival hypotheses from which to choose its successor. The first notable attempt to discuss the question was made by Laplace. f Laplace supposed gravity to be pro- duced by the impulsion on the attracted body of a " gravific fluid," which flows with a definite velocity toward the centre of attraction — say, the sun. If the attracted body or planet is in motion, the velocity of the fluid relative to it will be compounded of the absolute velocity of the fluid and the reversed velocity of the planet, and the force of gravity will

• Cf. "Whittaker, Analytical Dynamics, chapters ii, iii, xi. t Meeanique Celeste, Livre x, chap, vii, § 22.

Middle of the Nineteenth Century. 233

act in the direction thus determined, its magnitude being unaltered by the planet's motion. This amounts to supposing that gravity is subject to an aberrational effect similar to that observed in the case of light. It is easily seen that the modi- fication thus introduced into Newton's law may be represented by an additional perturbing force, directed along the tangent to the orbit in the opposite sense to the motion, and pro- portional to the planet's velocity and to the inverse square of the distance from the sun. By considering the influence of this force on the secular equation of the moon's motion, Laplace found that the velocity of the gravific fluid must be at least a hundred million times greater than that of light.

The assumptions made by Laplace are evidently in the highest degree questionable; but the generation immediately succeeding, overawed by his fame, seems to have found no way of improving on them. Under the influence of Weber's ideas, however, astronomers began to think of modifying Newton's law by^ adding a term involving the velocities of the bodies. Tisserand* in 1872 discussed the motion of the planets round the sun on the supposition that the law of gravitation is the same as Weber's law of electrodynamic action, so that the force is

jp = «/_^r n . -?.r — J

•  «.»  ix    nv^*  i     i«r^«i» 
  

where / denotes the constant of gravitation, ra the mass of the planet, // the mass of the sun, r the distance of the planet from the sun, and h the velocity of propagation of gravitation. The equations of motion may be rigorously integrated by the aid of elliptic functions!; but the simplest procedure is to write

• Comptes Rendus, Ixxv (1872), p. 760. Of. also Comptes Rendus, ex (1890), p. 313, and Holzmiiller, Zeitschrif t f iir Math. u. Phys., 1870, p. 69.

t This had been done in an inaugural dissertation by Seegers, Gottingen, 1864.

234 The Mathematical Electricians of the

and, regarding F\ as a perturbing function, to find the variation of the constants of elliptic motion. Tisserand showed that the perturbations of all the elements are zero or periodic, and quite insensible, except that of the longitude of perihelion, which has a secular part. If A be assumed equal to the velocity of light, the effect would be to rotate the major axis of the orbit of Mercury in the direct sense 14" in a century.

Now, as it happened, a discordance between theory and observation was known to exist in regard to the motion of Mercury's perihelion ; for Le Verrier had found that the attrac- tion of the planets might be expected to turn the perihelion 527" in the direct sense in a century, whereas the motion actually observed was greater than this by 38". It is evident, however, that only f of the excess is explained by Tisserand's adoption of "Weber's law; and it seemed therefore that this suggestion would prove as unprofitable as Le Terrier's own hypothesis of an intra-mercurial planet. But it was found later* that f of the excess could be explained by substituting Eiemann's electrodynamic law for Weber's, and that a com- bination of the laws of Biemann and Weber would give exactly the amount desired.f

After the publication of his memoir on the law of force between electrons, Weber turned his attention to the question of diamagnetism, and developed Faraday's idea regarding the explanation of diamagnetic phenomena by the effects of electric currents induced in the diamagnetic bodies.^ Weber remarked that if, with Ampere, we assume the existence of molecular circuits in which there is no ohmic resistance, so that currents can flow without dissipation of energy, it is quite natural to suppose that currents would be induced in these molecular

• By Maurice Levy, Comptes Eendus, ex (1890), p. 545.

t The consequences of adopting the electrodynamic law of Clausius (for which see later) were discussed by Oppenheim, Zur Frage nach der Fortpflanzungs- geschwindigJceit der Gravitation, Wien, 1895.

I Leipzig Berichte, i (1847), p. 346 ; Ann. d. Phys. Ixxiii (1848), p. 241 ; translated Taylor's Scientific Memoirs, v, p. 477 ; Abhandl. der K. Sachs. Ges. i (1852), p. 483; Ann. d. Phys. Ixxxvii (1852), p. 145; trans. Tyndall and Francis' Scientific Memoirs, p. 163.

Middle of the Nineteenth Century. 235

circuits if they were situated in a varying magnetic field ; and he pointed out that such induced molecular currents would confer upon the substance the properties characteristic of dia magnetism.

The difficulty with this hypothesis is to avoid explaining too much ; for, if it be accepted, the inference seems to be that all bodies, without exception, should be diamagnetic. Weber escaped from this conclusion by supposing that in iron and other magnetic substances there exist permanent molecular currents, which do not owe their origin to induction, and which, under the influence of the impressed magnetic force, set themselves in definite orientations. Since a magnetic field tends to give such a direction to a pre-existing current that its course becomes opposed to that of the current which would be induced by the increase of the magnetic force, it follows that a substance stored with such pre-existing currents would display the phenomena of paramagnetism: t The bodies ordinarily called paramagnetic are, according to this hypothesis, those bodies in which the paramagnetism is strong enough to mask the diamagnetism.

The radical distinction which Weber postulated between the natures of paramagnetism and diamagnetism accords with many facts which have been discovered subsequently. Thus in 1895 P. Curie showed* that the magnetic susceptibility per gramme- molecule is connected with the temperature by laws which are different for paramagnetic and diamagnetic bodies. For the former it varies in inverse proportion to the absolute tempe- rature, whereas for diamagnetic bodies it is independent of the temperature.

The conclusions which followed from the work of Faraday and Weber were adverse to the hypothesis of magnetic fluids ; for according to that hypothesis the induced polarity would be in the same direction whether due to a change of orientation of pre-existing molecular magnets, or to a fresh separation of magnetic fluids in the molecules. " Through the discovery of

• Annales de Chimie (7) v (1845), p. 289.

236 The Mathematical Electricians of the

diamagnetism," wrote Weber* in 1852, "the hypothesis of electric molecular currents in the interior of bodies is cor- roborated, and the hypothesis of magnetic fluids in the interior of bodies is refuted." The latter hypothesis is, moreover, unable to account for the phenomena shown by bodies which are strongly magnetic, like iron : for it is found that when the magnetizing force is gradually increased to a very large value, the magnetization induced in such bodies does not increase in proportion, but tends to a saturation value This effect cannot be explained on the assumptions of Poisson,but is easily deducible from those of Weber; for, according to Weber's theory, the magnetizing force merely orients existing magnets ; and when it has attained such a value that all of them are oriented in the same direction, there is nothing further to be done,

Weber's theory in its original form is, however, open to some objection. If the elementary magnets are supposed to be free to orient themselves without encountering any resistance, it is evident that a very small magnetizing force would suffice to turn them all parallel to each other, and thus would produce immediately the greatest possible intensity of induced magnetism. To overcome this difficulty, Weber assumed that every displace- ment of a molecular circuit is resisted by a couple, which tends to restore the circuit to its original orientation. This assump- tion fails, however, to account for the fact that iron which has been placed in a strong magnetic field does not return to its original condition when it is removed from the field, but retains a certain amount of residual magnetization.

Another alternative was to assume a frictional resistance to the rotation of the magnetic molecules ; but if such a resistance existed, it could be overcome only by a finite magnetizing force ; and this inference is inconsistent with the observation that some degree of magnetization is induced by every force, however feeble.

The hypothesis which has ultimately gained acceptance is that the orientation is resisted by couples which arise from the

• Ann. d. Phys. lxxxvii(1852), p. 145 ; Tyndall and Francis' Sci. Mem., p. 163.

Middle of the Nineteenth Century. 237

mutual action of the molecular magnets themselves. In the unmagnetized condition the molecules " arrange themselves so as to satisfy their mutual attraction by the shortest path, and thus form a complete closed circuit of attraction," as D. E. Hughes wrote* in 1883 ; when an external magnetizing force is applied, these small circuits are broken up ; and at any stage of the process a molecular magnet is in equilibrium under the joint influence of the external force and the forces due to the other molecules.

This hypothesis was suggested by Maxwell,t and has been since developed by J. A. Ewing;J its consequences may be illustrated by the following simple example§ : —

Consider two magnetic molecules, each of magnetic moment m, whose centres are fixed at a distance c apart. When undisturbed, they dispose themselves in the position of stable equilibrium, in which they point in the same direction along the line c. Now let an increasing magnetic force H be made to act on them in a direction at right angles to the line c. The magnets turn towards the direction of H ; and when H attains the value Sm/c3, they become perpendicular to the line c, after which they remain in this position, when H is increased further. Thus they display the phenomena of induc- tion initially proportional to the magnetizing force, and of saturation. If the magnetizing force H be supposed to act parallel to the line c, in the direction in which the axes originally pointed, the magnets will remain at rest. But if H acts in the opposite direction, the equilibrium will be stable only so long as H is less than ra/c3 ; when H increases beyond this limit, the equilibrium becomes unstable, and the magnets turn over so as to point in the direction of H\ when H is gradually decreased to zero, they remain in their new posi- tions, thus illustrating the phenomenon of residual magnetism.

• Proc. Roy. Soc. xxxv (1883), p. 178. f Treatise on Elect. $ May., § 443.

I Phil. Mag. xxx (1890), p. 205 ; Magnetic Induction in Iron atid other Metals,. 1891.

§ E. G. Gallop, Messenger of Math, xxvii (1897), p. 6.

238 The Mathematical Electricians of the

By taking a large number of such pairs of magnetic molecules, originally oriented in all directions, and at such distances that the pairs do not sensibly influence each other, we may construct a model whose behaviour under the influence of an external magnetic field will closely resemble the actual behaviour of ferromagnetic bodies.

In order that the magnets in the model may come to rest in their new positions after reversal, it will be necessary to suppose that they experience some kind of dissipative force which damps the oscillations ; to this would correspond in actual magnetic substances the electric currents which would be set up in the neighbouring mass when the molecular magnets are suddenly reversed ; in either case, the sudden reversals are attended by a transformation of magnetic energy into heat.

The transformation of energy from one form to another is a subject which was first treated in a general fashion shortly before the middle of the nineteenth century. It had long been known that the energy of motion and the energy of position of a dynamical system are convertible into each other, and that the amount of their sum remains invariable when the system is self-contained. This principle of conservation of dynamical energy had been extended to optics by Fresnel, who had assumed* that the energy brought to an interface by incident light is equal to the energy carried away from the interface by the reflected and refracted beams. A similar conception was involved in Eoget's and Faraday's defencef of the chemical theory of the voltaic cell ; they argued that the work done by the current in the outer circuit must be provided at the expense of the chemical energy stored in the cell, and showed that the quantity of electricity sent round the circuit is proportional to the quantity of chemicals consumed, while its tension is proportional to the strength of the chemical affinities concerned in the reaction. This theory was extended

*Cf. p. 133. tCf. p. 203.

Middle of the Nineteenth Century. 239

and completed by James Prescott Joule, of Manchester, in 1841. Joule, who believed* that heat is producible from mechanical work and convertible into it, measuredf the amount of heat evolved in unit time in a metallic wire, through which a current of known strength was passed; he found the amount to be proportional to the resistance of the wire multiplied by the square of the current- strength ; or (as follows from Ohm's law) to the current-strength multiplied by the difference of electric tensions at the extremities of the wire.

The quantity of energy yielded up as heat in the outer circuit being thus known, it became possible to consider the transference of energy in the circuit as a whole. " When," wrote Joule, " any voltaic arrangement, whether simple or compound, passes a current of electricity through any substance, whether an electrolyte or not, the total voltaic heat which is generated in any time is proportional to the number of atoms which are electrolyzed in each cell of the circuit, multiplied by the virtual intensity of the battery : if a decomposing cell be in the circuit, the virtual intensity of the battery is reduced in proportion to its resistance to electrolyzation." In the same year hej enhanced the significance of this by showing that the quantities of heat which are evolved by the combustion of the equivalents of bodies are proportional to the intensities of their affinities for oxygen, as measured by the electromotive force of a battery required to decompose the oxide electrolytically.

The theory of Koget and Faraday, thus perfected by Joule, enables us to trace quantitatively the transformations of energy in the voltaic cell and circuit. The primary source of energy is the chemical reaction : in a Daniell cell, ZnjZn SOJCu S04|Cu, for instance, it is the substitution of zinc for copper as the partner of the sulphion. The strength of the chemical affinities concerned is in this case measured by the difference of the heats of formation of zinc sulphate and copper sulphate ; and it is

*Cf. p. 33.

t Phil. Mag. xix (1841), p. 260 ; Joule's Scientific Papers i, p. 60. I Phil. Mag. xx (1841), p. 98 : cf. also Phil. Mag. xxii (1843), p. 204.

240 The Mathematical Electricians of the

this which determines the electromotive force of the cell.* The amount of energy which is changed from the chemical to the electrical form in a given interval of time is measured by the product of the strength of the chemical affinity into the quantity of chemicals decomposed in that time, or (what is the same thing) by the product of the electromotive force of the cell into the quantity of electricity which is circulated. This energy may be either dissipated as heat in conformity to Joule's law, or otherwise utilized in the outer circuit.

The importance of these principles was emphasized by Hermann von Helmholtz (b. 1821, d. 1894), in a memoir which was published in 1847, and which will be more fully noticed presently, and by W. Thomson (Lord Kelvin) in 1851f; the equations have subsequently received only one important modification, which is due to Helmholtz.:}: Helmholtz pointed out that the electrical energy furnished by a voltaic cell need not be derived exclusively from the energy of the chemical reactions : for the cell may also operate by abstracting heat- energy from neighbouring bodies, and converting this into electrical energy. The extent to which this takes place is determined by a law which was discovered in 1855 by Thomson. § Thomson showed that if E denotes the " available energy," i.e., possible output of mechanical work, of a system maintained at the absolute temperature T, then a fraction

TdE fidT

of this work is obtained, not at the expense of the thermal or

• The heat of formation of a gramme-molecule of ZnS04 is greater than the heat of formation of a gramme-molecule of CuSO* by about 50,000 calories ; and with divalent metals, 46,000 calories per gramme- molecule corresponds to ane.m.f. of one volt ; so the e.m.f. of a Daniell cell should be 50/46 volts, which is nearly the case.

t Kelvin's Math, and Phys. Papers, i, pp. 472, 490.

J Berlin Sitzungsber., 1882, pp. 22, 825 ; 1883, p. 647.

§ Quart. Journ. Math., April, 1855 ; Kelvin's Math, and Phys. Papers, i, p. 297, eqn. (7).

Middle of the Nineteenth Century. 241

chemical energy of the system itself, but at the expense of the thermal energy of neighbouring bodies. Now in the case of the voltaic cell, the principle of Eoget, Faraday, and Joule is expressed by the equation

^ = A,

where E denotes the available or electrical energy, which is measured by the electromotive force of the cell, and where X denotes the heat of the chemical reaction which supplies this energy. In accordance with Thomson's principle, we must replace this equation by

F \ 4- TdE ^=X + TdT'

which is the correct relation between the electromotive force of a cell and the energy of the chemical reactions which occur in it. In general the term A is much larger than the term T dEjdT ; but in certain classes of cells — e.g., concentration- cells — A is zero; in which case the whole of the electrical energy is procured at the expense of the thermal energy of the cells' surroundings.

Helmholtz's memoir of 1847, to which reference has already been made, bore the title, " On the Conservation of Force." It was originally read to the Physical Society of Berlin*; but though the younger physicists of the Society received it with enthusiasm, the prejudices of the older generation prevented its acceptance for the Annalen der Physik ; and it was eventually published as a separate treatise.f

In this memoir it was asserted* that the conservation of

• On July 23rd, 1847.

t Berlin, G. A. Reimer. English Translation in Tyndall & Francis' Scientific Memoirs, p. 114. The publisher, to Helmholtz's "great surprise," gave him an honorarium. Cf. Hermann von Helmholtz, by Leo Koenigsbeiger ; English translation by F. A. Welby.

j Helmholtz had been partly anticipated by "W. R. Grove, in his lectures on the Correlation of Physical Forces, which were delivered in 1843 and published in 1846. Grove, after asserting that heat is " purely dynamical " in its nature, and that the various " physical forces " may be transformed into each other, remarked : " The great problem which remains to be solved, in regard to the correlation of physical forces, is the establishment of their equivalent of power, or their measurable relation to a given standard."

P.

242 The Mathematical Electricians of the

energy is a universal principle of nature : that the kinetic and potential energy of dynamical systems may be converted into heat according to definite quantitative laws, as taught by Kumford, Joule, and Eobert Mayer* ; and that any of these forms of energy may be converted into the chemical, electro- static, voltaic, and magnetic forms. The latter Helmholtz examined systematically.

Consider first the energy of an electrostatic field. It will be convenient to suppose that the system has been formed by continually bringing from a very great distance infinitesimal quantities of electricity, proportional to the quantities already present at the various points of the system ; so that the charge is always distributed proportionally to the final distribution. Let e typify the final charge at any point of space, and V the final potential at this point. Then at any stage of the process the charge and potential at this point will have the values \e and A F, where A denotes a proper fraction. At this stage let charges ed\ be brought from a great distance and added to the charges \e. The work required for this is

so the total work required in order to bring the system from infinite dispersion to its final state is

fi

or

By reasoning similar to that used in the case of electrostatic distributions, it may be shown that the energy of a magnetic field, which is due to permanent magnets and which also contains bodies susceptible to magnetic induction, is

\ where p0 denotes the density of Poisson's equivalent magnetiza-

• Julius Robert Mayer (b. 1814, d. 1878), who was a medical man in Heilbronn, asserted the equivalence of heat and work in 1842, Annal. d. Chemie, xlii, p. 233 ; his memoir, like that of Helmholtz, was first declined by the editors of the Annalen der Physik. An English translation of one of Mayer's memoirs was printed in Phil. Mag. xxv (1863), p. 493.

Middle of the Nineteenth Century. 243

tion, for the permanent magnets only, and $ denotes the magnetic potential.*

Helmholtz, moreover, applied the principle of energy to systems containing electric currents. For instance, when a magnet is moved in the vicinity of a current, the energy taken from the battery may be equated to the sum of that expended as Joulian heat, and that communicated to the magnet by the electromagnetic force : and this equation shows that the current is not proportional to the electromotive force of the battery, i.e. it reveals the existence of Faraday's magneto-electric induction. As, however, Helmholtz was at the time un- acquainted with the conception of the electrokinetic energy stored in connexion with a current, his equations were for the most part defective. But in the case of the mutual action of a current and a permanent magnet, he obtained the correct result that the time-integral of the induced electromotive force in the circuit is equal to the increase which takes place in the potential of the magnet towards a current of a certain strength in the circuit.