In the latter part of the year 1888 the researches of Hertz* yielded more complete evidence of the similarity of electric waves to light. It was shown that the part of the radiation from an oscillator which was transmitted through an opening in a screen was propagated in a straight line, with diffraction effects. Of the other properties of light, polarization existed in the original radiation, as was evident from the manner in which it was produced ; and polarization in other directions was obtained by passing the waves through a grating of parallel metallic wires ; the component of the electric force parallel to the wires was absorbed, so that in the transmitted beam the electric vibration was at right angles to the wires. This effect obviously resembled the polarization of ordinary light by a plate of tourmaline. Refraction was obtained by passing the radiation through prisms of hard pitch. j-

• Ann. d. Phys.xxxvi (1889), p. 769; Electric Waves (Englished.), p. 172.

I 0. J. Lodge and J. L. Ho ward in the same year showed that electric radiation might be refracted and concentrated hy means of large lenses. Cf. Phil. Mag. xxvii (1889), p. 48.

364 The Followers of Maxwell.

The old question as to whether the light-vector is in, or at right angles to, the plane of polarization* now presented itself in a new aspect. The wave-front of an electric wave contains two vectors, the electric and magnetic, which are at right angles to each other. Which of these is in the plane of polarization ? The answer was furnished by Fitz Gerald and Trouton,f who found on reflecting Hertzian waves from a wall of masonry that no reflexion was obtained at the polarizing angle when the vibrator was in the plane of reflexion. The inference from this is that the magnetic vector is in the plane of polarization of the electric wave, and the electric vector is at right angles to the plane of polarization. An interesting development followed in 1890, when 0. Wiener^ succeeded in photographing stationary waves of light. The stationary waves were obtained by the composition of a beam incident on a mirror with the reflected beam, and were photographed on a thin film of transparent collodion, placed close to the mirror and slightly inclined to it. If the beam used in such an experiment is plane-polarized, and is incident at an angle of 45°, the stationary vector is evidently that perpendicular to the plane of incidence; but Wiener found that under these conditions the effect was obtained only when the light was polarized in the plane of incidence ; so that the chemical activity must be associated with the vector perpendicular to the plane of polarization — i.e., the electric vector.

In 1890 and the years immediately following appeared several memoirs relating to the fundamental equations of electro-magnetic theory. Hertz, after presenting§ the general

• Cf. pp. 168 et sqq.

f Nature, xxxix (1889), p. 391.

\ Ann. d. Phys. xl (1890), p. 203. Cf. a controversy regarding the results ; Comptes Rendus, cxii (1891), pp. 186, 325, 329, 365, 383, 456 ; and Ann. d. Phys. xli (1890), p. 154 ; xliii(1891), p. 177; xlviii (1893), p. 119.

§ Gott. Nach. 1890, p. 106; Aim. d. Phys. xl (1890), p. 577; Electric Waves (English ed.), p. 195. In this memoir Hertz advocated the form of the equations which Maxwell had used in his paper of 1868 (cf. supra, p. 287) in preference to the earlier form, which involved the scalar and vector potentials.

The Followers of Maxwell. 365

content of Maxwell's theory for bodies at rest, proceeded* to extend the equations to the case in which material bodies are in motion in the field.

In a really comprehensive and correct theory, as Hertz remarked, a distinction should be drawn between the quantities which specify the state of the aether at every point, and those which specify the state of the ponderable matter entangled with it. This anticipation has been fulfilled by later investigators ; but Hertz considered that the time was not ripe for such a complete theory, and preferred, like Maxwell, to assume that the state of the compound system — matter plus aether — can be specified in the same way when the matter moves as when it is at rest ; or, as Hertz himself expressed it, that " the aether contained within ponderable bodies moves with them."

Maxwell's own hypothesis with regard to moving systemsf amounted merely to a modification in the equation

B = - curl E,

which represents the law that the electromotive force in a closed circuit is measured by the rate of decrease in the number of lines of magnetic induction which pass through the circuit. This law is true whether the circuit is at rest or in motion ; but in the latter case, the E in the equation must be taken to be the electromotive force in a stationary circuit whose position momentarily coincides with that of the moving circuit; and since an electromotive force [w . B] is generated in matter by its motion with velocity w in a magnetic field B, we see that E is connected with the electromotive force E' in the moving ponderable body by the equation

E' = E + [w . B],

so that the equation of electromagnetic induction in the moving body is

B = - curl E' + curl [w . B].

• Ann. d. Phys. xli (1890), p. 369 ; Electric Waves (English ed.), p. 241.

The propagation of light through a moving dielectric had been discussed previously, on the basis of Maxwell's equations for moving bodies, by J. J. Thomson, Phil. Mag. ix (1880), p. 284 ; Proc. Camb. Phil. Soc. v (1885), p. 250.

tCf. p. 288.

366 The Followers of Maxwell.

Maxwell made no change in the other electromagnetic equations, which therefore retained the customary forms

D = f E'/47rc2, div D = 0, 47r(i 4 D) = curl H, Hertz, however, impressed by the duality of electric and magnetic phenomena, modified the last of these equations by assuming that a magnetic force 4?r [D . w] is generated in a dielectric which moves with velocity w in an electric field ; such a force would be the magnetic analogue of the electromotive force of induction. A term involving curl |D . w] is then introduced into the last equation.

The theory of Hertz resembles in many respects that of Heaviside,* who likewise insisted much on the duplex nature of the electromagnetic field, and was in consequence disposed to accept the term involving curl [D . w] in the equations of moving media. Heaviside recognized more clearly than his predecessors the distinction between the force E', which determines the flux D, and the force E, whose curl represents the electric current ; and, in conformity with his principle of duality, he made a similar distinction between the magnetic force H', which determines the flux B, and the force H, whose curl represents the " magnetic current." This distinction, as Heaviside showed, is of importance when the system is acted on by " impressed forces," such as voltaic electromotive forces, or permanent magnetization; these latter must be included in E' and H', since they help to give rise to the fluxes D and B ; but they must not be included in E and H, since their curls are not electric or magnetic currents ; so that in general

we have

E' = E + e, H' = + h,

where e and h denote the impressed forces.

Developing the theory by the aid of these conceptions, Heaviside was led to make a further modification. An im-

• Heaviside's general theory was published in a series of papers in the Electrician, from 1885 onwards. His earlier work was republished in his Electrical Papers (2 vols., 1892), and his Electromagnetic Theory (2 vols., 1894). Mention may be specially made of a memoir in Phil. Trans, clxxxiii (1892), p. 423.

The Followers of Maxwell. 367

pressed force is best defined in terms of the energy which it communicates to the system ; thus, if e be an impressed electric force, the energy communicated to unit volume of the electro- magnetic system in unit time is e x the electric current. In order that this equation may be true, it is necessary to regard the electric current in a moving medium as composed of the conduction-current, displacement-current, convection- current, and also of the term curl [D . w] , whose presence in the equation we have already noticed. This may be called the current of dielectric convection. Thus the total current is

S = D + i + pw + curl [D . w] ,

where pw denotes the conduction-current ; and the equation connecting current with magnetic force is curl (H' - h0) = 4?rS,

where h0 denotes the impressed magnetic forces other than that induced by motion of the medium.

We must now consider the advances which were effected during the period following the publication of Maxwell's Treatise in some of the special problems of electricity and optics.

We have seen* that Maxwell accounted for the rotation of the plane of polarization of light in a medium subjected to a magnetic field K by adding to the kinetic energy of the aether, which is represented by Jpe*, a term J<r (e . curl 9e/90), where cr is a magneto-optic constant characteristic of the substance through which the light is transmitted, and d/dO stands for Kxdl'dx + Kydldy + K-d/dz. This theory was developed further in 1879 by Fitz Gerald,f who brought it into closer connexion with the electromagnetic theory of light by identifying the curl of the displacement e of the aethereal particles with the electric displacement ; the derivate of e with respect to the time then corresponds to the magnetic force. Being thus in possession of a definitely electromagnetic theory of the magnetic rotation of

• Cf . p. 308.

t Phil. Trans., 1879, p. 691. Fitz Gerald's Scient. Writings, p. 45.

368 The Followers of Maxwell.

light, Fitz Gerald proceeded to extend it so as to take account of a closely related phenomenon. In 1876 J. Kerr* had shown experimentally that when plane-polarized light is regularly reflected from either pole of an iron electromagnet, the reflected ray has a component polarized in a plane at right angles to the ordinary reflected ray. Shortly after this discovery had been made known, Fitz Geraldf had proposed to explain it by means of the same term in the equations which accounts for the mag- netic rotation of light in transparent bodies. His argument was that if the incident plane-polarized ray be resolved into two rays circularly polarized in opposite senses, the refractive index will have different values for these two rays, and hence the intensities after reflexion will be different; so that on re- compounding them, two plane-polarized rays will be obtained — one polarized in the plane of incidence, and the other polarized at right angles to it.

The analytical discussion of Kerr's phenomenon, which was given by Fitz Gerald in his memoir of 1879, was based on these ideas ; the most essential features of the phenomenon were explained, but the investigation was in some respects imperfect.}

Anew and fruitful conception was introduced in 1879-1880, when H. A. Eowland§ suggested a connexion between the magnetic rotation of light and the phenomenon which had been discovered by his pupil Hall.|| Hall's effect may be regarded

• Phil. Mag. (5) iii (1877), p. 321.

t Proc. 11. S. xxv (1877), p. 447 ; Fitz Gerald's Sclent. Writings, p. 9.

J Cf. Larmor's remarks in his Report on the Action oj Magnetism on Light, Brit. Assoc. Kep., 1893 ; and his editorial comments in Fitz Gerald's Scientific Writings. Larmor traced to its source an inconsistency in the equations hy which Fitz Gerald had represented the boundary-conditions at an interface between the media. Fitz Gerald had indeed made the mistake, similar to that which was so often made hy the earlier writers on the elastic-solid theory of light, of forgetting that when a medium is assumed to be incompressible, the condition of in compressibility must be introduced into the variational equation of motion (as was done supra, p. 172). Larmor showed that when this correction was made, new terms (resembling the terms in p, supra, p. 172) made their appearance; and the inconsistency in the equations was thus removed.

§ Amer. Jour. Math, ii, p. 354, iii, p. 89; Phil. Mag. xi (1881), p. 254.

|| Cf. p. 321.

The Followers of Maxwell. 369

as a rotation of conduction-currents under the influence of a magnetic field ; and if it be assumed that displacement-currents in dielectrics are rotated in the same way, the Faraday effect may evidently be explained. Considering the matter from the analytical point of view, the Hall effect may be represented by the addition of a term k [K . S] to the electromotive force, where K denotes the impressed magnetic force, and S denotes the current : so Kowland assumed that in dielectrics there is an additional term in the electric force, proportional to [K . D], i.e. proportional to the rate of increase of [K . D]. Now it is universally true that the total electric force round a circuit is proportional to the rate of decrease of the total magnetic induction through the circuit : so the total magnetic induction through the circuit must contain a term proportional to the integral of [K . D] taken round the circuit : and therefore the magnetic induction at any point must contain a term proportional to curl [K . D]. We may therefore write

B = H + a curl [K . D],

where <r denotes a constant. But if this be combined with the customary electromagnetic equations

curl H = 47rD, curl E = - B, D = eE/47rc3,

and all the vectors except B be eliminated (K being treated as a constant), we obtain the equation

B - (c7/0 V2B + O/47r) curl

where 3/80 stands for (Kxd/fa + Kyd/dy + Kzd/dz) ; and this is identical with the equation which Maxwell had given* for the motion of the aether in magnetized media. It follows that the assumptions of Maxwell and of Eowland, different though they are physically, lead to the same analytical equations— at any rate so far as concerns propagation through a homogeneous medium.

The connexions of Hall's phenomenon with the magnetic rotation of light, and with the reflexion of light from magnetized

• Cf. p. 308.

2 B

370 The Followers of Maxwell.

metals, were extensively studied* in the years following the publication of Kowland's memoir: but it was not until the modern theory of electrons had been developed that a satisfactory representation of the molecular processes involved in magneto- optic phenomena was attained.

The allied phenomenon of rotary polarization in naturally active bodies was investigated in 1892 by Goldhammer.f It

• The theory of Basset (Phil. Trans, clxxxii (1891), p. 371) was, like Rowland's, based on the idea of extending Hall's phenomenon to dielectric media. An objec- tion to this theory was that the tangential component of the electromotive force was not continuous across the interface between a magnetized and an unmagnetized medium ; but Basset subsequently overcame this difficulty (Nature, Hi (1 895), p. 618 ; liii (1895), p. 130; Amer. Jour. Math, xix (1897), p. 60)— the effect analogous to Hall's being introduced into the equation connecting electric displacement with electric force, so that the equation took the form

E = (47rc2/€) D + ff [K . D].

Basset, in 1893 (Proc. Camb. Phil. Soc. viii, p. 68), derived analytical expressions which represent Kerr's magneto-optic phenomenon by substituting u complex quantity for the refractive index in the formulae applicable to transparent magnetized substances.

The magnetic rotation of light and Kerr's phenomenon have been investigated also by R. T. Glazebrook, Phil. Mag. xi (1881), p. 397 ; by J. J. Thomson, Recent Researches, p. 482 : by D. A. Goldhammer, Ann. d. Phys. xlvi (1892), p. 71 ; xlvii (1892), p. 345; xlviii (1893), p. 740; 1 (1893), p. 772 : by P. Drude, Ann. d. Phys. xlvi (1892), p. 353; xlviii (1893), p. 122; xlix (1893), p. 690; lii (1894)) p. 496 : by C. H. Wind, Verslagen Kon. Akad. Amsterdam, 29th Sept., 1894 : by Reiff, Ann. d. Phys. Ivii (1896), p. 281 : by J. G. Leathern, Phil. Trans, cxc (1897), p. 89; Trans. Camb. Phil. Soc. xvii (1898), p. 16: and by W. Voigt in many memoirs, and in his treatise, Magneto- und Elektro-optik. Larmor's report presented to the British Association in 1893 has been already mentioned.

In most of the later theories the equations of propagation of light in magnetized metals are derived from the two fundamental electromagnetic equations

curl H = 4?rS, - curl E = H ;

the total current S being assumed to consist of a part (the displacement-current) proportional to E, a part (the conduction -current) proportional to E, and a part proportional to the vector-product of E and the magnetization.

Various mechanical models of media in which magneto-optic phenomena take place have been devised at different times. W. Thomson (Proc. Lond. Math. Soc. vi (1875)) investigated the propagation of waves of displacement along a stretched chain whose links contain rotating fly-wheels : cf . also Larmor, Proc. Lond. Math. Soc. xxi. (1890), p. 423 ; xxiii (1891), p. 127 ; F. Hasenohrl, Wien Sitzungsberichte cvii, 2a (189S), p. 1015 ; W. Thomson (Kelvin), Phil. Mag. xlviii (1899), p. 236, and Baltimore Lectures ; and Fitz Gerald, Electrician, Aug. 4, 1899, Fitz Gerald's Scientific Writings, p. 481. t Journal de Physique (3) i, pp. 205, 345.

The Followers of Maxwell. 371

will be remembered* that in the elastic-solid theory of Boussinesq, the rotation of the plane of polarization of saccharine solutions had been represented by substituting the

equation

e' = Ae + B curl e

in place of the usual equation

e' = Ae.

Goldhammer now proposed to represent rotatory power in the electromagnetic theory by substituting the equation

E = (4ircVO D + k curl D, in place of the customary equation

E = (4ircVO D :

the constant k being a measure of the natural rotatory power of the substance concerned. The remaining equations are as

usual.

curl H = 47rD, - curl E = H

Eliminating H and E, we have

fi = (c2/£) V2D + (k/4w) V2 curl D.

For a plane wave which is propagated parallel to the axis of x, this equation reduces to

k_

47T

k

47T ~& '

and, as MacCullagh had shown in 1836,f these equations are competent to represent the rotation of the plane of polarization. In the closing years of the nineteenth century, the general theory of aether and electricity assumed a new form. But before discussing the memoirs in which the new conception was unfolded, we shall consider the progress which had been made since the middle of the century in the study of conduction in liquid and gaseous media.

*Cf. p. 186. tcf. p. 175.

2B2

( 372 )

CHAPTEE XI.

CONDUCTION IN SOLUTIONS AND GASES, FROM FARADAY TO J. J. THOMSON.

THE hypothesis which Grothuss and Davy had advanced* to explain the decomposition of electrolytes was open to serious objection in more than one respect. Since the electric force was supposed first to dissociate the molecules of the electrolyte into ions, and afterwards to set them in motion toward the electrodes, it would seem reasonable to expect that doubling the electric force would double both the dissociation of the molecules and the velocity of the ions, and would therefore quadruple the electrolysis — an inference which is not verified by observation. Moreover it might be expected, on Grothuss' theory, that some definite magnitude of electromotive force would be requisite for the dissociation, and that no electrolysis at all would take place when the electromotive force was below this value, which again is contrary to experience.

A way of escape from these difficulties was first indicated, in 1850, by Alex. Williamson,-)- who suggested that in compound liquids decompositions and recombinations of the molecules are continually taking place throughout the whole mass of the liquid, quite independently of the application of an external electric force. An atom of one element in the compound is thus paired now with one and now with another atom of another element, and in the intervals between these alliances the atom may be regarded as entirely free. In 1857 this idea was made by

• Cf. p. 78.

f Phil. Mag. xxxvii (1850), p. 350 ; Liebig's Annulen d. Chem. u. Pharni. Ixxvii (1851) p. 37.

Conduction in Solutions and Gase*, etc. 373

K. Clausius,* of Zurich, the basis of a theory of electrolysis. According to it, the electromotive force emanating from the electrodes does not effect the dissociation of the electrolyte into ions, since a degree of dissociation sufficient for the purpose already exists in consequence of the perpetual mutability of the molecules of the electrolyte. Clausius assumed that these ions are in opposite electric conditions; the applied electric force therefore causes a general drift of all the ions of one kind towards the anode, and of all the ions of the other kind towards the cathode. These opposite motions of the two kinds of ions constitute the galvanic current in the liquid.

The merits of the Williamson-Clausius hypothesis were not fully recognized for many years ; but it became the foundation of that theory of electrolysis which was generally accepted at the end of the century.

Meanwhile another aspect of electrolysis was receiving attention. It had long been known that the passage of a current through an electrolytic solution is attended not only by the appearance of the products of decomposition at the electrodes, but also by changes of relative strength in different parts of the solution itself. Thus in the electrolysis of a solution of copper sulphate, with copper electrodes, in which copper is dissolved off the anode and deposited on the cathode, it is found that the concentration of the solution diminishes near the cathode, and increases near the anode. Some experiments on the subject were made by Faradayf in 1835 ; and in 1844 it was further investigated by Frederic Daniell and W. A. Miller, J who explained it by asserting that the cation and anion have not (as had previously been supposed) the same facility of moving to their respective electrodes ; but that in many cases the cation appears to move but little, while the transport is effected chiefly by the anion.

• Ann. d. Phys. ci (1857), p. 338 ; Phil. Mag. xv (1858), p. 94. t Exper. Res. §§ 525-53C.

• Phil. Trans., 1844, p. 1. Cf. also Pouillet, Comptes Rendus xx (1845), p. 1544.

374 Conduction in Solutions and Gases,

This idea was adopted by W. Hittorf, of Minister, who, in the years 1853 to 1859, published* a series of memoirs on the migration of the ions. Let the velocity of the anions in the solution be to the velocity of the cations in the ratio v : u. Then it is easily seen that if (u + v) molecules of the electrolyte are decomposed by the current, and yielded up as ions at the electrodes, v of these molecules will have been taken from the fluid on the side of the cathode, and u of them from the fluid on the side of the anode. By measuring the concentration of the liquid round the electrodes after the passage of a current, Hittorf determined the ratio v/u in a large number of cases of electrolysis.!

The theory of ionic movements was advanced a further stage by F. W. KohlrauschJ (I. 1840, d. 1910), of Wurzburg. Kohlrausch showed that although the ohmic specific conduc- tivity k of a solution diminishes indefinitely as the strength of the solution is reduced, yet the ratio k/m, where m denotes the number of gramme-equivalents§ of salt per unit volume, tends to a definite limit, when the solution is indefinitely dilute. This limiting value may be denoted by A. He further showed that A may be expressed as the sum of two parts, one of which depends on the cation, but is independent of the nature of the anion; while the other depends on the anion, but not on the cation — a fact which may be explained by supposing that, in very dilute solutions, the twos ions move independently under the influence of the electric force. Let u and v denote the velocities of the cation and anion respectively, when the potential difference per cm. in the solution is unity : then the total current carried through a cube of unit volume is mE(u + v), where E denotes the electric charge carried by one gramme-

Ann. d. Phys. Ixxxix (1853), p. 177 ; xcviii (1856), p. 1 ; ciii (1858), p. 1 ;

cvi (1859), pp. 337, 513.

t The ratio v/(u + v) was termed by Hittorf the transport, number of the anion. J Ann d. Phys. vi (1879), pp. 1, 145. The chief results had been communicated to the Academy of Gottingen in 1876 and 1877.

§ A gramme-equivalent means a muss of the salt whose weight in grammes is the molecular weight divided by the valency of the ions.

from Faraday to J. J. Thomson. 375

equivalent of ion.* Thus mE (u + v) = total current = k = raA, or A = E (u + v). The determination of v/u by the method of Hittorf, and of (u + v) by the method of Kohlrausch, made it possible to calculate the absolute velocities of drift of the ions from experimental data.

Meanwhile, important advances in voltaic theory were being effected in connexion with a different class of investi- gations.

Suppose that two mercury electrodes are placed in a solution of acidulated water, and that a difference of potential, insufficient to produce continuous decomposition of the water, is set up between the electrodes by an external agency. Initially a slight electric current — the polarizing current,f as it is called — is observed; but after a short time it ceases; and after its cessation the state of the system is one of electrical equilibrium. It is evident that the polarizing current must in some way have set up in the cell an electromotive force equal and opposite to the external difference of potential ; and it is also evident that the seat of this electromotive force must be at the electrodes, which are now said to be polarized.

An abrupt fall of electric potential at an interface between two media, such as the mercury and the solution in the present case, requires that there should be a field of electric force, of considerable intensity, within a thin stratum at the interface > and this must owe its existence to the presence of electric charges. Since there is no electric field outside the thin stratum, there must be as much vitreous as resinous electricity present ; but the vitreous charges must preponderate on one side of the stratum, and the resinous charges on the other side ; so that the system as a whole resembles the two coatings of a con- denser with the intervening dielectric. In the case of the

• i.e. E is 96580 coulombs.

t The phenomenon of voltaic polarization was discovered by Hitter in 1803. Hitter explained it by comparing the action of the polarizing current to that of a current which is used to charge a condenser. Volta in 1805 put forward the alternative explanation, that the products of decomposition set tip a reverse electromotive force.

376 Conduction in Solutions and Gases,

polarized mercury cathode in acidulated water, there must be on the electrode itself a negative charge : the surface of this electrode in the polarized state may be supposed to be either mercury, or mercury covered with a layer of hydrogen. In the solution adjacent to the electrode, there must be an excess of cations and a deficiency of anions, so as to constitute the other layer of the condenser : these cations may be either mercury cations dissolved from the electrode, or the hydrogen cations of the'solution.

It was shown in 1870 by Cromwell Fleetwood Varley* that a mercury cathode, thus polarized in acidulated water, shows a tendency to adopt a definite superficial form, as if the surface- tension at the interface between the mercury and the solution were in some way dependent on the electric conditions. The matter was more fully investigated in 1873 by a young French physicist, then preparing for his inaugural thesis, Gabriel Lippmann.f In Lippmann's instrumental disposition, which is called a capillary electrometer, mercury electrodes are immersed in acidulated water : the anode HQ has a large surface, wkile^the cathode H has a variable surface S small in comparison. When the external electromotive force is applied, it is easily seen that the fall of potential at the large electrode is only slightly affected, while the fall of potential at the small electrode is altered by polarization by an amount practically equal to the external electromotive force. Lippmann found that the constant of capillarity of the interface at the small electrode was a function of the external electromotive force, and therefore of the difference of potential between the mercury and the electrolyte.

Let V denote the external electromotive force: we may, without loss of generality, assume the potential of £[„ to be zero, so that the potential of H is - V. The state of the system may be varied by altering either V or /S; we assume that these

• Phil. Trans, clxi (1871), p. 129.

f Comptes Rendus Ixxvi (1873), p. 1407. Phil. Mag. xlvii (1874), p. 281. Ann. de Chim. et de Phys. v (1875), p. 494, xii (1877), p. 265.

from Faraday to J . J . Thomson. 377

alterations may be performed independently, reversibly, and isothermally, and that the state of the large electrode H,} is not altered thereby. Let de denote the quantity of electricity which passes through the cell from 5"0 to H, when the state of the system is thus varied : then if E denote the available energy of the system, and y the surface-tension at H, we have

dE = ydS + Vde,

y being measured by the work required to increase the surface when no electricity flows through the circuit.

In order that equilibrium may be re-established between the electrode and the solution when the fall of potential at the cathode is altered, it will be necessary not only that some hydrogen cations should come out of the solution and be deposited on the electrode, yielding up their charges, but also that there should be changes in the clustering of the charged ions of hydrogen, mercury, and sulphion in the layer of the solution immediately adjacent to the electrode. Each of these circumstances necessitates a flow of electricity in the outer circuit : in the one case to neutralize the charges of the cations deposited, and in the other case to increase the surface-density of electric charge on the electrode, which forms the opposite sheet of the quasi-condenser. Let Sf (V) denote the total quantity of electricity which has thus flowed in the circuit when the external electromotive force has attained the value V. Then evidently

so

dE= {y+ Vf(V)\dS + VSf (V}dV.

Since this expression must be an exact differential, we have

so that - dy/d V is equal to that flux of electricity per unit of new surface formed, which will maintain the surface in a

378 Conduction in Solutions and Gases,

constant condition (V being constant) when it is extended. Integrating the previous equation, we have

Lippmann found that when the external electromotive force was applied, the surface-tension increased at first, until, when the external electromotive force amounted to about one volt, the surface-tension attained a maximum value, after which it diminished. He found that d-y/d F2 was sensibly independent of F, so that the curve which represents the relation between 7 and F is a parabola.*

The theory so far is more or less independent of assumptions as to what actually takes place at the electrode : on this latter question many conflicting views have been put forward. In 1878 Josiah Willard Gibbs,t of Yale (b. 1839, d. 1903), discussed the problem on the supposition that the polarizing current is simply an ordinary electrolytic conduction-current, which causes a liberation of hydrogen from the ionic form at the cathode. If this be so, the amount of electricity which passes through the cell in any displacement must be proportional to the quantity of hydrogen which is yielded up to the electrode in the displacement; so that dy/dV must be proportional to the amount of hydrogen deposited per unit area of the electrode.:}:

A different view of the physical conditions at the polarized electrode was taken by Helmholtz,§ who assumed that the ions of hydrogen which are brought to the cathode by the polarizing current do not give up their charges there, but remain in the vicinity of the electrode, and form one face of a quasi-condenser

• Lippman, Coniptes Eendus, xcv (1882), p. 686.

t Trans. Conn. Acad. iii (1876-1878), pp. 108, 343; Gibbs' Scientific Papers, i, p. 55.

J This is embodied in equation (690) of Gibbs' memoir.

§ Berlin Monatsber., 1881, p. 945 ; Wiss. Abh. i, p. 925 ; Ann. d. Phys. xvi. (1882), p. 31. Cf. also Planck, Ann. d. Phys. xliv (1891), p. 385.

from Faraday to J . J. Thomson. 379

of which the other face is the electrode itself.* If a denote the surface-density of electricity on either face of this quasi- condenser, we have, therefore,

de = - d(Sa) ; so a = dyfd V.

This equation shows that when dyldV is zero — i.e., when the surface-tension is a maximum — a must be zero ; that is to say, there must be no difference of potential between the mercury and the electrolyte. The external electromotive force is then balanced entirely by the discontinuity of potential at the other electrode J7"0 ; and thus a method is suggested of measuring the latter discontinuity of potential. All previous measurements of differences of potential had involved the employment of more than one interface ; and it was not known how the measured difference of potential should be distributed among these interfaces ; so that the suggestion of a means of measuring single differences of potential was a distinct advance, even though the hypotheses on which the method was based were somewhat insecure.