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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through I lie lull lexl of 1 1 us book on I lie web al|_-.:. :.-.-:: / / books . qooqle . com/| 3,Bl,ZEdhyG00gle Google 3,Bl,ZEdhyG00gle 3,Bl,ZEdhyG00gle 3,Bl,ZEdhyG00gle .yGoogle DUBLIN UNIVERSITY PRESS SERIES. A HISTORY THEOKIES OF AETHER AND ELECTRICITY FKOM THE AGE OF DESCARTES TO THE CLOSE OF THE NINETEENTH CENTURY. E. T. WH1TTAKEE, Ban.St.D.^Hubl.); F.S.S.; Seyat iiMMWr *f Inland. LONGMANS, GREEN, AND CO., 39 PATERNOSTER ROW, LONDON, NEW TOBK, BOMBAY, AND CALCUTTA. HODGES, FIGGIS, * CO., Ltd., DUBLIN. 1910. SiglizedbyGoOglc The author desirea to record his gratitude to Mr. W. W. Roose Ball, Fellow of Trinity College, Cambridge, and to Professor W. McF. Oee, F.E.S., of the Royal College of Science for Ireland ; these friends have read the proof-sheets, and have made many helpful suggestions and criticisms. Thanks are also due to the Board of Trinity College, Dublin, for the financial assistance which made possible the publication of the work. 236360 D,Bl,ZEdhyCOOgle SiglizedbyGoOglc CONTENTS. CHAPTER I. The Theoby of the Aethex in the Seventeenth Cbntuby Hatter and aether, .... The physical writings of Descartes, . Early history of magnetism : Petrus Peregrin ua, Gilbert, Dencartea, Ferroat attacks Descartes' theory of light : the principle of least Hooke's nndulat >ry theory : the advance of wave-fronts, Newton overthrows Hooke's theory of colours, Conception of the aether in the writings of Newton, Newton's theories of the periodicity of homogeneous light, fits of easy transmission, , . The velocity of light : Galileo, Roomer, Huygans' Traiti de la htmiert : his theories of the propagation o waves, and of crystalline optica, . Newton shows that rays obtained by double refraction have side* his objections to the modulatory theory. Electric and Magnetic Science, pbiob to the Introduction o tbb Potentials. The electrical researches of Gilbert : the theory of emanations. State of physical science in the first half of the eighteenth century, Gray discovers electric conduction : Desaguliers, . The electric fluid, ..... Du Fay distinguishes vitreous and resinous electricity, Nollet's effluent and affluent streams, . , He Leyden phial, ..... The one-fluid theory : ideas of Watson and Franklin, Final overthrow by Aepinus of the doctrine of effluvia, Priestley discovers the law of electrostatic force. dhyGoOgk Vlll Contents. Cavendish, . Michell disaovers the law of magnetic force, . . . The two-fluid theory : Coulomb, .... Limited mobility of the magnetic fluids, Poisson's mathematical theory of electrostatics, The equivalent surface- and volume-distributions of magnetism : Frisson's theory of magnetic induction, Green's Nottingham memoir, .... e fluid, CHAPTER III. Galvanism, fboh Galvani to Ohm. Sulzer's discovery, Galvanic phenomena, . Rival hypotheses regarding the gal' The voltaic pile, Nicholson and Carlisle decompose water voltaically, Davy's chemical theory of the pile, . Grothuss' chain, .... De La Rive's hypothesis, Berzelius' scheme of electro-chemistry, Early attempts to discover a connexion between magnetism, .... Oersted's experiment : his explanation of it, The law of Biot aud Savart, . The researches of Ampere on electrodynamics, Seebeck's phenomenon, ' Davy's researches on conducting power. Ohm's theory : electroscopic foroe, . CHAPTER IV. , The Ll-minifeeocs Medium, froji Bradley to Fresnel. Bradley discovert; aberration, ..... John Bernoulli's model of the aether, .... Maupertuis and the principle of least action, Views of Euler, Courtivron, Melvill, . Toung defends the modulatory theory, and explains the colours of thin plates, ....... Laplace supplies a corpuscular theory of double refraction, . dhyGoOglc Contents. Young proposes a dynamical theory of light in crystals, Researches of Malus on polarization, Recognition of biaxal crystals, Fresnel successfully explains diffraction, TTjk theory of the relative motion of aether and natter, Young suggests the transversal ity of the vibrations of light, Frosnel discusses the dynamics of transverse vibrations, Fresnel's theory of the propagation of light in crystal*, Hamilton predicts conical refraction, Fresnel 'h theory of reflexion, CHAPTER V. ;The Aether as am Elastic Solid. Astronomical objection to the elastic-solid theory : Stokes' hypothesis. ....... Navier and Cauchy discover the equation of vibration of an elastic Poisson distinguishes condensational and distortional waves, Cuuchy's first and second theories of light in crystals, Canchy's first theory of reflexion, ..... His second theory of reflexion, ..... The theory of reflexion of MacCullagh and Neumann, Green discovers the correct conditions at the boundaries. Green's theory of reflexion : objections to it, MacCullagh introduces a new type of elastic solid, W. Thomson's model of a rotationslly- elastic body, ' . Canchy's third theory of reflexion : the contractile aether, . Later work of W. Thomson and others on the contractile aether, . Green's first and second theories of light in crystals, Influence of Green, Researches of Stokes on the relation of the direction of vibration of light to its plane of polarization, The hypothesis of aeolotropic inertia, . '. Rotation of the plane of polarization of light by active bodies, : MacCullagh's theory of natural rotatory power, . J MacCullagh'i and Cauchy's theory of metallic reflexion, ' Extension of the elastic-solid theory to metals, . . 1 Lord Rayleigh's objection. . . . 1 Canchy's theory of dispersion, • ■ . . 1 s elastic-solid theory, . 3 3,Bl,ZEdhyG00gle x Contents, CHAPTER VI. Discovery of induced currents : lines of magnetic force, - • 189 Self-induction, . . . . . . .193 Identity of fricfcional and voltaic electricity : Faraday's views on the nature of electricity, ...... 194 Electro-chemistry, ....... 197 Controversy between the adherents of the chemical and contact hypotheses, ....... SOI The properties of dielectrics, . . .... 206 Theory of dielectric polarization : Faraday, W. Thomson, and Mossotti, . . . . . . . 211 The connexion between magnetism and light, . . 213 Airy's theory of magnetic rotatory polarization, . . . 214 Faraday's Thought! on Bay- Vibratiam, . . . .217 Researches of Faraday and Pluoker on diamagnettsm, 218 CHAPTER VH. The Mathematical Electricians op the Middle of the Nineteenth Cehtuby. F. Neumann's theory of induced currents : the electrodynamic potential, . . . . . . .222 W. Weber's theory of electrons, ..... 225 Riemann'a law, ....... 231 V'T'i-oposals to modify the law of gravitation, .... 232 Weber's theory of paramagnetism and diamagnetism : later theories, 234 Joule's law : energetics of the voltaic cell, .... 239 Researches of Helmholtz on electrostatic and electrodynamic energy, 242 W. Thomson distinguishes tho circuital and irrotational magnetic vectors, ........ 244 His theory of magnecrystallic action, .... 245 His formula for the energy of a magnetic field, . . . 247 Extension of this formula to the case of fields produced by currents, 249 Kirchhoff identifies Ohm's elecbroscopic force with electrostatic potential, ....... 251 The discharge of a Leyden jar : W. Thomson's theory, ■ ■ 253 The velocity of electricity and the propagation of telegraphic signals, 254 Clausius* law of force between electric charges : crucial experiments, 261 Nature of the current, ...... 263 The thermo-electric researches of Peltier and W. Thomson, . 264 3,Bl,ZEdhyG00gle CHAPTER VIII. Maxwell. Gauss and Riemaim on the propagation of electric actions, Analogies suggested by W. Thomson, Maxwell's hydrodynamical analogy, . . . The vector potential, .... Linear and rotatory interpretations of magnetism, . Maxwell's mechanical model of the electromagnetic field, Electric displacement, .... Similarity of electric vibrations to thoae of light. Connexion of refractive index and specific inductive capacity, Maxwell's memoir of 1864 The propagation of electric disturbances in crystals and in Anomalous dispersion, .... The Max well -Sellmeier theory of dispersion, Imperfections of the electromagnetic theory of light. The theory of L. Lorenz, .... Maxwell's theory of stress in the electric field, The pressure of radiation, .... Maxwell's theory of the magnetic rotation of light, . CHAPTER IX. \^r Models of the Aether. Analogies in which a rotatory character is attributed to magnetism, Models in which magnetic force is represented as a linear velocity, Researches of W. Thomson, Bjerknes, and Leahy, on pulsating and oscillating bodies, ...... MacCoJlagh's quasi-elastic solid as a model of tho electric medium, The Hall effect, Models of Riemann and Fitz Gerald, .... Vortex-atoms, ....... The vortex-sponge theory of the aether : researches of W.Thomson, FitsGerald, and Hicks, ...... CHAPTER X. The Followers of Maxwell. Helmholtz and H. A. Lorentz supply an electromagnetic theory of reflexion, ....... Crucial experiments of Helmholtz and Schiller, 3,Bl,ZEdhyG00gle xii Contents. Convection -currents : Rowland's experiments, . 339 The moving charged sphere: researches of J. J. Thomson, Kite Gerald, and Heaviside, ....... 340 Conduction of rapidly -alternating currents, .... 344 Fitz Gerald devises the magnetic radiator, .... 346 Pointing's theorem, ....... 347 Poynting and J. J. Thomson develop the theory of moving lines of force, . . . . . . 349 Mechanical momentum in the electromagnetic field, . 352 New derivation of Maxwell's equations by Hertz, . . 353 Hertz's assumptions and Weber's theory, .... 356 Experiments of Hertz on electric waves, .... 357 The memoirs of Hertz and Heaviside on fields in which material bodies are in motion, ...... 365 The current of dielectric convection, ..... 367 Kerr's magneto-optic phenomenon, . ' 363 Rowland's theory of magneto -optics, .... 369 The rotation of the plane of polarization in naturally active bodies, 370 CHAPTER XI. Conduction in Solutions and Gases, from Faraday to J. J. Thomson. The hypothesis of Williamson and Clausina, Migration of the ions, ..... The researches of Hittorf and Kohlrausch, . Polarization of electrodes, ..... Electro capillarity, ...... Single differences of potential, .... Helmholtz' theory of concentration -cells, Arrhenius' hypothesis, ..... The researches of Nerast, ..... Earlier investigations of the discharge in rarefied gases, Faraday observes the dark space, .... Researches of Pliicker, Hittorf, Goldstein, and Varley, on tb cathode rays, ...... Crookes and the fourth state of matter, Objections and alternatives to the cbarged-particle theory < cathode rays, ...... Giese's and Schuster's ionic theory of conduction in gases, . J. J. Thomson measures the velocity of cathode rays, dhyGoogle Contents. xiii Pup Discovery of X-rays : hypotheses regarding them, . 401 Further researches of J. J. Thomson on cathode rays : the ratio m/e, 404 Vitreous and resinous electricity, ..... 406 Determination of the ionic charge by J. J. Thomson, 407 Beequerers radiation : discovery of radio-active substances, . 40S CHAPTER XII. The Theory of Axtheb and Elkotkonb in the Closing Yeabs op the nineteenth century. Stokes' theory of aethersal motion near moving bodies, . 411 Astronomical phenomena in which the velocity of light is involved, 413 Crucial experiments relating to the optica of moving bodies, 416 Lorentz' theory of electrons, ...... 419 The current of dielectric convection : Rontgen'n experiment, 426 The electronic theory of dispersion, ..... 428 Deduction of Fresnel's formula from the theory of electrons, . 430 Experimental verification of Lorentz' hypothesis, . . 431 Fitz Gerald's explanation of Michelson's experiment, 432 Lorentz' treatise of 1896, . . . . . 433 Expression of the potentials in terms of the electronic charges, 436 Farther experiments on the relative motion of earth and aether, . 437 Extension of Lorentz' transformation : Larmor discovers its connexion with FitzGerald's hypothesis of contraction, 440 Examination of the supposed primacy of the original variables : fisity relative to the aether : the principle of relativity, 444 The phenomenon of Zeeman, ..... 449 Connexion of Zeeman's effect with the magnetic rotation of light, . 452 The optical properties of metals, ..... 454 The electronic theory of metals, ..... 456 Thermionic*, . .464 3,Bl,ZEdhyG00gle dhyGoogle MEMORANDUM ON NOTATION. Yxcroas are denoted by letters in clarendon type, as E. The three components of a vector E are denoted by E„ Er, E,; and the magnitude of the vector is denoted by E, so that S* m £* + E* + E*. The vector product of two vectors E and H, which is denoted by [E . H], is the vector whose components are {E,M, - EJT„ EM. - M*K» EJB\ - K^*)- I*» direction is at right angles to the direction of E and H, and its magnitude is represented by twice the area of the triangle formed by them. The ualar product of E and H is .E,N, + E,Br + EJS,. It 'is denoted by (B . E). *=? + -r* i* denoted by div E. oy Zt J The vector whose components are fZE, ZE, ZE. ZE, dEt Z£.\ \Hy~~Z*' Zy~~Zx' ~Zx~ as'} is denoted by curl E. If V denote a scalar quantity, the vector whose components are i zv zv zv\ The symbol V is used to denote the vector operator whose Differentiation with respect to the time ia frequently indicated by a dot placed over the symbol of the variable which is differentiated. 3,Bl,ZEdhyG00gle 3,Bl,ZEdhyG00gle THEORIES OF AETHER AND ELECTRICITY. THE THEORY OF THE AETHER IN THE SEVENTEENTH CEHTDET. The observation of the heavens, which has been pursued con- tinually from the earliest ages, revealed to the ancients the regularity of the planetary motions, and gave rise to the conception of a universal order. Modern research, building on this foundation, has shown how intimate is the connexion between the different celestial bodies. They are formed of the same kind of matter ; they are similar in origin and history ; and across the vast spaces which divide them they hold perpetual intercourse. Until the seventeenth century the only influence which was known to be capable of passing from star to star was that of light. Newton added to this the force of gravity ; and it is now recognized that the power of communicating across vacuous regions is possessed also by the electric and magnetic attractions. It is thus erroneous to regard the heavenly bodies as isolated in vacant space ; around and between them is an incessant conveyance and transformation of energy. To the vehicle of this activity the name aether has been given. The aether is the solitary tenant of the universe, save for that infinitesimal fraction of space which is occupied by ordinary matter. Hence arises a problem which has long engaged attention, and is not yet completely solved : "What relation subsists between the medium which fills the interstellar void and the condensations of matter that are scattered throughout it? B 3,Bl,ZEdhyG00gle *'*2 : :* : /: '•' : : ' -ffli Theory of the -Aether The history of this problem may be traced back continuously to the earlier half of the seventeenth century. It first emerged clearly in that reconstruction of ideas regarding the physical universe which was effected by Rene" Descartes. Descartes was born in 1596, the son of Joachim Descartes, Counsellor to the Parliament of Brittany. As a young man he followed the profession of arms, and served in the campaigns of Maurice of Nassau, and the Emperor ; but hie twenty-fourth year brought a profound mental crisis, apparently not unlike those which have been recorded of many religious leaders ; and he resolved to devote himself thenceforward to the study of philosophy. The age which preceded the birth of Descartes, and that in which he lived, were marked by events which greatly altered the prevalent conceptions of the world. The discovery of America, the circumnavigation of the globe by Drake, the over- throw of the Ptolemaic system of astronomy, and the invention of the telescope, all helped to loosen the old foundations and to make plain the need for a new structure. It was this that Descartes set himself to erect. His aim was the most ambitious that can be conceived ; it was nothing less than to create from the beginning a complete system of human knowledge. Of such a system the basis must necessarily be metaphysical ; and this part of Descartes' work is that by which he is most widely known. But his efforts were also largely devoted to the mechanical explanation of nature, which indeed he regarded as one of the chief ends of Philosophy* The general character of his writings may be illustrated by a comparison with those of his most celebrated contemporary. + Bacon clearly denned the end to be sought for, and laid down the method by which it was to be attained ; then, recognizing that to discover all the laws of nature is a task beyond the * Of the work* which bear on our present subject, the Bioplriqui and the Xttiort* were published at Leyden in 1638, and the iViisei/jia PhiloiepMas at Amtterdam in 1614, aii yean before the death of its author. t The principal philosophical works of Bacon were written about eighteen years before those of Desejiries. 3,Bl,ZEdhyG00gle in the Seventeenth Century. 3 powers of one man or one generation, he left to posterity the work of filling in the framework which he had designed. Descartes, on the other hand, desired to leave as little as possible for his successors to do ; his was a theory of the universe, worked out as far as possible in every detail. It is, however, impossible to derive such a theory inductively unless there are at hand sufficient observational data on which to base the induction ; and as such data were not available in the age of Descartes, he was compelled to deduce phenomena from preconceived principles and causes, after the fashion of the older philosophers. To the inherent weakness of this method may be traced the errors that at last brought his scheme to ruin. The contrast between the systems of Bacon and Descartes is not unlike that between the Roman republic and the empire of Alexander. In the one case we have a career of aggrandizement pursued with patience for centuries ; in the other a growth of fungus-like rapidity, a speedy dissolution, and an immense influence long exerted by the disunited fragments. The grandeur of Descartes' plan, and the boldness of its execution, stimulated scientific thought to a degree before unparalleled; and it was largely from its ruins that later philosophers constructed those more valid theories which have endured to our own time. Descartes regarded the world as an immense machine, operating by the motion and pressure of matter. "Give me matter and motion," he cried, " and I will construct the universe." A peculiarity which distinguished his system from that which afterwardB sprang from its decay was the rejection of all forms of action at a distance ; he assumed that force cannot be com- municated except by actual pressure or impact. By this assumption he was compelled to provide an explicit mechanism in order to account for each of the known forces of nature — a task evidently much more difficult than that which lies before those who are willing to admit action at a distance as an ultimate property of matter. Since the sun interacts with the planets, in sending them DslzEdhyGoOgle 4 The Theory of the Aether light and heat and influencing their motions, it followed from. Descartes' principle that interplanetary apace must be a plenum, occupied by matter imperceptible to the touch but capable of serving as the vehicle of force and light. This conclusion in turn determined the view which he adopted on the all-important question of the nature of matter. Matter, in the Cartesian philosophy, is characterized not by impenetrability, or by any quality recognizable by the senses. but Bimply by extension ; extension constitutes matter, and matter constitutes space. The basis of all things is a primitive, elementary, unique type of matter, boundless in extent and infinitely divisible. In the process of evolution of the universe 'three distinct forms of this matter have originated, correspond- ing respectively to the luminous matter of the sun, the transparent matter of interplanetary space, and the dense, opaque matter of the earth. " The first is constituted by what has been scraped off the other particles of matter when they were rounded ; it moves with so much velocity that when it meets other bodies the force of its agitation causes it to be broken and divided by them into a heap of small particles that are of such a figure as to fill exactly all the holes and small interstices which they find around these bodies. The next type includes most of the rest of matter ; its particles are spherical, and are very small compared with the bodies we see on the earth ; but nevertheless they have a finite magnitude, so that they can be divided into others yet smaller. There exists in addition a third type exemplified by some kinds of matter — namely, those which, on account of their size and figure, cannot be so easily moved as the preceding. I will endeavour to show that all the bodies of the visible world are composed of these three forms of matter, as of three distinct elements ; in fact, that the sun and the fixed stars are formed of the first of these elements, the interplanetary spaces of the second, and the earth, with the planets and comets, of the third. For, seeing that the sun and the fixed stars emit light, the heavens transmit it, and the earth,. the planets, and the comets reflect it, it appears to me that there 3,Bl,ZEdhyG00gle in the Seventeenth Century. 5 is ground for using these three qualities of luminosity, trans- parence, and opacity, in order to distinguish the three elements of the visible world.* According to Descartes' theory, the sun is the centre of an immense vortex formed of the first or subtlest kind of matter, t The vehicle of light in interplanetary space is matter of the second kind or element, composed of a closely packed assemblage of globules whose size is intermediate between that of the vortex-matter and that of ponderable matter. The globules of the second element, and all the matter of the first element, are constantly straining away from the centres around which they turn, owing to the centrifugal force of the vortices £ so that the globules are pressed in contact with each other, and tend to move outwards, although they do not actually so move.§ It is the transmission of this pressure which constitutes light ; the action of light therefore extends on all sides round the sun and fixed stars, and travels instantaneously to any distance.]) In the Dioptri$2te,% vision is compared to the perception of the presence of objects which a blind man obtains by the use of hiB stick ; the transmission of pressure along the stick from the object to the hand being analogous to the transmission o! pressure from a luminous object to the eye by the second kind of matter. Descartes supposed the " diversities of colour and light " to lie due to the different ways in which the matter moves.** In the MH6ores,\^ the various colours are connected with different rotatory velocities of the globules, the particles which rotate most rapidly giving the sensation of red, the slower ones of yellow, and the slowest of green and blue — the order of colours being taken from the rainbow. The assertion of the dependence of colour " Trindpia, Part iii, ) 52. t It if curious to speculate on the impression which would have been produced had the spiraliiy of neb nine been discovered before the overthrow of the Cartesian theory of vortices. * /Bid., f} 55-69. j Ibid., {63. | Hid., { 6*. H Diteom-ii prtmiir. " Prineipia, Part iv, 4 195. ft Diatom Huiiiimi. 3,Bl,ZEdhyG00gle 6 The Theory of the Aether on periodic time is a curious foreshadowing of one of the great discoveries of Newton. The general explanation of light on these principles was amplified by a more particular discussion of reflexion and refraction. The law of reflexion— that the angles of incidence and refraction are equal — had been known to the Greeks ; but the law of refraction — that the sines of the angles of incidence and refraction are to each other in a ratio depending on the media — was now published for the first time* Descartes gave it as his own ; but he seems to have been under considerable obligations to Willebrord Snell (b. 1591, d. 1626), Professor of Mathematics at Leyden, who had discovered it experimentally (though not in the form in which Descartes gave it) about 1621. Snell did not publish his result, but communicated it in manuscript to several persons, and Huygens affirms that this manuscript had been seen by Descartes. Descartes presents the law as a deduction from theory. This, however, he is able to do only by the aid of analogy ; when rays meet ponderable bodies, " they are liable to be deflected or stopped in the same way as the motion of a ball or a stone impinging on a body " ; for " it is easy to believe that the action or inclination to move, which I have said must be taken for light, ought to follow in this the same laws as motion."f Thus he replaces light, whose velocity of propagation he believes to be always infinite, by a projectile whose velocity varies from one medium to another. The law of refraction is then proved as follows; : — Let a ball thrown from A meet at B a cloth GB£, so weak that the ball is able to break through it and pass beyond, bub with its resultant velocity reduced in some definite proportion, say 1 : k. Then if BI be a length measured on the refracted ray equal to AB, the projectile will take k times as long to describe BI as it took to describe AB. But the component • Dioptriqut, Diieeuri leeond. f liii., Liteourt premier. % Ibid., Diicoun itiond. 3,Bl,ZEdhyG00gle in the Seventeenth Century. 7 of velocity parallel to the cloth must be unaffected by the impact; and therefore the projection BE of the refracted ray must be k times as long as the projection BC of the incident ray. So if i and r denote the angles of incidence and refraction, we have • BE , BC , . . Binr-j^-ft.^-tBini, or the sines of the angles of incidence and refraction are in a constant ratio ; this is the law of refraction. Desiring to include all known phenomena in .his system, Descartes devoted some attention to a class of effects which were at that time little thought of, but which were destined to play a great part in the subsequent development of Physics. The ancients were acquainted with the curious properties possessed by two minerals, amber (IjXtKTpov) and magnetic iron ore (17 XtBot VLayvtjn^). The former, when rubbed, attracts light bodies : the latter has the power of attracting The use of the magnet for the purpose of indicating direc- tion at sea does not seem to have been derived from classical antiquity ; but it was certainly known in the time of the Crusades. Indeed, magnetism was one of the few sciences which progressed during the Middle Ages ; for in the thirteenth century Petrus Peregrinus,• a native of Maricourt in Picardy, made a discovery of fundamental importance, Taking a natural magnet or lodestone, which had been rounded into a globular form, he laid it on a needle, and marked • HU Epittala irna H-rftti.il in 1209. 3,Bl,ZEdhyG00gle 8 The Theoty of the Aether the line along which the needle set itself. Then laying the needle on other parts of the stone, he obtained more lines in the same way. When the entire Burface of the stone had been covered with such lines, their general disposition became evident ; they formed circles, which girdled the Btone in exactly the Bame way as meridians of longitude girdle the earth ; and there were two points at opposite ends of the stone through which all the circles passed, just as all the meridians pass through the Arctic and Antarctic poles of the earth.* Struck by the analogy, Peregrinus proposed to call these two points the poles of the magnet : and he observed that the way in which magnets set themselves and attract each other depends solely on the position of their poles, as if these were the seat of the magnetic power. Such was the origin of those theories of poles and polarization which in later ages have played so great a part in Natural Philosophy. The observations of Peregrinus were greatly extended not long before the time of Descartes by William Gilberd or Gilbertf (b. 1540, d. 1603). Gilbert was born at Colchester: after studying at Cambridge, he took up medical practice in London, and had the honour of being appointed physician to Queen Elizabeth. In 1600 he published a work* on Magnetism and Electricity, with which the modern history of both subjects begins. Of Gilbert's electrical researches we shall speak later : in magnetism he made the capital discovery of the reason why magnets set in definite orientations with respect to the earth ; which is, that the earth is itself a great magnet, having one of its poles in high northern and the other in high southern latitudes. Thus the property of the compass was seen to be included in the general principle, that the north-seeking pole of • " Procul dubio otunes lineae hujusmodi In duo puncta C< orbes meridian! in duo concurrunt polos mundi oppositos." t Tbe form in the Colchester records is Gilberd. * Gulielmi Gilbert! de Magnate, Magneticisquecorporibiia, et de magna ma^nete tellure : London, 1600. An English translation by P. F. Mottelay was published 3,Bl,ZEdhyG00gle in the Seventeenth Century. 9 every magnet attracts the south-seeking pole of every other magnet, and repels its north-seeking pola Descartes attempted* to account for magnetic phenomena by his theory of vortices. A vortex of fluid matter was postulated round each magnet, the matter of the vortex entering by one pole and leaving by the other : this matter was supposed to act on iron and steel by virtue of a special resistance to its motion afforded by the molecules of those substances. Crude though the Cartesian system was in this and many other features, there is no doubt that by presenting definite conceptions of molecular activity, and applying them to so wide a range of phenomena, it stimulated the spirit of inquiry, and prepared the way for the more accurate theories that came after. In its own day it met with great acceptance i the confusion which had resulted from the destruction of the old order was now, as it seemed, ended by a reconstruction of knowledge in a system at once credible and complete. Nor did its influence quickly wane ; for even at Cambridge it was studied long after Newton had published his theory of gravitation ;t and in the middle of the eighteenth century Euler and two of the Bernoullis based the explanation of magnetism on the hypothesis of vortices.J Descartes' theory of light rapidly displaced the conceptions which had held sway in the Middle Ages. The validity of his explanation of refraction was, however, called in question by bis fellow-countryman Pierre de Fermat (b. 1601, d. 1665),§ and a controversy ensued, which was kept up by the Cartesians long after the death of their master. Fermat • Principia, Part IT, f 133 sqq. 1 Wilis tun has recorded that, having returned 10 Cambridge after his ordination in 1093, he resumed his studies there, " particularly the Mathema ticks, and the Cartesian Philosophy: which was alone in Vogue with us at that Time. But it was Dot long before I, with immense Pains, but mi Assistance, set myself with the utmost Zeal to the study of Sir Isaac Newton's wonderful Discoveries." — H'huton't Memoirs (1749), 1, p. 36. J Their memoirs shared a prize of the French Academy in 1743, and were printed in 17S2 in the JtMWii dei piieti qui out rtmpartf In prix it TAtad., tome v. t fienati Descartes Epistolae, Para tertia ; Amstelodami, 16S3. The Fermat correspondence is comprised in letters XKli to XLTI. DinlizBdbyGoOgle 10 The Theory of the Aether eventually introduced a new fundamental law, from which he proposed to deduce the paths of rays of light. This waa the celebrated Prindple of Least Time, enunciated* in the form, " Nature always actB by the shortest course." From it the law of reflexion can readily be derived, since the path described by light between a point on the incident ray and a point on the reflected ray is the shortest possible consistent with the con- dition of meeting the reflecting surfaces.f In order to obtain the law of refraction, Fermat assumed that " the resistance of the media is different," and applied his "method of maxima and minima " to find the path which would be described in the least time from a point of one medium to a point of the other. In 1661 he arrived at the solution.* "The result of my work," he writes, " has been the most extraordinary, the most unforeseen, and the happiest, that ever was ; for, after having performed all the equations, multiplications, antitheses, and other operations of my method, and having finally finished the problem, I have found that my principle gives exactly and precisely the same proportion for the refractions which Monsieur Descartes has established." His surprise waa all the greater, as he had supposed light to move more slowly in dense than in rare media, whereasDeecartes had (as will be evident from the demonstration given above) been obliged to make the contrary supposition. Although Fermat's result was correct, and, indeed, of high permanent interest, the principles from which it was derived were metaphysical rather than physical in character, and con- sequently were of little use for the purpose of framing a mechanical explanation of light. Descartes' theory therefore held the field until the publication in 1667g of the Mierograp/iia *Epiiit. zlii, written at Toulouse in August, 1657, to Monsieur de Is Cbambre; reprinted in (Evrrei dt Fenxat [ed. 1891), ii, p. 354. t That reflected light follow! the shortest path was no new result, for it hnd been affirmed (sod attributed to Hero of Alexandria) in the KnooAnia tat irrin&r of Heliodonu of I&viisn, a work of which several editions were published in the seventeenth cenhirr. J Epist . run, written at Toulouse on Jan. 1, 1632 ; reprinted in lEavrii d* Fermat. ii, p. 4S7 ; i, pp. 170, 173. $ The imprimatur of Viscount Brouncker, p.e.s., is dated Nor. 23, 1661. 3,Bl,ZEdhyG00gle in the Seventeenth Century. 1 1 of Robert Hooka (b. 1635, d. 1703), one of the founders of the Royal Society, and at one time its Secretary. Hooke, who was both an observer and a theorist, made two experimental discoveries which concern our present subject ; but in both of these, as it appeared, he had been anticipated. The first' was the observation of the iridescent colours which are seen when light falls on a thin layer of air between two glass plates or lenses, or on a thin film of any transparent substance. These are generally known as the " colours of thin plates," or " Newton's rings " ; they had been previously observed by Boyle.t Hooke's second experimental discovery ,J made after the date of the ificrographia, was that light in air is not propagated exactly in straight lines, but that there is some illumination within the geometrical shadow of an opaque body. This observation had been published in 1665 in a posthumous workg of Francesco Maria Grimaldi (b. 1618, d. 1663), who had given to the phe- nomenon the name diffraction. Hooke's theoretical investigations on light were of great importance, representing as they do the transition from the Cartesian system to the fully developed theory of undulations. He begins by attacking DeBcartes' proposition, that light is a tendency to motion rather than an actual motion. " There is," he observes,!! " no luminous Body but has the parts of it in motion more or less " ; and this motion is " exceeding quick." Moreover, since some bodies (eg. the diamond when rubbed or heated in the dark) shine for a considerable time without being wasted away, it follows that whatever is in motion is not per- manently lost to the body, and therefore that the motion must be of a to-and-fro or vibratory character. The amplitude of the vibrations must be exceedingly small, since some luminous bodies (e.g. the diamond again) are very hard, and so cannot yield or bend to any sensible extent. • Mitrtgraphia, p. 47. t Boyle'* Vtrkt (ed. 1772), i, p. 742. t Hooke's I'ostliumoiis Werkt, p. 188. $ Fhytiea-Hathtnt it liiiaim, coloriiui, tl iride. Bologna, I66S ; booki, prop.i. I Sticrtgrtiphia, p. 55. 3,Bl,ZEdhyG00gle 12 The Theory of the Aether Concluding, then, that the condition associated with the emission of light by a luminous body is a rapid vibratory motion of very small amplitude, Hooke next inquires how light travels through apace. " The next thing we are to consider," he says, " is the way or manner of the trajection of this motion through the interpos'd pellucid body to the eye : And here it will be easily granted — " First, that it must be a body susceptible and impartible of this motion that will deserve the name of a Transparent ; and next, that the parte of Buch a body must be homogeneous, or of the same kind. " Thirdly, that the constitution and motion of the parts must be such that the appulse of the luminous body may be commu- nicated or propagated through it to the greatest imaginable distance in the least imaginable time, though I see no reason to affirm that it must be in an instant. " Fourthly, that the motion is propagated every way through an Homogeneous mediwm by direct or straight lines extended every way like Rays from the centre of a Sphere. " Fifthly, in an Homogeneous medium this motion is propa- gated every way with equal velocity, whence necessarily every pulse or vibration of the luminous body will generate a Sphere, which will continually increase, and grow bigger, just after the same manner (though indefinitely swifter) as the waves or rings on the surface of the water do swell into bigger and. bigger circles about a point of it, where by the sinking of a Stone the motion was begun, whence it necessarily follows, that all the parts of these Spheres undulated through an Homogeneous medium cut the Rays at right angles." Here we have a fairly definite mechanical conception. It resembles that of Descartes in postulating a medium as the vehicle of light ; but according to the Cartesian hypothesis the disturbance is a statical pressure in this medium, while in Hooke's theory it is a rapid vibratory motion of small amplitude. In the above extract Hooke introduces, moreover, the idea of the wave-surface, or locus at any instant of a disturbance gene- 3,Bl,ZEdhyG00gle in the Seventeenth Century. 13 rated originally at a point, and affirms that it is a sphere, whose centre ia the point in question, and whose radii are the rays of light issuing from the point. Hooke's next effort was to produce a mechanical theory of refraction, to replace that given by Descartes. " Because," he says, " all transparent mediums are not Homogeneous to one another, therefore we will next examine how this pulse or motion will be propagated through differingly transparent mediums. And here, according to the most acute and excellent Philosopher De& Cartes, I suppose the sine of the angle of inclination in the firat medium to be to the sine of refraction in the second, as the density of the first to the density of the second. By density, I mean not the density in respect of gravity (with which the refractions or transparency of mediums hold no proportion), but in respect only to the trajecHon of the Bays of light, in which respect they only differ in this, that the one propagates the pulse more easily and weakly, the other more slowly, but more strongly. But as for the pulses themselves, they will by the refraction acquire another property, which we shall now endeavour to explicate. " We will suppose, therefore, in the first Figure, ACFD to be £ a physical Ray, or ASCand DEFbobe two mathematical Rays, trajected from a very remote point of a luminous body through 3,Bl,ZEdhyG00gle 14 The Theory of the Aether an Homogeneous transparent medium LL, and DA, EB, FC, to be small portions of the orbicular impulses which must therefore cut the Rays at right angles ; these Kays meeting with the plain surface NO of a medium that yields an easier transitu* to the propagation of light, and falling obliquely on it, they will in the medium MM be refracted towards the perpendicular of the surface. And because this medium is more easily trajected tlum the former by a third, therefore the point C of the orbicular pulse FC will be moved to H four spaces in the same time that F, the other end of it, is moved to three spaces, therefore the whole refracted pulse to H shall be oblique to the refracted BayH CEK&n&GL" * Although this is not in all respects successful, it represents a decided advance on the treatment of the same problem by Descartes, which rested on a mere analogy. Houke tries to determine what happens to the wave-front when it meets the interface between two media ; and for this end he intro- duces the correct principle that the side of the wave-front which first meets the interface will go forward in the second medium with the velocity proper to that medium, while the other side of the wave-front which is still in the first medium is still moving with the old velocity : so that the wave-front will be deflected in the transition from one medium to the other. This deflection of the wave-front was supposed by Hooke to be the origin of the prismatic colours. He regarded natural or white light as the simplest type of disturbance, being consti- tuted by a simple and uniform pulse at right angles to the direction of propagation, and inferred that colour is generated by the distortion to which this disturbance is subjected in the process of refraction. " The Eay,"* he says, "is dispersed, split, and opened by its Refraction at the Superficies of a second medium, and from a line is opened into a diverging Superficies, and so obliquated, whereby the appearances of Colours are produced." ■Hooks, Putt&mMiu IForki, p. 82. ■D,Bl,ZEdhyG00gle in the Seventeenth Century. 15 " Colour," he says in another place* " is nothing but the disturbance of light by the communication of the pulse to other transparent mediums, that is by the refraction thereof." His precise hypothesis regarding the different colours wasf "that Blue is an impression on the Retina of an oblique and confus'd pulse of light, whose weakest part precedes, and whose strongest follows. And, that red is an impression on the Eetina of an oblique and confus'd pulse of light, whose strongest part precedes, and whose weakest follows." Hooke's theory of colour was completely overthrown, within a few years of its publication, by one of the earliest discoveries of Isaac Newton (b. 1642, d. 1727). Newton, who was elected a Fellow of Trinity College, Cambridge, in 1667, had in the beginning of 1666 obtained a triangular prism, " to try therewith the celebrated Phaenoniena of Colours." For this purpose, " having darkened my chamber, and made a small hole in my window-shuts, to let in a convenient quantity of the Sun's light, I placed my PriBme at his entrance, that it might be thereby refracted to the opposite wall. It was at first a very pleasing divertisement, to view the vivid and intense colours produced thereby ; hut after a while applying myself to consider them more circumspectly, I became surprised to see them in an oblmg form, which, according to the received laws of Refraction, I expected should have been circular." The length of the coloured spectrum was in fact about five times as great as its breadth. This puzzling fact he set himself to study ; and after more experiments the true explanation was discovered — namely, that ordinary white light is really a mixture of rays of every variety of colour, and that the elongation of the spectrum is due to the differences in the refractive power of the glass for these different rays. " Amidst these thoughts," he tells us,J " I was forced from ■To the Royal Society, February IS, 1671-1. t it terw0 raphia, p. 64. 1 Pbil. nun., Mo. 80, February 19, 1671-2. 3,Bl,ZEdhyG00gle 16 The Theory of the Aether Cambridge by the intervening Plague " ; this waa in 1666, and his memoir on the subject was not presented to the Royal Society until five years later. In it he propounds a theory of colour directly opposed to that of Hooke. " Colours," he says, "are not Qualifications of light derived from Refractions, or Reflections of natural Bodies (as 'tis generally believed), but Original and connate properties, which in divers Kays are divers. Some Rays are disposed to exhibit a red colour and no other : some a yellow and no other, some a green and no other, and so of the reet. Nor are there only Rays proper and particular to the more eminent colours, but even to all their intermediate gradations. " To the same degree of Refrangibility ever belongs the same colour, and to the same colour ever belongs the same degree of Refrangibility." " The species of colour, and degree of Refrangibility proper to any particular sort of Rays, is not mutable by Refraction, nor by Reflection from natural bodies, nor bv any other cause, that I could yet observa When any one sort of Rays hath been well parted from those of other kinds, it hath afterwards obstinately retained its colour, notwithstanding my utmost endeavours to change it." The publication of the new theory gave rise to an acute controversy. As might have been expected, Hooke was foremost among the opponents, and led the attack with some degree of asperity. When it is remembered that at this time Newton was at the outset of his career, while Hooke was an older man, with an established reputation, such harshness appears par- ticularly ungenerous; and it is likely that the unpleasant consequences which followed the announcement of his first great discovery had much to do with the reluctance which Newton ever afterwards showed to publish his results to the world. In the course of the discussion Newton found occasion to explain more fully the views which he entertained regarding the nature of light. Hooke charged him with holding the 3,Bl,ZEdhyG00gle in the Seventeenth Century. 17 doctrine that light is a material substance. Now Newton had, as a matter of fact, a great dislike of the more imaginative kind of hypotheses ; he altogether renounced the attempt to construct the universe from its foundations after the fashion of Descartes, and aspired to nothing more than a formulation of the laws which directly govern the actual phenomena. His theory of gravitation, for example, is strictly an expression of the results of observation, and involves no hypothesis as bo the cause of the attraction which subsists between ponderable bodies ; and his own desire in regard to optics was to present a theory free from speculation as to the hidden mechanism of light. Accordingly, in reply to Hooke's criticism, he protested" that his views on colour were in no way hound up with any particular conception of the ultimate nature of optical processes. Newton was, however, unable to carry out hie plan of connecting together the phenomena of light into a coherent and reasoned whole without having recourse to hypotheses. The hypothesis of Hooke, that light consists in vibrations of an aether, he rejected for reasons which at that time were perfectly cogent, and which indeed were not successfully refuted for over a century. One of these was the incompetence of the wave- theory to account for the rectilinear propagation of light, and another was its inability to embrace the facta — discovered, as we shall presently see, by Huygens, and Brat interpreted correctly by Newton himself — of polarization. On the whole, he seems to have favoured a scheme of which the following may be taken as a summaryf: — All space is permeated by an elastic medium or aether, which is capable of propagating vibrations in the same way as the * Phil. Trans, vii, 1672, p. 6086. t Cf. Newton's memoir in Phil. Trans, vii, 1672 ; his memoir presented to the Royal Society in December, 1676, which is printed in Birch, iii, p. 247; his Optitki, especially Queries IS, 19, 20, 21, 23, 29; the Scholium at the end of tbeFrineipia; and a letter to Boyle, written in February, 1678-9, which is printed in Horaley's JfeinCani Optra, p. 385. In the Prineipia, Book I., section ii v, the analogy between rays of light and streams of corpuscles is indicated; hut Newton does not commit himself to any theory of light based on this. 3,Bl,ZEdhyG00gle 18 The Theory of the Aether air propagates the vibrations of sound, but with far greater velocity. This aether pervades the pores of all material bodies, and is the cause of their cohesion ; its density varies from one body to another, being greatest in the free interplanetary spaces. It is not necessarily a single uniform substance : but just as air contains aqueous vapour, bo the aether may contain various "aethereal spirits," adapted to produce the phenomena of electricity, magnetism, and gravitation. The vibrations of the aether cannot, for the reasons already mentioned, be supposed in themselves to constitute light. Light is therefore taken to be " something of a different kind, propagated from lucid bodies. They, that will, may suppose it an aggregate of various peripatetic qualities. Others may suppose it multitudes of unimaginable small and swift corpuscles of various sizes, springing from shining bodies at great distances one after another ; but yet without any sensible interval of time, and continually urged forward by a principle of motion, which in the beginning accelerates them, till the resistance of the aethereal medium equals the force of that principle, much after the manner that bodies let fall in water are accelerated till the resistance of the water equals the force of gravity. But they, that like not this, may suppose light any other corporeal emanation, or any impulse or motion of any other medium or aethereal spirit diffused through the main body of aether, or what else they can imagine proper for this purpose. To avoid dispute, and make this hypothesis general; let every man here take his fancy ; only whatever light be, I suppose it consists of rays differing from one another in contingent circumstances, as bigness, form, or vigour. "* In any case, light and aether are capable of mutual inter- action; aether is in fact the intermediary between light and ponderable matter. When a ray of light meets a stratum of aether denser or rarer than that through which it has lately been passing, it is, in general, deflected from its rectilinear " Eoynl Society, Dec. 9, 1575. 3,Bl,ZEdhyG00gle in the Seventeenth Century. 19 course ; and differences of density of the aether between one material medium and another account on these principles for the reflexion and refraction of light. The condensation or rarefaction of the aether due to a material body extends to some little distance from the surface of the body, so that the inflexion due to it is really continuous, and not abrupt; and this further explains diffraction, which Newton took to be " only a new kind of refraction, caused, perhaps, by the external aether's beginning to grow rarer a little before it came at the opake body, than it was in free spaces." Although the regular vibrations of Newton's aether were not supposed to constitute light, its irregular turbulence seems to have represented fairly closely Wb conception of heat. He supposed that when light is absorbed by a material body, vibrations are set up in the aether, and are recognizable as the heat which is alwayB generated in Buch cases. The conduction of heat from hot bodies to contiguous cold ones he conceived to be effected by vibrations of the aether propagated between them ; and he supposed that it is the violent agitation of aethereal motions which excites incandescent substances to emit light. Assuming with Newton that light is not actually con- stituted by the vibrations of an aether, even though such vibrations may exist in close connexion with it, the most definite and easily conceived supposition is that rays of light are streams of corpuscles emitted by luminous bodies. Although this was not the hypothesis of DeBcartes himself, it was so thoroughly akin to his general scheme that the scientific men of Newton's generation, who were for the most part deeply imbued with the Cartesian philosophy, instinctively selected it from the wide choice of hypotheses which Newton had offered them ; and by later writers it was generally associated with Newton's name. A curious argument in its favour was drawn from a phenomenon which had then been known for nearly half a century : Vincenzo Cascariolo, a shoemaker of Bologna, had discovered, about 1630, that a substance, which afterwards C 2 3,Bl,ZEdhyG00gle 20 The Theory of the Aether received the name of Bologna stone or Bologna phosphorus, has* the property of shining in the dark after it has been exposed for some time to sunlight ; and the storage of light which seemed to be here involved was more easily explicable on the corpuscular theory than on any other. The evidence in this quarter, however, pointed the other way when it was found that phosphorescent substances do not necessarily emit the same kind of light as that which was used to stimulate them. In accordance with his earliest discovery, Newton considered colour to be an inherent characteristic of light, and inferred that it must be associated with some definite quality of the corpuscles or aether-vibrations. The corpuscles corresponding to different colours would, he remarked, like sonorous bodies of different pitch, excite vibrations of different types in the aether ; and " if by any means those [aether-vibrations] of unequal bignesses be separated from one another, the largest beget a Sensation of a Red colour, the least or shortest of a deep Violet, and the intermediate ones, of intermediate colours ; much after the manner that bodies, according to their several sizes, shapes, and motions, excite vibrations in the Air of various bignesses, which, according to those bignesses, make several Tones in Sound.'" This sentence is the first enunciation of the great principle that homogeneous light is essentially periodic in its nature, and that differences of period correspond to differences of colour. The analogy with Sound is obvious ; and it may be remarked in passing that Newton's theory of periodic vibrations in an elastic medium, which he developed! in connexion with the explanation of Sound, would alone entitle him to a place among those who have exercised the greatest influence on the theory of light, even if he had made no direct contribution to the latter Bubject. • Phil. Trail*, vji (1672), p. 5088. t Newton'i Frineipia, Book ii., Props, iliii.-l. 3,Bl,ZEdhyG00gle in the Seventeenth Century. 21 Newton devoted considerable attention to the colours of thin plates, and determined the empirical laws of the phenomena with great accuracy. In order to explain them, he supposed that " every ray of light, in its passage through any refracting surface, is put into a certain transient constitution or state, which, in the progress of the ray, returns at equal intervals, and disposes the ray, at every return, to be easily transmitted through the next refracting surface, and, between the returns, to be easily reflected by it."" The interval between two consecutive dispositions to easy transmission, or •"length of fit," he supposed to depend on the colour, being greatest for red light and least for violet. If then a ray of homogeneous light falls on a thin plate, its fortunes as regards transmission and reflexion at the two surfaces will depend on the relation which the length of fit bears to the thickness of the plate ; and on this basis he built up a theory of the colours of thin plates. It is evident that Newton's " length of fit " corresponds in some measure to the quantity which in the undulatory theory is called the wave-length of the light ; but the suppositions of easy transmission and reflexion were soon found inadequate to explain all Newton's experimental results — at least without making other and more complicated additional assumptions. At the time of the publication of Hooke's Micrographia, and Newton's theory of colours, it was not known whether light is propagated instantaneously or not. An attempt to settle the question experimentally had been made many years previously by Galileo, f who had stationed two men with lanterns at a considerable distance from each other ; one of them was directed to observe when the other uncovered his light, and exhibit hiB own the moment he perceived it. But the interval of time required by the light for its journey was too small to be perceived in this way ; and the discovery was • Opticki, Boole ii., Prop. 12. t Biicorii t dimoilrtxitmi mattmaticht, p. 43 of the Elzevir edition of 1638. 3,Bl,ZEdhyG00gle 22 The Theory of Ike Aether ultimately made by an astronomer. It was observed in 1675 by Olof Roemer* (5. 1644, d. 1710) that the eclipses of the first satellites of Jupiter were apparently affected by an unknown disturbing cause ; the time of the occurrence of the phenomenon was retarded when the earth and Jupiter, in the course of their orbital motions, happened to be moat remote from each other, and accelerated in the contrary case. Roemer explained this by supposing that light requires a finite time for its pro- pagation from the satellite to the earth ; and by observations of eclipses, he calculated the interval required for its passage from the sun to the earth (the light-equation, as it is called) to be 11 minuteB.f Shortly after Roemer's discovery, the wave-theory of light was greatly improved and extended by Christiaan Huygens (b. 1629, d. 1695). Huygens, who at the time was living in Paris, communicated his results in 1678 to Cassini, Roemer, De la Hire, and the other physicists of the French Academy, and prepared a manuscript of considerable length on the subject. This he proposed to translate into Latin, and to publish in that language together with a treatise on the Optics of Telescopes ; but the work of translation making little progress, after a delay of twelve years, he decided to print the work on wave-theory in its original form. In 1690 it appeared at Leyden,{ under the title Traiti de la htmiere oit soni expliqu^es Us causes de ce qui luy arrive dans la riflerion et dans la refraction. Et parti- * Mem. derAcad.*. (1666-1699), p. 67&. t It was soon recognized that Boemei's value tu too large ; and the astronomers of the succeeding half -century reduced it to 7 minutes. Delanibre, by an Investigation whose detail* appear to have been completely destroyed, published in 1817 tha value 49B-2", from a discussion of eclipses of Jupiter's satellites during the previous ISO yean. Gluenapp, in an inaugural dissertation published in 1875, discussed the eclipses of the first satellite between 1848 and 1870, and derived, by different assumptions, values between 496' and 501", tlie moat probable value being oOO-B'. Sampson, in 1909, derived 498-84' from his own leadings of the Harvard Observations, and 49B-79* from the Harvard readings, with probable errors of about ± 001". The inequalities of Jupiter's surface give- rise to some difficulty in exact determinations. t Huygens had by this time returned to Holland. 3,Bl,ZEdhyG00gle in the Seventeenth Century. 23 culierement darts I'dtrange refraction dv, cristal d'Islande. Par C.H.D.Z* The truth of Hooke's hypothesis, that light is essentially a form of motion, seemed to Huygens to be proved by the effects observed with burning-glasses ; for in the combustion induced at the focus of the glass, the molecules of bodies are dissociated ; which, as he remarked, must be taken as a certain sign of motion, if, in conformity to the Cartesian philosophy, we seek the cause of all natural phenomena in purely mechanical actions. The question then arises as to whether the motion is that of a medium, as is supposed in Hooke's theory, or whether it may be compared rather to that of a flight of arrows, as in the corpuscular theory. Huygens decided that the former alter- native is the only tenable one, since beams of light proceeding in directions inclined to each other do not interfere with each other in any way. Moreover, it had previously been shown by Torricelli that light is transmitted as readily through a vacuum as through air ; and from this Huygens inferred that the medium or aether , in which the propagation takes place must penetrate all matter, ' and be present even in all so-called vacua. The process of wave-propagation he discussed by aid of a principle which was nowt introduced for the first time, and has since been generally known by his name. It may he stated thus : Consider a wave-front* or locus of disturbance, as it exists at a definite instant t0 : then each surface-element of the wave-front may be regarded as the source of a secondary wave, which in a homogeneous isotropic medium will be propagated outwards from the surface-element in the form of a sphere whose radius at any subsequent instant t is proportional to (Mt) ; and thB wave-front which represents the whole distur- * i.e. CbrutiMD Huygena de Zuylichem. The custom of indicating u&mee by initial* ni not unusual in that age. t Traiii dt la lnm„ p. 17. lit may bo remarked that Huygens' "wmvei" are really what modern writer*, following Hooke, call " pnlaea " ; Huygan* never considered true wate-traina having the property of periodicity. 3,Bl,ZEdhyG00gle 24 The Theory of the Aether bance at the instant t is simply the envelope of the secondary waves which arise from the various surface elements of the original wave-front* The introduction of this principle enabled Huvgens to succeed where Hooke and other contemporary wave-theoristst had failed, in achieving the explanation of refraction and reflexion. His method was to combine his own principle with Hooke's device of following separately the fortunes of the right-hand and left-hand sides of a wave-front when it reaches the interface between two media. The actual explana- tion for the case of reflexion is as follows : — Let AB represent the interface at which reflexion takes place, AHC the incident wave-front at an inBtant t0, OMB the position which the wave-front would occupy at a later instant t if the propagation were not interrupted by reflexion. Then by Huygens' principle the secondary wave from A ifi at the instant t a sphere MNS of radius equal to AG : the disturbance from IT, after meeting the interface at K, will generate a secondary wave TV of radius equal to KM, and similarly the secondary wave corresponding to any other element of the original wave- * The justification for thia tu given long afterward! by Freanel, Annalet ds chimir, iii. , + e.g. Ignaee Gaston Pardiea and Pierre Ango, the latter of whom published a work od Optica at Faria'in 1682. «:tyG00£>lc in the Seventeenth Century. 25 front can be found. It is obvious that the envelope of these Becondary waves, which constitutes the final wave-front, will he a plane BN, which will he inclined to AB at the same angle as AC. This gives the law of reflexion. The law of refraction is established by similar reasoning, on the supposition that the velocity of light depends on the medium in which it is propagated Since a ray which passes from air to glass is bent inwards towards the normal, it may be inferred that light travels more slowly in glass than in air. Huygens offered a physical explanation of the variation in velocity of light from one medium to another, by supposing that transparent bodies consist of hard particles which interact with the aethereal matter, modifying its elasticity. The opacity of metals he explained by an extension of the same idea, supposing that some of the particles of metals are hard {these account for reflexion) and the rest soft : the latter destroy the luminous motion by damping it. The second half of the Thiorie de la lumibre is concerned with a phenomenon which had been discovered a few years pre- viously by a Danish philosopher, Erasmus Bartholin (b. 1625, d. 1698). A Bailor had brought from Iceland to Copenhagen a number of beautiful crystals which he had collected in the Bay of Roerford. Bartholin, into whose hands they passed, noticed* that any small object viewed through one of these crystals appeared double, and found the immediate cause of this in the fact that a ray of light entering the crystal gave rise in general to two refracted rays. One of these rays was subject to the ordinary law of refraction, while the other, which was called the extraordinary ray, obeyed a different law, which Bartholin did not succeed in determining. The matter had arrived at this stage when it was taken up by Huygens. Since in his conception each ray of light corresponds to the propagation of a wave-front, the two rays in Iceland spar must correspond to two different wave-fronts propagated * Eiperiineuta crittalli Iiiandici dUdiaeiattiei ; 1569. D,Bl,ZEdhyG00gle 26 The Theory of the Aether simultaneously. In this idea he found no difficulty ; as he says : " It is certain that a space occupied by more than one kind of matter may permit the propagation of several kinds of waves, different in velocity ; for this actually happens in air mixed with aethereal matter, where sound-waves and light-waves are propagated together." Accordingly he supposed that a light-disturbance generated at any spot within a crystal of Iceland spar spreads out in the form of a wave-surface, composed of a sphere and a spheroid having the origin of disturbance as centre. The spherical wave- front corresponds to the ordinary ray, and the spheroid to the extraordinary ray ; and the direction in which the extraordinary ray is refracted may be determined by a geometrical construc- tion, in which the spheroid takes the place which in the ordinary construction is taken by the sphere. Thus, let the plane of the figure be at right angles to the intersection of the wave-front with the surface of the crystal ; let AB represent the trace of the incident wave-front ; and suppose that in unit time the disturbance from B reaches the interface at T. In this unit-interval of time the disturbance from A will have spread out within the crystal into a sphere and spheroid : so the wave-front corresponding to the ordinary ray will be the tangent-plane to the sphere through the line whose traco is T, while the wave-front corresponding to the extraordinary ray will be the tangent-plajne to the spheroid through the same line. The points of contact Jf 3,Bl,ZEdhyG00gle in the Seventeenth Century, 27 and M will determine the directions AN and AM of the two refracted rays* within the crystal. Huygens did not in the Tk4orie de la lumifrre attempt a detailed physical explanation of the spheroidal wave, but communicated one later in a letter to Papin,t written in December, 1690. " As to the kinds of matter contained in Iceland crystal," he says, " I suppose one composed of small spheroids, and another which occupies the interspaces around these spheroids, and which serves to bind them together. Besides these, there is the matter of aether permeating all the crystal, both between and within the - parcels of the two kinds of matter just mentioned ; for I suppose both the little spheroids, and the matter which occupies the intervals around them, to be composed of small fixed particles, amongst which are diffused in perpetual motion the still finer particles of the aether. There is now uo reason why the ordinary ray in the crystal should not be due to waves propa- gated in this aethereal matter. To account for the extraordinary refraction, I conceive another kind of waves, which have for vehicle both the aethereal matter and the two other kinds of matter constituting the crystal. Of these latter, I suppose that the matter of the small spheroids transmits the waves a little more quickly than the aethereal matter, while that around the spheroids transmits these waves a little more slowly than the same aethereal matter. . . . These same waves, when they travel in the direction of the breadth of the spheroids, meet with more of the matter of the spheroids, or at least pass with less obstruction, and so are propagated a little more quickly in this sense than in the other ; thus the light-disturbance is propagated as a spheroidal sheet." Huygens made another discoveryj of capital importance when * The void ray in the wire- theory it always applied to the liae which goes from the centre of a wave (i.e. the origin of the disturbance) to a point on its surface, whatever may be the inclination of tbu line to the surface -element on wroth it abate ; for tfaii line has the optical properties of the "raya" of the emiseinn theory. t Huygens' (Bnvrtt, ed. 1905, z., p. ITT. ; Wont dt la lumiirt, p. 83. 3,Bl,ZEdhyG00gle 28 Theory of the Aether in the Seventeenth Century. experimenting with the Iceland crystal. He observed that the two rays which are obtained by the double refraction Of a single ray afterwards behave in a way different from ordinary light which has not experienced double refraction ; and in particular, if one of these rays is incident on a second crystal of Iceland spar, it gives rise in some circumstances to two, and in others to only one, refracted ray. The behaviour of the ray at this second refraction can be altered by simply rotating the second crystal about the direction of the ray as axis ; the ray under- going the ordinary or extraordinary refraction according as the principal section of the crystal is in a certain direction or in the direction at right angles to this. The first stage in the explanation of Huygens' observation was reached by Newton, who in 1717 showed* that a ray obtained by double refraction differs from a ray of ordinary light in the same way that a long rod whose cross-section is a rectangle differs from a long rod whose cross-Bection is a circle : in other words, the properties of a ray of ordinary light are the same with respect to all directions at right angles to its direction of propagation, whereas a ray obtained by double refraction must be supposed to have sides, or properties related to special directions at right angles to its own direction. The refraction of such a ray at the surface of a crystal depends on the relation of its sides to the principal plane of the crystal. That a ray of light should possess such properties seemed to Newtonf an insuperable objection to the hypothesis which regarded waves of light as analogous to waves of sound. On this point he was in the right : his objections are perfectly valid against the wave-theory as it was understood by his contemporarieBj, although not against the theory§ which was put forward a century later by Young and Fresnel. * The second edition of Newton's Optkki, Query 26. t Optitkt, Query 28. X In which the oscillation* are performed in the direction in which the ware advances. j In which the oscillations are performed in a direction at right angles to that in which the wave advances. 3,Bl,ZEdhyG00gle 29 ) ELECTRIC AND MAGNETIC BCIENCE PRIOR TO THE INTRODUCTION OF THE POTENTIALS. The magnetic discoveries of Peregrinus and Gilbert, and the vortex-hypothesis by which Descartes had attempted to explain them * had raised magnetism to the rank of a separate science by the middle of the seventeenth century. The kindred science of electricity was at that time in a less developed state ; but it had been considerably advanced by Gilbert, whose researches in this direction will now be noticed. For two thousand years the attractive power of amber had been regarded aa a virtue peculiar to that substance, or possessed by at most one or two others. Gilbert provedf this view to be mistaken, showing that the same effects are induced by friction in quite a large class of bodies; among which he mentioned glass, sulphur, sealing-wax, and various precious stones. A force which was manifested by so many different kinds of matter seemed to need a name of its own; and accordingly Gilbert gave to it the name electric, which it has ever since retained. Between the magnetic and electric forces Gilbert remarked many distinctions. The lodestone requires no stimulus of friction such as is needed to stir glass and sulphur into activity. The lodestone attracts only magnetizable substances, whereas electrified bodies attract everything. The magnetic attraction between two bodies is not affected by interposing a sheet of paper, or a linen cloth, or by immersing the bodies in water ; whereas the electric attraction is readily destroyed by screens. Lastly, the magnetic force tends to arrange bodies in definite * Cf. pp. 7-9. f $i Ifagtittt, lib. ii., cop. 2. 3,Bl,ZEdhyG00gle 30 Electric and Magnetic Science orientations ; while the electric force merely tends to heap them together in shapeless clusters. These facts appeared to Gilbert to indicate that electric phenomena are due to something of a material nature, which under the influence of friction is liberated from the glass or amber in which under ordinary circumstances it is imprisoned. In support of this view he adduced evidence from other quarters. Being a physician, he was well acquainted with the doctrine that the human body contains various humours or kinds of moisture — phlegm, blood, choler, and melancholy, — which, as they predominated, were supposed to determine the temper of mind; and when he observed that electritiable bodies were almost all hard and transparent, and therefore (according to the ideas of that time) formed by the consolidation of watery liquids, he concluded that the common menstruum of these liquids must be a particular kind of humour, to the possession of which the electrical properties of bodies were to be referred. Friction might be supposed to warm or otherwise excite or liberate the humour, which would then issue from the body as an effluvium and form an atmosphere around it. The effluvium must, he remarked, be very attenuated, for its emission cannot be detected by the senses. The existence of an atmosphere of effluvia round every electrified body might indeed have been inferred, according to Gilbert's ideas, from the single fact of electric attraction. For lie believed that matter cannot act where it is not ; and hence if a body acts on all surrounding objects without appearing to touch them, something must have proceeded out of it unseen. The whole phenomenon appeared to him to be analogous to the attraction which is exercised by the earth on falling bodies. For in the latter case he conceived of the atmospheric air as the effluvium by which the earth draws all things downwards to itself. Gilberts theory of electrical emanations commended itself generally to such of the natural philosophers of the seventeenth century as were interested in the subject ; among whom were 3,Bl,ZEdhyG00gle prior to the Introduction of the Potentials. 31 numbered Niccolo Cabeo (b. 1585, d. 1650), an Italian Jesuit who was perhaps the first to observe that electrified bodies repel as well as attract ; the English royalist exile, Sir Kenelm Digby (6. 1603, d. 1665); and the celebrated Robert Boyle (&. 1627, d. 1691). There were, however, some differences of opinion as to the manner in which the effluvia acted on the small bodies and set them in motion towards the excited electric; Gilbert himself had supposed the emanations to have an inherent tendency to reunion with the parent body ; Digby likened their return to the condensation of a vapour by cooling ; and other writers pictured the effluvia as forming vortices round the attracted bodies in the Cartesian fashion. There is a well-known allusion to Gilbert's hypothesis in Newton's Opticka * " Let him also tell me, how an electrick body can by friction emit an exhalation so rare and subtle,! and yet so potent, as by its emission to cause no sensible diminution of the weight of the electrick body, and to be expanded through a sphere, whose diameter is above two feet, and yet to be able to agitate and carry up leaf copper, or leaf gold, at a distance of above a foot from the electrick body ? " It is, perhaps, somewhat surprising that the Newtonian doctrine of gravitation should not have proved a severe blow to the emanation theory of electricity ; but Gilbert's doctrine was now so firmly established as to be unshaken by the overthrow of the analogy by which it had been originally justified It was, however, modified in one particular about the beginning of the eighteenth century. In order to account for the fact that electrics are not perceptibly wasted away by excitement, the earlier writers had supposed all the emanations to return ultimately to the body which had emitted them ; but the corpuscular theory of light accustomed philosophers to the idea of emissions so subtle as to cause no perceptible loss ; and • Query 22. t" Subtlety," aojs Johnion, "which in Ut original import menus exility of particle*, )■ taken in ita metaphorical meaning for nicety of distinction. " 3,Bl,ZEdhyG00gle 32 Electric and Magnetic Science after the time of Newton the doctrine of the return of the electric effluvia gradually lost credit. Newton died in 1727. Of the expositions of Mb philosophy which were published in his lifetime by his followers, one at least deserves to be noticed for the sake of the insight which it affords into the state of opinion regarding light, heat, and electricity in the first half of the eighteenth century. This was the Physices elementa mathematica experimentis conj&rmata of Wilhelm Jacob s'Gravesande (J. 1688, d. 1742), published at Leyden in 1720. The Latin edition was afterwards reprinted several times, and was, moreover, translated into French and English : it seems to have exercised a considerable and, on the whole, well-deserved influence on contemporary thought. s'Gravesande supposed light to consist in the projection of corpuscles from luminous bodies to [the eye ; the motion being very swift, as is Bhown by astronomical observations. Since many bodies, e.g. the metals, become luminous when they -are heated, he inferred that every substance possesses a natural store of corpuscles, which are expelled when it is heated to incandescence ; conversely, corpuscles may become united to a material body ; as happens, for instance, when the body is exposed to the rays of a fire. Moreover, since the heat thus acquired is readily conducted throughout the substance of the body, he concluded that corpuscles can penetrate all substances, however hard and dense they be. Let us here recall the ideas then current regarding the nature of material bodies. From the time of Boyle (1626-1691) it had been recognized generally that substances perceptible to the senses may be either dements or compounds or mixtures ; the compounds being chemical individuals, distinct from mere mixtures of elements. But the substances at that time accepted as elements were very different from those which are now known by the name. Air and the calces* of the metals figured in the list, while almost all the chemical elements now recognized were 3,Bl,ZEdhyG00gle prior to the Introduction oj the Potentials. 33 omitted from it ; some of them, such aa oxygen and hydrogen, because they were as yet undiscovered, and others, such as the metals, because they were believed to be compounds. Among the chemical elements, it became customary after the time of Newton to include light-corpuscles.* That some- thing which is confessedly imponderable should ever have been admitted into this class may at first sight seem surprising. But it must be remembered that questions of ponderability counted for very little with the philosophers of the period. Three- quarters of the eighteenth century had passed before Lavoisier enunciated the fundamental doctrine that the total weight of the substances concerned in a chemical reaction is the same after the reaction as before it As soon as this principle came to be universally applied, light parted company from the true elements in the scheme of chemistry. We must now consider the views which were held at this time regarding the nature of heat. These are of interest for our present purpose, on account of the analogies which were set up between heat and electricity. The various conceptions which have been entertained concerning heat fall into one or other of two classes, according as heat is represented as a mere condition producible in bodies, or as a distinct species of matter. The former view, which is that universally held at the present day, was advocated by the great philosophers of the seventeenth century. Bacon maintained it in the Novum (hganuin : " Calor," he wrote, " est motus expansivus, cohibitus, et nitens per partes minores."t BoyleJ affirmed that the " Nature of Heat " consists in " a various, vehement, and intestine commotion of the Farts among themselves." Hooke§ declared that " Heat is a property of a body arising from the motion or agitation of its parts." And Newton|| asked : " Do not * Newton himeelf (Opticii, p. 349} euepected that light •corpuaclee and ponderable matter might be transmuted into each other: much later, BoMOTich {Throria, pp. 216, 217) regarded the matter of light ■■ a principle or element in the conatitutiun of natural bodies. t Nov. Org., Lib. ii., Aphor. ix. I Iftehmrieai Production of Htat and Void, t, MinBgraphia, p. 37. || Optirhi. D 3,Bl,ZEdhyG00gle 34 Electric and Magnetic Science all fixed Bodies, when heated beyond a certain Degree, emit light and Bhine ; and is not this Emission performed by the vibrating Motion of their Parts ? " and, moreover, suggested the converse of this, namely, that when light is absorbed by a material body, vibrations are set up which are pereeived by the senses as heat The doctrine that heat is a material substance was main- tained in Newton's lifetime by a certain school of chemists. The moat conspicuous member of the school was "Wilhelm Homberg (ft, 1652, d. 1715) of Paris, who" identified heat and light with the sulphureous principle, which he supposed to be one of the primary ingredients of all bodies, and to be present even in the inter- planetary spaces. Between this view and that of Newton it might at first seem as if nothing but sharp opposition was to be expected, f But a few years later the professed exponents of the Principia and the Opticks began to develop their Bystem under the evident influence of Homberg'B writings. This evolution may easily be traced in s'GraveBande, whose starting-point is the admittedly Newtonian idea that heat bears to light a relation similar to that which a state of turmoil bears to regular rectilinear motion ; whence, conceiving light as a projection of corpuscles, he infers that in a hot body the material particles and the light-corpuscles! are in a state of agitation, which becomes more violent as the body is more intensely heated. s'Gravesande thus holds a position between the two opposite camps. On the one hand he interprets heat as a mode of motion ; but on the other he associates it with the presence of a particular kind of matter, which he further identifies with the matter of light. After this the materialistic hypothesis made * M6ni. de l'Auod., 1706, p. 89. t Though it reminds ua of a curious conjecture of Newton's : "Is not the strength and vigour of the action between light and sulphureous bodies one reason why sulphureous bodies take fire more readily and burn more vehemently than other bodies do?" 1 1 have thought it best to translate s'Qravesanda's ignii by "light- corpuscles." This is, I think, fully justified by such of his statements as Quanrio ignit fur Unit! rtctai ocidca iioitroi intrtti, *x metu qvtm fibril in fuh&o eeuli nmmumcal idtain Ivminit txcitat. 3,Bl,ZEdhyG00gle prior to the. Introduction of the Potentials, 35 rapid progress. It was frankly advocated by another member of the Dutch school, Hermann Boerhaave" (b. 1668, d. 1738), Professor in the University of Leyden, whose treatise on chemistry was translated into English in 1727. Somewhat later it was found that the heating effects of the rays from incandescent bodies may be separated from their luminous effects by passing the rays through a plate of glass, which transmits the light, but absorbs the heat. After this discovery it was no longer possible to identify the matter of heat with the corpuscles of light ; and the former was consequently accepted as a distinct element, under the name of calorie.^ In the latter part of the eighteenth and early part of the nineteenth centuries? caloric was generally conceived as occupying the interstices between the particles of ponderable matter — an idea which fitted in well with the observation that bodies commonly expand when they are absorbing heat, but which was less com- petent to explain the factg that water expands when freezing. The latter difficulty was overcome by supposing the union between a body and the caloric absorbed in the process of melting to be of a chemical nature; so that the consequent changes in volume would be beyond the possibility of prediction. As we have already remarked, the imponderability of heat did not appear to the philosophers of the eighteenth century to be a sufficient reason for excluding it from the list of chemical -elements ; and in any case there was considerable doubt as to whether calorie was ponderable or not. Some experimenters believed that bodies were heavier when cold than when hot ; others that they were heavier when hot than when cold. The ■century was far advanced before Lavoisier and Rumford finally * JJnertioave followed Homberg in supposing the metier of beet to be present in all so-called vacuous spaces. t Seheele in 1777 supposed caloric to be a compound of oxygen and phlogiston, and light tu be oxygen combined with a greater proportion of phlogiston. J In spite of the experiments of Benjamin Thompson, Count Eumfurd {6. 1753, d. 1814), in the closing years of (he eighteenth ceuluiy. These should have sufficed to re-establish the older conception of heat. j Thi-> bad been known since the time of Boyle. D : 3,Bl,ZEdhyG00gle 36 Electric and Magnetic Science proved that the temperature of a body is without sensible influence on its weight. PerhapB nothing in the history of natural philosophy is more amazing than the vicissitudes of the theory of heat. The true hypothesis, after having met with general acceptance throughout a century, and having been approved by a succession of illus- trious men, was deliberately abandoned by their successors in favour of a conception utterly false, and, in some of its developments, grotesque and absurd. We must now return to s'Gravesande's book. The pheno- mena of combustion he explained by assuming that when a body is sufficiently heated the light-corpuscles interact with the material particles, some constituents being in consequence sepa- rated and carried away with the corpuscles as flame and smoke. This view harmonizes with the theory of calcination which had been developed by Becher and his pupil Stahl at the end of the seventeenth century, according to which the metals were sup- posed to be composed of their calces and an element phlogiston. The process of combustion, by which a metal is changed into its calx, was interpreted as a decomposition, in which the phlogiston separated from the metal and escaped into the atmosphere ; while the conversion of the calx into the metal was regarded as a union with phlogiston * s'Gravesande attributed electric effects to vibrations induced in effluvia, which he supposed to be permanently attached to- such bodies as amber. " Glass," he asserted, " contains in it, and has about its surface, a certain atmosphere, which is excited by Friction and put into a vibratory motion; for it attracts and •The correct idea of combustion bad been advanced by Hooke. "The disso- lution of inflammable bodies," he asserts in the Xierographia, " ia performed by a substance inherent in and mixed with the air, that is like, if not (he very aame with, that which is fixed in saltpetre.'' But this statement met with tittle favour at the lime, and the doctrine of the compound nature of metals survived in full vigour until the discovery of oxygen by Prientley and Scheele in 1771-5. In 1776 Lavoisier reaffirmed Hooka's principle that a metallic calx is not the metal minus phlogiston, but the metal plus oxygen; and this idea, which carried with it the recognition of thu elementary nature of metals, was generally accepted by the end of the eighteenth century. 3,Bl,ZEdhyG00gle prior to the Introduction of the Potentials. 37 repels light Bodies. The smallest parts of the glass are agitated by the Attrition, and by reason of their elasticity, their motion is vibratory, which is communicated to the Atmosphere above- mentioned : and therefore that Atmosphere exerts its action the further, the greater agitation the Parts of the Glass receive when a greater attrition is given to the glass." The English translator of s'Gravesande's work was himself destined to play a considerable part in the history of electrical science. Jean Theophile Desaguliers (b. 1683, d. 1744) was an Englishman only by adoption. His father had been a Huguenot pastor, who, escaping from France after the revocation of the Edict of Nantes, brought away the boy from La Rochelle, concealed, it is said, in a tub. The young Desaguliers was afterwards ordained, and became chaplain to that Duke of Chandos who was so ungratefully ridiculed by Pope. In this situation he formed friendships with some of the natural philosophers of the capital, and amongst others with Stephen Gray, an experimenter of whom little is known* beyond the fact that he was a pensioner of the Charterhouse. In 1729 Gray communicated, as he says,f " to Dr. Desaguliers and some other Gentlemen " a discovery he had lately made, " showing that the Electrick Vertue of a Glass Tube may be conveyed to any other Bodies so as to give them the same Property of attracting and repelling light Bodies as the Tube ' does, when excited by rubbing : and that this attractive Vertue might be carried to Bodies that were many Feet distant from the Tube." This was a result of the greatest importance, for previous workers had known of no other way of prodncing the attractive emanations than by rubbing the body eoncerned.J It was found * Those who are interested in the literary history of the eighteenth century will recall the controversy u to whether the veraee on tlia death of Stephen Gray were written by Anna Williams, whose name they bore, or by her patron Johnoon. f Phil. Trans, xzxvu {1731}, pp. 13, 227, 285, 397. 1 Otto Ton Ouericke (A. 1602, d. 1686) had, aa a matter of fact, observed the conduction of electricity along a linen thread ; but this experiment duet not seem to hare been followed up. Cf. Experimenta iiota magdeburgiai, 1672. DinlizBdnyGoOgk 38 Electric and Magnetic Science that only a limited class of substances, among which the metals were conspicuous, had the capacity of acting as channels for the transport of the electric power ; to these Desaguliera, whp~eon- tinued the experiments after Gray's death in 1736, gaveS* the name non-eltctrica or conductors. After Gray's discovery it was no longer possible to believe that the electric effluvia are inseparably connected with the bodies from which they are evoked by rubbing ; and it became necessary to admit that these emanations have an independent existence, and can be transferred from one body to another. Accordingly we find them recognized, under the name of the electric fluid,f as one of the substances of which the world is constituted. The imponderability of this fluid did not, for the- reasons already mentioned, prevent its admission by the side of light and caloric into the list of chemical elements. The question was actively debated as to whether the electric fluid was an element sui generis, or, as some suspected, was another manifestation of that principle whose operation is seen in the phenomena of heat. Those who held the latter view urged that the electric fluid and heat can both be induced by friction, can both induce combustion, and can both be transferred from one body to another by mere contact ; and, moreover, that the best conductors of heat are also in general the best con- ductors of electricity. On the other hand it was contended that the electrification of a body does not cause any appreciable rise- in its temperature; and an experiment of Stephen Gray's brought to light a yet more striking difference. Gray,J in 1729, made two oaken cubes, one solid and the other hollow, and showed that when electrified in the same way they produced exactly similar effects ; whence he concluded that it was only the surfaces which had taken part in the phenomena. Thus while heat is disseminated throughout the substance of a body, the electric fluid resides at or near its surface. In the middle of ■Phil. Trans, ili. (1739), pp. 186, 193, 200, 209: Lintrtatim ttnermag Electricity, 1742. t The Cartemna defined a fluid to bo ■ body « hose minute parta are in a continual agitation. J Pliil. Trans, ixxvii., p. 36. 3,Bl,ZEdhyG00gle prior to Ike Introduction of the Potentials. 30 the eighteenth century it was generally compared to an envelop- ing atmosphere. "The electricity which a non-electric of great length (for example, a hempen string 800 or 900 feet long) receives, runs from one end to the other in a sphere of electrical Effluvia," says Desaguliers in 1740 p^and a report of the French Academy in 1733 says rf " Around an electrified body there is formed a vortex of exceedingly fine matter in a state of agitation, which urges towards the body such light substances as he within its sphere of activity. The existence of this vortex is more than a mere conjecture ; for when an electrified body is brought close to the face it causes a sensation like that of encountering a cobweb. "J The report from which this is quoted was prepared in connexion with the discoveries of Charles-Francois du Fay (b. 1698, d. 1739), superintendent of gardens to the King of France. Du Fay§ accounted for the behaviour of gold leaf when brought near to an electrified glass tube by supposing that at first the vortex of the tube envelopes the gold-leaf, and so attracts it towards the tube. But when contact occurs, the gold-leaf acquires the electric virtue, and bo becomes surrounded by a vortex of its own. The two vortices, striving to extend in contrary senses, repel each other, and the vortex of the tube, being the stronger, drives away that of the gold-leaf. " It is then certain," says du Fay, i " that bodies which have become electric by contact are repelled by those which have rendered them electric ; but are they repelled likewise by other electrified bodies of all kinds ? And do electrified bodies differ from each other in no respect save their intensity of electrification ? An examination of this matter has led me to a discovery which I should never have foreseen, and of which I believe no one hitherto has had the least idea." • Pin. Trane. xli., p. 636. t Hist, de L'Acad., 1733, p. 6. t Thii ofawrvition hart been made tint by Hawkabee at the beginning of the I Hem. de 1'Acad. dee Sciences, 1733, pp. S3, 73, 233, 457 ; 1734, pp. Ml, «03; J 737, p. 86 ; Phil. Traiie. kitIU. (1734), p. 2J8. I Hen. de l-Acad., 1733, p. 464. 3,Bl,ZEdhyG00gle 40 Electric and Magnetic Science He found, in fact, that when gold-leaf which had been electrified by contact with excited glass was brought near to an excited piece of copal* an attraction was manifested between them. " I had expected," he writes, " quite the opposite effect, eince, according to my reasoning, the copal and gold-leaf, which were both electrified, should have repelled each other." Proceeding with his experiments he found that the gold-leaf, when electrified and repelled by glass, was attracted by all electrified resinous substances, and that when repelled by the latter it was attracted by the glass. " We see, then," he continues, " that there are two electricities of a totally different nature — namely, that of transparent solids, such as glass, crystal, &c, and that of bituminous or resinous bodies, such as amber, copal, sealing-wax, &c. Each of them repels bodies which have contracted an electricity of the same nature as its own, and attracts those whose electricity is of the contrary nature. We see even that bodies which are not themselves electrics can acquire either of these electricities, and that then their effects are similar to those of the bodies which have communicated it to them." To the two kinds of electricity whose existence was thus demonstrated, du Fay gave the names vitreous and resinous, by which they have ever since been known. An interest in electrical experiments seems to have spread from du Fay to other members of the Court circle of Louis XV ; and from 1745 onwards the Memoirs of the Academy contain a series of papers on the subject by the Abbe- Jean-Antoine Nollet (b. 1700, d. 1770), afterwards preceptor in natural philosophy to the Royal Family. Nollet attributed electric phenomena to the movement in opposite directions of two currents of a fluid, " very subtle and inflammable," which he supposed to be present in all bodies under all circumstances.t When an electric is excited by friction, part of this fluid escapes from its pores, forming an effluent stream; and this Iosb is repaired by an * A hard transparent resin, used in the preparation of Tarnish. t Cf. Nollet'a Kechtrehti, 1719, p. 245. 3,Bl,ZEdhyG00gle prior to the Introduction of the Potentials. 41 affluent stream of the same fluid entering the body from outside. Light bodies hi the vicinity, being caught in one or other of these streams, are attracted or repelled from the excited electric. Nollet's theory was in great vogue for some time ; but six or seven years after its first publication, its author came across a work purporting to be a French translation of a book printed originally in England, describing experiments said to have been made at Philadelphia, in America, by one Benjamin Franklin. "He could not at first believe," as Franklin tells us in his Autobiography, " that such a work came from America, and said it must have been fabricated by his enemies at Paris to decry his system. Afterwards, having been assured that there really existed such, a person as Franklin at Philadelphia, which he had doubted, he wrote and published a volume of letters, chiefly addressed to me, defending his theory, and denying the verity of my experiments, and of the positions deduced from them." We must now trace the events which led up to the discovery which so perturbed Nbllet. In 1745 Pieter van Musschenbroek (b. 1692, d. 1761), Professor at Leyden, attempted to find a method of preserving electric charges from the decay which was observed when the charged bodies were surrounded by air. With this purpose he tried the effect of surrounding a charged mass of water by an envelope of Borne non-conductor, e.g., glass. In one of his experiments, a phial of water was suspended from a gun- barrel by a wire let down a few inches into the water through the cork; and the gun-barrel, suspended on silk lines, was applied bo near an excited glass globe that some metallic fringes inserted into the gun-barrel touched the globe in motion. Under these circumstances a friend named Cunaeus, who happened to grasp the phial with one hand, and touch the gun- barrel with the other, received a violent shock ; and it became evident that a method of accumulating or intensifying the electric power had been discovered.* .ids year by Ewnld Georg 3,Bl,ZEdhyG00gle 42 Electric and Magnetic Science Shortly after the disoovery of the Leyden phial, as it Was named by Nollet, had become known in England, a London apothecary named William Watson {*. 1715, d. 1787)* noticed that when the experiment is performed in this fashion the observer feels the shock " in no other parts of his body but his arms and breast"; whence he inferred that in the act of discharge there is a transference of something which takes the shortest or best-conducting path between the gun-barrel and the phial. This idea of transference seemed to him to bear some similarity to Hollet'B doctrine of afflux and efflux; and there can indeed be little doubt that the Abbe's hypothesis, though totally false in itself, furnished some of the ideas from which WatBon, with the guidance of experiment, constructed a correct theory. In a memoUft^read to the Boyal Society in October, 1746, he propounded the doctrine that electrical actions are due to the presence of an " electrical aether," which in the charging or discharging of a Leyden jar is transferred, but is not created or destroyed. The excitation of an electric, according to this view, consists not in the evoking of anything from within the electric itself without compensation, but in the accumulation of a Burplus of electrical aether by the electric at the expense of some other body, whose stock is accordingly depleted. All bodies were supposed to possess a certain natural store, which could be drawn upon for this purpose. " I have shewn," wrote Watson, " that electricity is the effect of a very subtil and elastic fluid, occupying all bodies in contact with the terraqueous globe ; and that every-where, in its natural state, it is of the same degree of density ; and that glass and other bodies, which we denominate electrics per se+ have the power, by certain known operations, of taking this fluid from one body, and conveying it to another, in a quantity sufficient to be obvious to all our senses; and that, under * Watson afterward* rose to eminence in the medical profession, and was knighted. t Phil. Trans. xuV.. p. 718. It may here be noted that it was Watson who- improved the phial by coating it nearly to the top, both inside and outside, with 3,Bl,ZEdhyG00gle prior to the Introduction of the Potentials. 4$ certain circumstances, it was possible to render the electricity in some bodies more rare than it naturally is, and, by communi- cating this to other bodies, to give them an additional quantity, and make their electricity more dense." In the same year in which Watson's theory was proposed, a certain Dr. Spence, who had lately arrived in America from Scotland, was showing in Boston some electrical experiments. Among his audience was a man who already at forty years of age was recognized as one of the leading citizens of the English colonies in America, Benjamin Franklin of Philadelphia [b. 1706, d. 1790). Spence's experiments " were," writes Franklin,* " imperfectly performed, as he waB not very expert; but, being on a subject quite new to me, they equally surprised and pleased me." Soon after this, the "Library Company" of Philadelphia (an institution founded by Franklin himself) received from Mr. Peter Collinson of London a present of a glass tube, with some account of its use. In a letter written to Collinson on July 11th, 1747,t Fianklin described experiments made with this tube, and certain deductions which he had drawn from them. If one person A, standing on wax so that electricity cannot pass from him to the ground, rubs the tube, and if another person B, likewise standing on wax, passes his knuckle along near the glass so as to receive its electricity, then both A ami B will be capable of giving a spark to a third person C standing on the floor; that is, they will be electrified. If, however, A and B touch each other, either during or after the rubbing, they will not be electrified. This observation suggested to Franklin the same hypothesis that (unknown to him) had been propounded a few months previously by Watson : namely, that electricity is an element present in a certain proportion in all matter in its normal condition ; so that, before the rubbing, each of the persons A, B, and C has an equal share. The effect of the rubbing is to * Franklin' a Auto&iogrtip&y. t Franklin** ft» Szprrimtntt and Obtrvationi on Electricity, letter ii. D,Bl,ZEdhyG00gle 44 Electric and Magnetic Science transfer some of A'a electricity to the glass, whence it is transferred to B. Thus A has a deficiency and B a superfluity of electricity ; and if either of them approaches C, who has the normal amount, the distribution will be equalized by a spark. If, however, A and B are in contact, electricity flows between them so as to re-establish the original equality, and neither is then electrified with reference to C. Thus electricity is not created by rubbing the glass, but only transferred to the glass from the rubber, so that the rubber loses exactly as much as the glass gains ; the total quantity of electricity in any insulated system is invariable. This assertion is usually known as the principle of conservation of electric charge. The condition of A and B in the experiment can evidently be expressed by plus and minus signs : A having a deficiency - e and B a superfluity + e of electricity. Franklin, at the commencement of his own experiments, was not acquainted with du Fay's discoveries ; but it is evident that the electric fluid of Franklin is identical with the vitreous electricity of du Fay, and that du Fay's resinous electricity is, in Franklin's theory, merely the deficiency of a stock of vitreous electricity supposed to be possessed naturally by all ponderable bodies. In Franklin's theory we are spared the necessity for admitting that two quasi-material bodies can by their union annihilate each other, as vitreous and resinous electricity were supposed to do, Some curiosity will naturally be felt as to the considerations which induced Franklin to attribute the positive character to vitreous rather than to resinous electricity. They seem to have been founded on a comparison of the brush discharges from conductors charged with the two electricities; when the electricity was resinous, the discharge was observed to spread over the surface of the opposite conductor " as if it flowed from it." Again, if a Leyden jar whose inner coating is electrified vitreously is discharged silently by a conductor, of whose pointed ends one is near the knob and the other near the outer coating, the point which is near the knob is seen in the dark to be illumi- D,Bl,ZBdhyG00gle prior to the Introduction of the Potentials. 45 nated with a star or globule, while the point which is near the outer coating is illuminated with a pencil of rays; which suggested to Franklin that the electric fluid, going from the inside to the outside of the jar, enters at the former point and issues from the latter. And yet again, in some cases the flame of a wax taper is blown away from a brass ball which is discharging vitreous electricity, and towards one which is discharging resinous electricity. But Franklin remarks that the interpretation of these observations is somewhat conjectural, and that whether vitreous or resinous electricity is the actual electric fluid is not certainly known. Regarding the physical nature of electricity, Franklin held much the same ideas as his contemporaries ; he pictured it as an elastic* fluid, consisting of " particles extremely subtile, since it can permeate common matter, even the densest metals, with such ease and freedom as not to receive any perceptible resistance." He departed, however, to some extent from the conceptions of his predecessors, who were accustomed to ascribe all electrical repulsions to the diffusion of effluvia from the excited electric to the body acted on ; so that the tickling sensation which is experienced when a charged body is brought near to the human face was attributed to a direct action of the effluvia on the skin. This doctrine, which, as we shall see, practically ended with Franklin, hears a suggestive resemblance to that which nearly a century later was introduced by Faraday ; both explained electrical phenomena without intro- ducing action at a distance, by supposing that something which forms an essential part of the electrified system is present at the spot where any electric action takes place ; but in the older theory this something was identified with the electric fluid itself, while in the modern view it is identified with a state of stress in the aether. In the interval between the fall of one school and the rise of the other, the theory of action at a distance was dominant. The germs of the last-mentioned theory may be found in * Le., repulsive of iia own particles. D,Bl,ZEdhyG00gle 46 Electric and Magnetic Science Franklin's own writings. It originated in connexion with the explanation of the Leyden jar, a matter which is discussed in Ids third letter to Collinson, of date September let, 1747. In charging the jar, he says, a quantity of electricity is taken away from one Bide of the glass, by means of the coating in contact with it, and an equal quantity is communi- cated to the other side, by means of the other coating. The glass itself he supposes to be impermeable to the electric fluid, so that the deficiency on the one side can permanently coexist with the redundancy on the other, so long as the two sides are not connected with each other; but when a con- nexion is set up, the distribution of fluid is equalized through the body of the experimenter, who receives a shock. Compelled by this theory of the jar to regard glass as impenetrable to electric effluvia, Franklin was nevertheless well aware* that the interposition of a glass plate between an electrified body and the objects of its attraction does not shield the latter from the attractive influence. He was thus driven to suppose! that the surface of the glass which is nearest the excited body is directly affected, and is able to exert an influence through the glass on the opposite surface ; the latter surface, which thus receives a kind of secondary or derived excitement, is responsible for the electric effects beyond it This idea harmonized admirably with the phenomena of the jar ; for it was now possible to hold that the excess of electricity on the inner face exercises a repellent action through the substance of the glass, and so causes a deficiency on the outer faces by driving away the electricity from it.J Franklin had thus arrived at what was really a theory of action at a distance between the particles of the electric fluid ; and this he was able to support by other experiments. "Thus," he writes,! " the stream of a fountain, naturally dense and con- tinual, when electrified, will separate and spread in the form of a brush, every drop endeavouring to recede from every other ilbid., 1760, J 34. j Letter v. 3,Bl,ZEdhyG00gle prior to the Introduction of the Potentials. 47 drop.' In order to account for the attraction between oppositely charged bodies, in one of which there is an excess of electricity as compared with ordinary matter, and in the other an excess of ordinary matter as compared with electricity, he assumed that " though the particles of electrical matter do repel each other, they are strongly attracted by all other matter " ; so that " common matter is as a kind of spunge to the electrical fluid." These repellent and attractive powers he assigned only to the actual (vitreous) electric fluid; and when later on the mutual repulsion of resinously electrified bodies became known to him * it caused him considerable perplexity, f As we shall see, the difliculty was eventually removed by Aepinus. In spite of his belief in the power of electricity to act at a distance, Franklin did not abandon the doctrine of effluvia. " The form of the electrical atmosphere," he saysj "is that of the body it surrounds. This shape may be rendered visible in a still air, by raising a smoke from dry rosin dropt into a hot tea- spoon under the electrified body, which will be attracted, and spread itself equally on all sides, covering and concealing the body, And this form it takes, because it is attracted by all parts of the surface of the body, though it cannot enter the substance already replete. Without this attraction, it would not remain round the body, but dissipate in the air." He olaerved, however, that electrical effluvia do not seem to affect, or be affected by, the air ; since it is possible to breathe freely in the neighbourhood of electrified bodies ; and moreover a current of dry air does not destroy electric attractions and repulsions.! Begarding the suspected identity of electricity with the matter of heat, as to which Nollet liad taken the affirmative position, Franklin expressed no opinion. " Common fire," he • Ha refers to it in hit Paper read to the Royal Society, December 13, 1 75s. f-Cf. letters imii and xixnii, dated 1761 and 1762. t Jftn E*p»riiMHl~, 17(0, Jl5- i Letter Tii, 1751. 3,Bl,ZEdhyG00gle 48 Electric and Magnetic Science writes,* "is in all bodies, more or less, as well as electrical fire. Perhaps they may be different modifications of the same element; or they may be different elements. The latter is by some suspected. If they are different things, yet they may and do subsist together in the same body." Franklin's work did not at first receive from European philosophers the attention which it deserved ; although Watson generously endeavoured to make the colonial writer's merits known, t and inserted some of Franklin's letters in one of his own papers communicated to the Royal Society. But an account of Franklin's discoveries, which had been printed in England, happened to fall into the hands of the naturalist Buffon, who was so much impressed that he secured the issue of a French transla- tion of the work ; and it was this publication which, as we have seen, gave such offence to Nollet. The success of a plan proposed by Franklin for drawing lightning from the clouds soon engaged public attention everywhere; and in a short time the triumph of the one-fiuid theory of electricity, as the hypothesis of Watson and Franklin is generally called, was complete. Nollet, who was obdurate, "lived to see himself the last of his sect, except Monsieur B — of Paris, his eleve and immediate disciple." J The theory of effluvia was finally overthrown, and replaced by that of action at a distance, by the labours of one of Franklin's continental followers, Francis Ulrich Theodore Aepinus§ (p. 1724, d. 1802). The doctrine that glass is impermeable to electricity, which had formed the basis of Franklin's theory of theLeyden phial, was generalized by Aepinus|| and his co-worker Johann Karl Wilcke (6. 1732, d. 1796) into the law that all non-conductors are impermeable to the * Letter v. t Phil. Trans, ilvii, p. 202. Watson agreed with Nollet in rejecting Franklin's theory of the impermeability of glass. J Franklin'a Autobiography. } This philosopher's surname had been heUenized from its original form Stuck to aitaroi by one of his ancestors, a distinguished theologian. | F. Y. T. Aepinus TlHlamtn Theoriat Eltclricitatit tt Mirgnttiimi : fit. Petersburg, 1759. 3,Bl,ZEdhyG00gle prior to the Introduction of the Potentials. 49 electric fluid. That this applies even to air they proved by constructing a machine analogous to the Leyden jar, in which, however, air took the place of glass as the medium between two oppositely charged surfaces. The Buccess of this experi- ment led Aepinus to deny altogether the existence of electric effluvia surrounding charged bodies:* a position which he regarded as strengthened by Franklin's observation, that the electric field in the neighbourhood of an excited body is not destroyed when the adjacent air is blown away. The electric fluid must therefore be supposed not to extend beyond the excited bodies themselves. The experiment of Gray, to which we have already referred, showed that it does not penetrate far into their substance; and thus it became necessary to suppose that the electric fluid, in its state of rest, is con- fined to thin layers on the surfaces of the excited bodies. This being granted, the attractions and repulsions observed between the bodies compel us to believe that electricity acts at a distance across the intervening air. Since two vitreously charged bodies repel each other, the force between two particles of the electric fluid must (on Franklin's one-fluid theory, which Aepinus adopted) be repulsive : and since there is "an attraction between oppositely charged bodies, the force between electricity and ordinary matter must be attractive. These assumptions had been made, as we have seen, by Franklin; but in order to account for the repulsion between two resinously charged bodies, Aepinus introduced a new supposition — namely, that 1 he particles of ordinary matter repel each other. This, at first, startled his contemporaries ; but, as he pointed out, the " unelectrified" matter with which we are acquainted is really matter saturated with its natural quantity of the electric fluid, and the forces due to the matter and fluid balance each other ; or perhaps, . as he suggested, a slight want of equality between these forces might give, as a residual, the force of gravitation. Assuming that the attractive and repellent forces increase as ' * Tbia was alio m«int*ined about the tame time by Qiacomo Battlata Beccaiia of Turin)*. 1716, a". 1781). E 3,Bl,ZEdhyG00gle 50 Electric and Magnetic Science the distance between the acting charges decreases, Aepinus applied hU theory to explain a phenomenon which had been more or less indefinitely observed by many previous writers, and specially studied a short time previously by John Canton* (ft. 1718, d. 1772) and by Wilckef— namely, that if a conductor is brought into the neighbourhood of an excited body without actually touching it, the remoter portion of the conductor acquires an electric charge of the same kind as that of the excited body, while the nearer portion acquires a charge of the opposite kind. This effect, which is known as the induction of electric charges, had been explained by Canton himself and by FranklinJ in terms of the theory of electric effluvia. Aepinus showed that it followed naturally from the theory of action at a distance, by taking into account the mobility of the electric fluid in conductors ; and by discussing different cases, so far as was possible with the means at his command, he laid the foundations of the mathematical theory of electrostatics. Aepinus did not succeed in determining the law according to which the force "between two electric charges varies with the distance between them ; and the honour of having first accom- plished this belongs to Joseph Priestley (b. 1733, d. 1804), the discoverer of oxygen. Priestley, who was a friend of Franklin 'b, had been informed by the latter that he had found cork balls to be wholly unaffected by the electricity of a metal cup within which they were held ; and Franklin desired Priestley to repeat and ascertain the fact. Accordingly, on December 21st, 1766, Priestley instituted experiments, which showed that, when a hollow metallic vessel is electrified, there is no charge on the inner surface (except near the opening), and no electric force in the air inside. From this he at once drew the correct conclusion, which was published in 1767.§ " May we not infer," he says, "from • Phil. Tinni. xlviii (Uo3), p. 350. t Ditputatio phj/iica experimintalU it tltctricitatibui contrariii : Eoilock, 1757. ; In liia paper read to ihe Royal Society on Deo. 18t.li, 1755. ij J. Priestley, The Hillary and Freunt Stals of BketrUity, with QrigmtU Kiperimemt; London, 1767: page 732. That electrical attraction follows itm law of the in verso square liiul been suspected liy Daniel Bernoulli in 1760: Cf. Sofia's Experiment*, AiU Etketica, IT, p. 214. 3,Bt,ZEdhyG00gle prior to the Introduction of the Potentials. 51 this experiment that the attraction of electricity ia subject to ' the same laws with that of gravitation, and is therefore according to the squares of the distances ; since it is easily demonstrated that were the earth in the form of a shell, a body in the inside of it would not be attracted to one side more than another ? " This brilliant inference seems to have been insufficiently studied by the scientific men of the day ; and, indeed, its author appears to have hesitated to claim for it the authority of a com- plete and rigorous proof. Accordingly we find that the question of the law of force was not regarded as finally settled for eighteen years afterwards.* By Franklin's law of the conservation of electric charge, and Priestley's law of attraction between charged bodies, electricity was raised to the position of an exact science. It is impossible to mention the names of these two friends in such a connexion without reflecting on the curious parallelism of their lives. In both men there was the same combination of intellectual bold- ness and power with moral earnestness and public spirit. Both of them carried on a long and tenacious struggle with the reac- tionary influences which dominated the English Government in the reign of George III ; and both at last, when overpowered in the confliet, reluctantly exchanged their native flag for that of the United States of America. The names of both have been held in honour by later generations, not more for their scientific discoveries than for their services to the cause of religious, intellectual, and political freedom. The most celebrated electrician of Priestley's contemporaries in London was the Hon. Henry Cavendish (b. 1731, d. 1810), whose interest in the subject was indeed hereditary, for his father, Lord Charles Cavendish, had assisted in Watson's experi- ments of 1747-t In 1771 CavendishJ presented to the Royal Society an " Attempt to explain some of the principal phenomena at Electricity, by means of an elastic fluid." The hypothesis , * In 1789 Dr. John Robitou (A. 1739, A. 1805}, of Edinburgh, endeavoured to determine toe law of force by direct experiment, and found it to be that of the inverse 2'06Ul power of the distance. t J'hil. Trans, xlv, p. 67 (1750). \ Phil. Tram, lii, p. 584 (1771). E 2 3,Bl,ZEdhyG00gle 52 Electric and Magnetic Science adopted is that of the one-fluid theory, in much the same form as that of Aepinus. It was, as he tells us, discovered indepen- dently, although he became acquainted with Aepinus' work before the publication of his own paper. In this memoir Cavendish makes no assumption regarding the law of force between electric charges, except that it is " inversely as some leas power of the distance than the cube " ; but he evidently inclines to believe in the law of the inverse square. Indeed, he shows it to be " likely, that if the electric attraction or repulsion is inversely as the square of the distance, almost all the redundant fluid in the body will be lodged close to the surface, and there pressed close together, and the rest of the body will be saturated "; which approximates closely to the discovery made four years previously by Priestley. Cavendish did, as a matter of fact, rediscover the inverse square law shortly afterwards; but, indifferent to fame, he neglected to communicate to others this and much other work of importance. The value of his researches was not realized until the middle of the nineteenth century, when William Thomson (Lord Kelvin) found in Caven- dish's manuscripts the correct value for the ratio of the electric charges carried by a circular disk and a sphere of the same radius which had been placed in metallic connexion. Thomson urged that the papers should be published ; which came to pass* in 1879, a hundred years from the date of the great discoveries which they enshrined. It was then seen that Cavendish had anticipated his successors in several of the ideas which will presently be discussed — amongst others, those of electrostatic capacity and specific inductive capacity. In the published memoir of 1771 Cavendish worked out the consequences of his fundamental hypothesis more completely than Aepinus ; and, in fact, virtually introduced the notion of electric potential, though, in the absence of any definite assump- tion as to the law of force, it was impossible to develop this idea . to any great extent. 3,Bl,ZEdhyG00gle prior to the Introduction of the Potentials. 53 One of the investigations with which Cavendish occupied himself was a comparison between the conducting powers of different materials for electrostatic dischargee. The question had been first raised by Beccaria, who had shown* in 1753 that when the circuit through which a discharge is passed contains tubes of water, the shock is more powerful when the cross-section of the tubes is increased. Cavendish went into the matter much more thoroughly, and was able, in a memoir presented to the Royal Society in 1775,t to say: "It appears from some experiments, of which I propose shortly to lay an account before this Society, that iron wire conducts about 400 million times better than rain or distilled water — that is, the electricity meets with no more resistance in passing through a piece of iron wire 400,000,000 inches long than through a column of water of the same diameter only one inch long. Sea*water, or a solution of one part of salt in 30 of water, conducts 100 times, or a saturated solution of sea-salt about 720 times, better than rain-water." The promised account of the experiments was published in the volume edited in 1879. It appears from it that the method of testing by which Cavendish obtained these, results was simply that of physiological sensation; but the figures given in the comparison of iron and sea-water are remarkably exact While the theory of electricity was being established on a sure foundation by the great investigators of the eighteenth century, a no less remarkable development was taking place in the kindred science of magnetism, to which our attention must now be directed. The law of attraction between magnets was investigated at an earlier date than the corresponding law for electrically charged lwdies. Newton, in the Prindpia,* says : " The power of gravity is of a different nature from the power of magnetism. For the magnetic attraction is not as the matter attracted. Some bodies are attracted more by the magnet, otherB less ; most bodies not at all. The power of magnetism, in one and the same • G. B. Iltccaria, Dell" elittriciia- ♦■Phil. Tram. livi(1776), p. 19f 3,Bl,ZEdhyG00gle 54 Electric and Magnetic Science body, may be increased and diminished ; and is sometimes far Btronger, for the quantity of matter, than the power of gravity ; and in receding from the magnet, decreases not in the duplicate, but almost in the triplicate proportion of the distance, as nearly as I could judge from Borne rude observations." The edition of the JPrincipia which was published in 1742 by Thomas Le Seur and Francis Jacquier contains a note on this corollary, in which the correct result is obtained that the directive couple exercised on one magnet by another is proportional to the inverse cute of the distance. The first discoverer of the law -of force between magnetic "I poles was John Michell (6. 1724, d. 1793), at that time a young Fellow of Queen's College, Cambridge,* who in 1750 published A Treatise of Artificial Magnets ; in which is shown an easy and expeditions method of making them superior to the best natural ones. In this he states the principles of magnetic theory as follows! : — " Wherever any Magnetism, is found, whether in the Magnet itself, or any piece of Iron, etc, excited by the Magnet, there are always found two Poles, which are generally called North and South ; and the North Pole of one Magnet always attracts the South Pole, and repels the North Pole of another : and vice versa." This is of course adopted from Gilbert. "Each Pole attracts or repels exactly equally, at equal distances, in every direction." This, it may be oltserved, over- throws the theory of vorticeB, with which it is irreconcilable. " The Magnetical Attraction and Repulsion are exactly equal to each other." This, obvious though it may seem to us, was really a most important advance, for, as he remarks, " Most people, who •Michell hud Likrn bis degree only t«o years previously. Later in life be wa! On terms of friendship with Priestley, Cavendish, and William Herachel ; it was he who taught Herachel the art of grinding mirrors for telescopes. The plan of determining the density of the earth, which una tarried out by Cavendish in 1798, and is generally known as the " Cavendish Experiment," was due to Michell. Michell was the 0 rat inventor of the tiirsion-beiance; ha also made many valuable eon tribn lions to Astronomy. In 1767 he became Rector of Thornbill, Yorka, and lived there until his deatb. tLoc. cit., p. 17. 3,Bl,ZEdhyG00gle prior to the Introduction of the Potentials. 55 have mention'tl any thing relating to this property of the Magnet, have agreed, not only that the Attraction and Repulsion of Magnets are not equal to each other, but that also, they do not observe the same rule of increase and decrease." " The Attraction and Repulsion of Magneto decreases, as the Squares of the distances from the respective poles increase." This great discovery, which is the basis of the mathematical theory of Magnetism, was deduced partly from his own observa- tions, and partly from those of previous investigators {e.g. Dr. Brook Taylor and F. Muschenbroek), who, as he observes, had made accurate experiments, but had failed to take into account all the considerations necessary for a sound theoretical discussion of them. After Michel! the law of the inverse square was maintained by Tobias Mayer' of Gottingen (b. 1723, d. 1762), better known as the author of Lunar Tables which were long in use ; and by the celebrated mathematician, Johann Heinrich Lambertf (6. 1728, d. 1777). The promulgation of the one-fluid theory of electricity, in "* the middle of the eighteenth century, naturally led to attempts to construct a similar theory of magnetism ; this was effected in 1759 by AepinusJ, who supposed the " poles " to be places at which a magnetic fiuid was present in amount exceeding or falling short of the normal quantity. The permanence of magnets was accounted for by supposing the fluid to be entangled in their pores, so as to be with difficulty displaced. The particles of the fluid were assumed to repel each other, and to attract the particles of iron and steel ; but, as Aepinus saw, in order to satis- factorily explain magnetic phenomena it was necessary to assume also a mutual repulsion among the material particles of the magnet. Subsequently two imponderable magnetic fluids, to which •Enticed in Gottiagtr Gtlthrter Anztiger, 1760: cf. Aspinui, Nov. Comm. Acad. Petrop., 1788, and Mityer's Opera Itudila, herauag. * afterwards published, and ascribed to Laplace, in a memoir by Legend™ on the Attractions of Spheroid!, which will be found in the Mem. par divfri eTttW, published in 1785. 3,Bl,ZEdhyG00gle prior to the Introduction of the Potentials. 61 at any point can be simply expressed as the derivates of the function which is obtained by adding together the masses of all the particles of an attracting system, each divided by its distance from the point; and Laplace had shown* that this function V satisfies the equation <£Z ^1 ?v ae» + ay* + a* in space free from attracting matter. Poisson himself showed later, in 1813, t that when the point (x, y, z) is within the substance of the attracting body, this equation of Laplace must be replaced by VV d*V &V — + —, ,- + -z-.- - - 4jTp, a*.'1 d>/ as5 where p denotes the density of the attracting matter at the point. In the present memoir Poisson called attention to the utility of this function V in electrical investigations, remarking that its value over the surface of any conductor must be constant. The known formulae for the attractions of spheroids show that when a charged conductor is spheroidal, the repellent force acting on a small charged body immediately outside it will be directed at right angles to the surface of the spheroid, and will be proportional to the thickness of the surface-layer of electricity at this place. Poisson suspected that this theorem might be true for conductors not having the spheroidal form — a result which, as we have seen, had been already virtually given by Coulomb ; and Laplace suggested to Poisson the following proof, applicable to the general case. The force at a point immediately outside the conductor can be divided into a part s due to the part of the charged surface immediately adjacent to the point, and a part 8 due to the rest of the surface. At a point close to this, but just inside the con- ductor, the force &fcw.ill still act; but the force a will evidently • Him. At, l'Aiad., 1782 (published in 17B5), p. 113. f Bull, de In Sou. Philomathique. iii. (1813;, p. 388. 3,Bl,ZEdhyG00gle 62 Electric and Magnetic Science be reversed in direction. Since the resultant force at the latter point vanishes, we must have S=s; so the resultant force at the exterior point is 2s. But s is proportional to the charge per unit area of the surface, as is seen by considering the ease of an infinite plate ; which establishes the theorem. When several conductors are in presence of each other, the distribution of electricity on their surfaces may be determined by the principle, which Poisson took as the basis of his work, that at any point in the interior of any one of the conductors, the resultant force due to all the surface-layers must be zero. He discussed, in particular, one of the classical problems of electrostatics— namely, that of determining the surface-density on two charged conducting spheres placed at any distance from each other. The solution depends on Double Gamma Functions in the general case; when the two spheres are in contact, it depends on ordinary Gamma Functions. Poisson gave a solution in terms of definite integrals, which is equivalent to that in termB of Gamma Functions ; and after reducing his results to numbers, compared them with Coulomb's experiments. - The rapidity with which in a single memoir Poisson passed from the barest elements of the subject to such recondite problems as those just mentioned may well excite admiration. His success is, no doubt, partly explained by the high state of development to which analysis had been advanced by the great mathematicians of the eighteenth century ; but even after allowance has been made for what is due to his predecessors, Poiason's investigation must be accounted a splendid memorial i_ of his genius. Some years later Poisson turned his attention to magnetism ; and, in a masterly paper* presented to the French Academy in 1824, gave a remarkably complete theory of the subject. His starting-point is Coulomb's doctrine of two imponderable magnetic fluids, arising from the decomposition of a neutral fluid, and confined in their movements to the individual elements * Mi-m. de 1' Ac-id., v, p. 247. 3,Bl,ZEdhyG00gle prior to the Introduction of the Potentials. 63 of the magnetic body, bo as to be incapable of passing from one element to the next Suppose that an amount m of the positive magnetic fluid is located at a point (x y, z) ; the. components of the viagnctic intensity, or force exerted on unit magnetic pole, at a point (S, «. ?) will evidently be -4© -m -40) where r denotes ((£ - xf + (n - yf + (£ - *)*]*. Hence if we consider next a magnetic element in which equal quantities of the two magnetic fluids are displaced from each other parallel to^ the avaxis, the components of the magnetic intensity at (£, if. £) will be the negative derivates, with respect to f, q, £ respectively, of the function dx\rj where the quantity A, which does not involve (f, u, £), may be called the magnetic moment of the element : it may be measured by the couple required to maintain the element in equilibrium at a definite angular distance from the magnetic meridian. If the displacement of the two fluids from each other in the element is not parallel to the axis of x, it is easily seen that the expression corresponding to the last is where the vector (A, B, C) now denotes the magnetic moment of the element. Thus the magnetic intensity at an «xternal point (£, n, r) due to any magnetic body has the components f_3F _dV dV\ { Bf" ~V ~k) where integrated throughout the Bubstance of the magnetic body, and 3,Bl,ZEdhyG00gle 64 Electric and Magnetic Science where the vector (A, B,0)otI represents the magnetic moment per unit-volume, or, as it is generally called, the magnetization. The function V was afterwards named hy Green the magnetic potential. Poisson, by integrating by parts the preceding expression for the magnetic potential, obtained it in the form '-¥ dS). - - - div I dx dy dz," the first integral being taken over the surface S of the magnetic body, and the second integral being taken throughout its volume. This formula shows that the magnetic intensity produced by the body in external space is the same as would be produced by a fictitious distribution of magnetic fluid, consisting of a layer over its surface, of Burface-charge (I , SB) per element dJS, together with a volume-distribution of density - div I through- out its substance. These fictitious magnetizations are generally known as Prison's equivalent surface- and volume-distributions of magnetism. Poisson, moreover, perceived that at a point in a very small cavity excavated within the magnetic body, the magnetic potential has a limiting value which is independent of the shape of the cavity as the dimensions of the cavity tend to zero ; but that this is not true of the magnetic intensity, which in such a small cavity depends on the shape of the cavity. Taking the cavity to be spherical, he showed that the magnetic intensity within it is grad T4 frl,t where I denotes the magnetization at the place. • If the components of a vector a are denoted by (*«, a,, a,), the quantity arb, + a,bt +■ aih is called the scalar product of two vectora a and b, and u denoted by (» . b). The quantity iT + a~ + 57 'a called the divtrgtnc* of the vector a, and ii denoted by ilir a. t The vector what* component* are - — , - -^-, — — is denoted by grad V. 3,Bl,ZEdhyG00gle prior to the Introduction of the Potentials, 65 This memoir also contains a discussion of the magnetism temporarily induced in soft iron and other magnetizable metals by the approach of a permanent magnet. Poiason accounted for the properties of temporary magnets by assuming that they contain embedded in their substance a great number of small spheres, which are perfect conductors for the magnetic fluids ; so that the resultant magnetic intensity in the interior of one of these small spheres must be zero. He showed that such a sphere, when placed in a Held of magnetic intensity F," must acquire a magnetic moment of amount -~ F * the volume of the sphere, in order to counteract within the sphere the force F. Thus if k, denote the total volume of these spheres contained within a unit volume of the temporary magnet, the magnetization will be I, where %wl = k, F, and F denotes the magnetic intensity within a spherical cavity excavated in the body. This iB Poisson's law of induced magnetism. It is known that some Bubstances acquire a greater degree of temporary magnetization than others when placed in the same circumstances : Poisson accounted for this by supposing that the quantity kp varies from one substance to another. But the experimental data show that for soft iron kp must have a value very near unity, which would obviously be impossible if A, is to mean the ratio of the volume of spheres contained within a region to the total volume of the region. f The physical inter- pretation assigned by Poisson to his formulae must therefore he rejected, although the formulae themselves retain their value. Poisson's electrical and magnetical investigations were generalized and extended in 1828 by George GreenJ (b. 1793, d. 1841). Green's treatment iB based on the properties of the function already used by Lagrange, Laplace, and Poisson, which * In the present work, vectors will generally lie distinguished by heavy type. t Thie objection, was advanced by Harwell in {430 of hie Treat™. An attempt to overcome it was made by Betti : of. p. 377 of hi* Ltaoni an thi Pottntia.'. J An may en the application efmathmatiral analyrii te thi thtorui «/ gleclrieity mi magnttUtn, Nottingham, 1828; reprinted in The Mtdhimaticai Paptrt u/ Urn lati Qterji Gretn, p. I. F 3,Bl,ZEdhyG00gle 66 Electric and Magnetic Science. represents the sum of all the electric or magnetic charges in the field, divided by their respective distances from some given point : to this function Green gave the name potential, by which it has always since been known.* Near the beginning of the memoir is established the celebrated formula connecting surface and volume integrals, which is now generally called Green's TJieorem., and of which Poisaon's result on the equivalent surface- and volume-distribn- tions of magnetization is a particular application. By using this theorem to investigate the properties of the potential, Green arrived at many reaulta of remarkable beauty and interest. We need only mention, as an example of the power of his method, the following : — Suppose that there is a hollow conducting shell, bounded by two closed surfaces, and that a number of electrified bodies are placed, some within and some without it ; and let the inner surface and interior bodies be called the interior system, and the outer surface and exterior boliies be called the exterior system. Then all the electrical phenomena of the interior Bystem, relative to attractions, repulsions, and densities, will be the same as if there were no exterior Bystem, and the inner surface were a perfect conductor, put in communication with the earth ; and all those of the exterior system will be the same as if the interior system did not exist, and the outer surface were a perfect conductor, containing a quantity of electricity equal to the whole of that originally contained in the shell itself and in all the interior bodies. It will be evident that electrostatics bad by this time attained a state of development in which further progress could be hoped for only in the mathematical superstructure, unless experiment should unexpectedly bring to light phenomena of an entirely new character. This will therefore be a convenient place to pause and consider the rise of another branch of electrical philosophy. * Euler in 1744 {Da mtlhodu wttmiimrfi . . .) had spoken of thevti petentutlii— what would now be called the potential energy — poue*sed by an elastic body when bent. dhyGoogle ( 67 ) CHAPTER III. GALVANISM, FROM GALVANI TO OHM. Until the last decade of the eighteenth century, electricians were occupied solely with statical electricity. Their attention was then turned in a different direction. * In a work entitled Retkerches sur Conginc des seidiinents tup-tables et disagriables, which was published* in 1752, Johann Georg Sulzer (b. 1720, d. 1779) had mentioned that, if two pieces of metal, the one of lead and the other of silver, be joined together in such a manner that their edges touch, and if they be placed on the tongue, a taste is perceived " similar to that of vitriol of iron," although neither of these metals applied separately gives any trace of such a taste. " It is not probable," he says, " that this contact of the two metals causes a solution of either of them, liberating particles which might affect the tongue ; and we must therefore conclude that the contact Bets up a vibration in their particles, which, by affecting the nerves of the tongue, produces the taste in question." This observation was not suspected to have any connexion with electrical phenomena, and it played no part in the Incep- tion of the next discovery, which indeed was suggested by a . mere accident Luigi Galvaui, born at Bologna in 1737, occupied from 1775 onwards a chair of Anatomy in his native city. For many years before the event which made him famous he had been studying the susceptibility of the nerves to irritation ; and, having been * formerly a pupil of Beccaria, he was also interested in electrical experiments. One day in the latter part of the year 1780 he ' had, as he tells us,f " dissected and prepared a frog, and laid it on a table, on which, at some distance from the frog, was an electric machine. It happened by chance that one of my * Mem. de 1" Acad, de Berlin, 1752, p. 3S6. t Aloyui Galvaui, l)t Pinim Eltetricilatii m Mot* Mmctitari : CommenCarii ii (1791). p- 363. dhyGoogle 68 Galvanism, from Galvani to Ohm. assistants touched the inner crural nerve of the frog with the point of a scalpel ; whereupon at once the muscles of the limbs were violently convulsed. "Another of those who used to help me in electrical experi- ments thought he had noticed that at this instant a spark was drawn from the conductor of the machine. I myself was at the time occupied with a totally different matter; but when he drew my attention to this, I greatly desired to try it for myself, and discover its hidden principle. So I, too, touched one or other of the crural nerves with the point of the scalpel, at the same time that one of those present drew a spark ; and the same phenomenon waB repeated exactly as before."* After this, Galvani conceived the idea of trying whether the electricity of thunderstorms would induce muscular contractions equally well with the electricity of the machine. Having successfully experimented with lightning, he " wished," as he writes,t " to try the effect of atmospheric electricity in calm weather. My reason for this was an observation I had made, that frogs which had been suitably prepared for these experi- ments and fastened, by brass hooks in the spinal marrow, to the iron lattice round a certain hanging-garden at my house, exhibited convulsions not only during thunderstorms, but sometimes even when the sky was quite serene. I suspected these effects to be due to the changes which take place during the day in the electric state of the atmosphere ; and so, with some degree of confidence, I performed experiments to test the point ; and at different hours for many days I watched frogs which I had disposed for the purpose ; but could not detect any motion in their muscles. At length, weary of waiting in vain, I pressed the brass hooks, which were driven into the spinal marrow, against the iron lattice, in order to see whether contractions could be excited by varying the incidental circum- * According to a story which has often been repeated, but -which rerta on no sufficient evidence, the frog was one of a number which had been procured for the Signora Galvani, who, being in poor health, had been recommended to take a soup. made of these animals as a restorative. f Loo. oit., p. 377. 3,Bl,ZEdhyG00gle Galvanism, from Galvani to Ohm. 69 stances of the experiment. I observed contractions tolerably often, but they did not seem to bear any relation to the changes in the electrical state of the atmosphere. " However, at this time, when as yet I had not tried the experiment except in the open air, I came very near to adopt- ing a theory that the contractions are due to atmospheric electricity, which, having slowly entered the animal and accu- mulated in it, is suddenly discharged when the hook comes in contact with the iron lattice. For it is easy in experimenting to deceive ourselves, and to imagine we Bee the things we wish to see. " But I took the animal into a closed room, and placed it on an iron- plate ; and when I pressed the hook which was fixed in the spinal marrow against the plate, behold I the same spasmodic contractions as before. I tried other metals at different hours on various days, in several places, and always with the same result, except that the contractions were more violent with some metals than with others. After this I tried various bodies which are not conductors of electricity, such as glass, gums, resins, stones, and dry wood ; but nothing happened. This was somewhat surprising, and led me to suspect that electricity is inherent in the animal itself. This suspicion was strengthened by the observation that a kind of circuit of subtle nervous fluid (resembling the electric circuit which is manifested in the Leyden jar experiment) is completed from the nerves to the muscles when the contractions are produced. " For, while I with one hand held the prepared frog by the hook fixed in its spinal marrow, so that it stood with its feet on a silver box, and with the other hand touched the lid of the box, or its sides, with any metallic body, I was surprised to see the frog become strongly convulsed every time that I applied this artifice."* Galvani thus ascertained that the limbs of the frog are con- vulsed whenever a connexion is made between the nerves and muscles by a metallic arc, generally formed of more than one •Thii obiemitioc was mide in 1786. 3,Bl,ZEdhyG00gle 70 Galvanism, from Galvani to Ohm. I kind of metal ; and he advanced the hypothesis that the convul- ■ sions are caused by the transport of a peculiar fluid from the ' nerves to the muscles, the arc acting as a conductor. To this fluid the names Galvanism and Avimal Electricity were soon generally applied. Galvani himself considered it to be the same as the ordinary electric fluid, and, indeed, regarded the entire phenomenon as similar to the discharge of a Leyden jar. *' The publication of Galvani's views soon engaged the attention of the learned world, and gave rise to an animated controversy between those who supported Galvani's own view, those who believed galvanism to be a fluid distinct from ordinary electricity, and a third school who altogether refused to attribute the effects , to a supposed fluid contained in the nervous system. The leader of the last-named party was Alcssandro Yolta (b. 1745, d. 1827), Professor of Natural Philosophy in the University of Pavia, who in 1792 put forward the view* that the stimulus in Galvani's. experiment is derived essentially from the connexion of two different metals by a moist body. "The metals used in the experiments, being applied to the moist bodies of animals, can by themselves, and of their proper virtue, excite and dislodge the electric fluid from its state of rest ; so that the organs of the animal act only passively." At first he inclined to combine thia theory of metallic stimulus with a certain degree of belief in such a fluid as Galvani had supposed ; but after the end of 1793 he denied the existence of animal electricity altogether. From this standpoint Volta continued his experiments and worked out his theory. The following quotation from a letterf which he wrote later to Gren, the editor of the Neves Journal d. PkysUe, sets forth his view in a more developed form : — " The contact of different conductors, particularly the metallic, including pyrites and other minerals, as well as charcoal, which I call dry conductors, or of the Jirst class, with moist conductors, or conductors of the second class, agitates or disturbs the electric fluid, or gives it a certain impulse. Do not ask in what manner : it is enough that it is a principle, and a general principle. Thia • Phi!. Trans., 1793, pp. 10, 27. t PbiL Hag. iv (1766), pp. 60, 163, 306. 3,Bl,ZEdhyG00gle Galvanism, from Galvani to Ohm, 71 impulse, whether produced by attraction or any other force, is different or unlike, both in regard to the different metals and to the different moist conductors; so that the direction, or at least the power, with which the electric fluid is impelled or excited, is different when the conductor A is applied to the conductor B, or to another C. In a perfect circle of conductors, where either one of the second class is placed between two different from each other of the first class, or, contrariwise, one of the first class is placed between two of the second class different from each other, an electric stream is occasioned by the predominating force either to the right or to the left— a circulation of this fluid, which ceases only when the circle is broken, and which is renewed when the circle is again rendered complete." Another philosopher who, like Volta, denied the existence of a fluid peculiar to animals, but who took a somewhat different view of the origin of the phenomenon, was Giovanni Fabroni, of Florence (6. 1752, d. 1822), who* having placed two plates of different metals in water, observed that one of them was partially oxidized when they were put in contact ; from which he rightly concluded that some chemical action is inseparably connected with galvanic effects. The feeble intensity of the phenomena of galvanism, which compared poorly with the striking displays obtained in electro- statics, was responsible for some falling off of interest in them towards the end of the eighteenth century ; and the last years of their illustrious discoverer were clouded by misfortune. Being attached to the old order which was overthrown by the armies of the French Revolution, he refused in 1798 to take the oath of allegiance to the newly constituted Cisalpine Republic, and was deposed from his professorial chair. A profound melancholy, which had been induced by domestic bereavement, was aggra- vated by poverty and disgrace ; and, unable to survive the loss of all he held dear, he died broken-hearted before the end of the year.f * Pbi!. Journal, 4tO. iii. 308; it. 120; Journal tie Physique, vi. 348. t A decree of reinstatement bid been granted, but had nut come into operation at the time of GalTani'a death. 3,Bl,ZEdhyG00gle 72 Galvanism, from Gatvani to Ohm. Scarcely more than a year after the death of Galvani, the new science suddenly regained 'the eager attention of philo- sophers. This renewal of interest was due to the discovery by Volta, in the early spring of 1800, of a means of greatly increasing the intensity of the effects. Hitherto all attempts to magnify the action by enlarging or multiplying the apparatus had ended in failure. If a long chain of different metals was used instead of only two, the convulsions of the frog were no more violent But Volte now showed" that if any nnmber of couples, each consisting of a zinc disk and a copper disk in contact, were taken, and if each couple was separated from the next by a disk of moist- ened pasteboard (so that the order was copper, zinc, pasteboard, copper, zinc, pasteboard, 4c), the effect of the pile thus formed was much greater than that of any galvanic apparatus previously introduced. When the highest and lowest disks were simul- taneously touched by the fingers, a distinct shock was felt ; and this could be repeated again and again, the pile apparently possessing within itself an indefinite power of recuperation. It thus resembled a Leyden jar endowed with a power of automati- cally re-establishing its state of tension after each explosion ; with, in fact, "an inexhaustible charge, a perpetual action or impulsion on the electric fluid." Volta unhesitatingly pronounced the phenomena of the pile to be in their nature electrical. The circumstances of Galvani's original discovery liad prepared the minds of philosophers for this belief, which was powerfully supported by the similarity of the physiological effects of the pile to those of the Leyden jar, and by the observation that the galvanic influence was conducted only by those bodies — e.g. the ruetals — which were already known to be good conductors of static electricity. But Volta now supplied a still more convincing proof. Taking a disk of copper and one of zinc, 'he held each by an insulating handle and applied them to each other for an instant. After the disks had been separated, they were brought into contact with a deli- ■l'hil. Tian*., 1800, p. 408. 3,Bl,ZEdhyG00gle Galvanism, from Galvani to Ohm. 73 cate electroscope, which indicated by the divergence of its straws that the disks were now electrified — the zinc had, in fact, acquired a positive and the copper a negative electric charge." Thus the mere contact of two different metals, such as those employed in / the pile, was shown to be sufficient for the production of effects ' undoubtedly electrical in character. On the basis of this result Volta in the same year (1800) put forward a definite theory of the action of the pile. Suppose first that a disk of zinc is laid on a disk of copper, which in turn rests on an insulating support. The experiment juat described shows that the electric fluid will be driven from the copper to the zinc. We may then, according to Volta, represent the state or " tension " of the copper by the number - \, and that of the zinc by the number + \, the difference being arbitrarily taken as unity, and the sum being (on account of the insulation) zero. It will be seen that Volta's idea of " tension " was a mingling of two ideas, which in modern electric theory are clearly distin- guished from each other — namely, electric charge and electrie potential. Now let a disk of moistened pasteboard be laid on the zinc, and a disk of copper on this again. Since the uppermost -copper is not in contact with the zinc, the contact-action does not take place between them ; but since the moist pasteboard is a conductor, the copper will receive a charge from the zinc. Thus the states will now be represented by - \ for the lower copper, + \ for the zinc, and + \ for the upper copper, giving a zero sum as before. If, now, another zinc disk is placed on the top, the states will be represented by - 1 for the lower copper, 0 for the lower zinc and upper copper, and + 1 for the upper zinc. In this way it is evident that the difference between the numbers indicating the tensions of the uppermost and lowest "Abraham Bonnet lb. 1760, 4. 1799) bad previously shown (A>ur Eijiniintnti ia Eiertritily, 1789, pp. 86-102) that many bodiee, when leparuted aftor contact, ' are oppositely electrified ; he conceived that different budiva have different attrac- tioBM or capacities f»r electricity. 3,Bl,ZEdhyG00gle L 74 Galvanism, from Galvani to Ohm. disks in the pile will always be equal to the number of pairs of metallic disks contained in it. If the pile is insulated, the sum of the numbers indicating the states of all the disks must be zero; but if the lowest disk is connected to earth, the tension of this disk will be zero, and the numbers indicating the states of all the other disks will be increased by the same amount, their mutual differences remaining unchanged. The pile as a whole is thus similar to a Leyden.jar; when the experimenter touches the uppermost and lowest disks, he receives the shock of its discharge, the intensity being proportional to the number of diBks. The moist layers played no part in Volta's theory beyond that of conductors* It was soon found that when the moisture is acidified, the pile is more efficient; but this was attributed solely to the superior conducting power of acids. Yolta fully understood and explained the impossibility of constructing a pile from disks of metal alone, without making use of moist substances. As he showed in 1801, if disks of various metals are placed in contact in any order, the extreme metals will be in the same state as if they touched each other directly without the intervention of the others ; so that the whole is equivalent merely to a single pair. When the metals are arranged in the order silver, copper, iron, tin, lead, zinc, each of them becomes positive with respect to that which precedes it, and negative with respect to that which follows it ; but the moving force from the silver to the zinc is equal to the sum of the moving forces of the metals comprehended between them in the seriea. When a connexion was maintained for some time between the extreme disks of a pile by the human body, sensations were experienced which Beemed to indicate a continuous activity in the entire system. Volta inferred that the electric current persists during the whole time that communication by con- * Volta had inclined, in his earlier experiments on galvaniiru, to locate thereat of power at the interfaces of the metsJi with the moist conductor*. Cf. hia letter to Gren, I'hil. Mag. iv (1799), p. 62. • D,Bl,ZEdhyG00gle Galvanism, front Galvani to Ohm. 75 ductors exists all round the circuit, and that the current is suspended only when this communication is interrupted. " This endless circulation or perpetual motion of the electric fluid," he says, "may seem paradoxical, and may prove inexplicable; bat it is none the less real, and we can, so to speak, touch and handle it." Volta announced his discovery in a letter to Sir Joseph Banks, dated from Como, March 20th, 1800. Sir Joseph, who was then President of the Royal Society, communicated the news to William Nicholson {h. 1753, d. 1815), founder of the Journal which is generally known by his name, and his friend Anthony Carlisle (b. 1768, d. 1840), afterwards a distinguished surgeon. On the 30th of the following month, Nicholson and Carlisle set up the first pile made in England. In repeating Volta's experiments, having made the contact more secure at the upper plate of the pile by placing a drop of water there, they noticed* a disengagement of gas round the con- < ducting wire at this point ; whereupon they followed up the matter by introducing a tube of water, into which the wires from the terminals of the pile were plunged. Bubbles of an inflammable gas were liberated at one wire, while the other wire became oxidised ; when platinum wires were UBed, oxygen and hydrogen were evolved in a free state, one at each wire. This effect, which was nothing less than the electric decom- position of water into its constituent gases, was obtained on / May 2nd, 1800.f Although it had long been known that frictional electricity is capable of inducing chemical action,* the discovery of Nicholson and Carlisle was of the first magnitude. It was at once extended by William Uruickshank, of Woolwich (b. 1745, • Xir&iUen'i Jturuii (4to), iv, 179 (1800) ; Phil. Hag. vii, 337 11800). t It in obtained independently fuur months later by J. W. Bitter. t Beccaria [LtUm MP eltttrieiino, Bologna, 1758, p. 282) had reduced mercury and other metala from their oxides by discharged ot frictiuniil electricity ; and Priestley hud obtained an inflammable gas from certain organic liquids in the one way. Careiidiah in 1781 bad established the constitution of water by electrically exploding hydrogen and oxygen. 3,Bl,ZEdhyG00gle 76 Galvanism, from Galvani lo Ohm. d. 1800), who* showed that solutions of metallic salts are also decomposed by the current; and William Hyde Wollaston (b. 1766, d. 1828) seized on it as a testf of the identity of the electric currents of Yolta with those obtained by the discharge of frictional electa) city. He found that water could be decom- J posed by currents of either type, and inferred that all differences between them could be explained by supposing that voltaic electricity &n commonly obtained is "less intense, but produced in much, larger quantity." Later in the same year (1801), Martin van Marum (b. 1750, d. 1837) and Christian Heinrich Pfaff (l>. 1773, d. 18">2) arrived at the same conclusion by carrying out on a. large scalej Volta's plan of using the pile to V cliarge batteries of Leytlen jara. The discovery of Nicholson and Carlisle made a great impression on the mind of Humphry Davy (6. 1778, d. 1829), a young Cornishman who about this time was appointed Professor of Chemistry at the Royal Institution in London. Davy at once began to experiment with Voltaic piles, and in November, 1800,§ showed that they give no current when the water between the J pairs of plates is pure, and that their power of action is " in great measure proportional to the power of the conducting fluid substance between the double plates to oxydate the zinc." Tins result, as lie immediately perceived, did not harmonize well with Volta's views on the source of electricity in the pile, but was, on the other hand, in agreement with labroni's idea that galvanic effects are always accompanied by chemical action. After a series of experiments he definitely ' concluded that " the galvanic pile of Volta acts only when the conducting substance between the plates is capable of oxydating the zinc ; and that, in proportion as a greater quantity of oxygen enters into combination with the zinc in a given time, so in proportion is the power of the pile to decompose water and to give the shock greater. It seems therefore reasonable • McheUeu't Journal (4to), iv (1800), pp. 187,2*5: Phil. Hag., vii (1800), p. 337. T Phil. Mag., 1801, p. 427. J Phil. Mag., xii (1802), p. 161. ( Nicholtoit', Journal (4toj, iv (1800) ; Davy'i Work; it, p. 16ft. 3,Bl,ZEdhyG00gle Galvatiism, from Galvani (o Ohm, 77 to conclude, though with our present quantity of facts we are unable to explain the exact mode of operation, that the ■/ oxydatiou of the zinc in the pile, and the chemical changes connected with it, are somehoio the cause of the electrical effects » it produces." This principle of oxidation guided Davy in designing many new types of pile, with elements chosen from the whole range of the known metals. Davy's chemical theory of the pile was supported by Wollaston* and by NicholBon.t the latter of whom urged that the existence of piles in which only one metal is used (with more than one kind of fluid) is fatal to any theory which places the seat of the activity in the contact of dissimilar metal;-. Davy afterwards proposed^ a theory of the voltaic pile which combines ideas drawn from both the " contact " and ■ " chemical " explanations. Ho supposed that l>efore the circuit is closed, the copper and zinc disks in each contiguous pair assume opposite electrostatic states, in consequence of inherent " electrical energies " possessed by the metals ; and when a J communication is made between the extreme disks by a wire, the opposite electricities annihilate each other, as in the dis- charge of a Leyden jar. If the liquid (which Davy compared to the glass of a Leyden jar) were incapable of decomposition, the current would cease after this discharge. But the liquid in the pile is composed of two elements which have inherent attractions for electrified metallic surfaces : hence arises chemical action, which removes from the disks the outermost layers of molecules, whose energy is exhausted, and exposes new metallic surfaces. The electrical energies of the copper and zinc are consequently again exerted, and the process of electro- motion continues. Thus the contact of metals is the cause which disturbs the equilibrium, while the chemical changes continually restore the conditions under which the contact energy can be exerted. In this and other memoirs Davy asserted that chemical •Phi) Trans-, 1801, p. 427. t Xienaltn'i Journal, i (1802), p. 142. JPbU.Tnuu., 1807, p. 1. 3,Bl,ZEdhyG00gle 78 Galvanism, from Galvani to Ohm. J affinity is essentially of an electrical nature. " Chemical and electrical attractions," he declared* "are produced by the same cause, acting in one case on particles, in the other on masses, of matter; and the same property, under different modifications, is the cause of all the phenomena exhibited by different voltaic combinations." The further elucidation of this matter came chiefly from r- researches on electro-chemical decomposition, which we must now consider. A phenomenon which had greatly surprised Nicholson and Carlisle in their early experiments was the appearance of the products of galvanic decomposition at places remote from each other. The first attempt to account for this was made in 1806 by Theodor von Grothusst (b. 1785, d. 1822) and by Davy,} who advanced a theory that the terminals at which water is decomposed have attractive and repellent powers ; that the pole whence resinous electricity issues has the property of attracting hydrogen and the metals, and of repelling oxygen and acid substances, while the positive terminal has the power of attract- ing oxygen and repelling hydrogen ; and that these forces are sufficiently energetic to destroy or suspend the usual operation of chemical affinity in the water-molecules nearest the terminals. The force due to each terminal was supposed to diminish with the distance from the terminal. When the molecule nearest one of the terminals has been decomposed by the attractive and repellent forces of the terminal, one of its constituents is liberated there, while the other constituent, by virtue of electrical forces (the oxygen and hydrogen being iu opposite electrical states), attacks the next molecule, which is then decomposed. The surplus constituent from this attacks the next molecule, and so on. Thus a chain of decompositions and recompositions was supposed to be Bet up among the molecules intervening between the terminals, ■ Phil. Train., 1828, p. 333. fAnn. de Chim., Iriii (180G), p. 6*. ; Bukerian lecture for 1806, Phil. Trans., 1807, Ji. 1. A theory similar to that of Grothusa and Davy vat communicated by Peter Mark Itoget (*. 177B, d. 1869) in 180T to the Philosophical Society of Manchester : cf. Roget's Gafoanitm, $ 106. 3,Bl,ZEdhyG00gle Galvanism, Jrom Galvani to Ohm. 79 The hypothesis of Grothuss and Davy was attacked in 1825 by Anguste De La Rive' (b. 1801, d. 1873) of Geneva, on the ground of its failure to explain what happens when different liquids are placed in series in the circuit. If, for example, a solution of zinc sulphate is placed in one compartment, and water in another, and if the positive pole is placed in the solution of zinc sulphate, and the negative pole in the water, De La Rive found that oxide of zinc is developed round the latter; although decomposition and recbmposjtion of zinc sulphate could not take place iu the water, which contained none of it. Accordingly, he supposed the constituents of the decomposed liquid to be bodily transported across the liquids, in close union with the moving electricity. In the electrolysis of water, one current of electrified hydrogen was supposed to leave the positive pole, and become decomposed into hydrogen and electricity at the negative pole, the hydrogen being there liberated as a gas. Another current in the same way carried electrified oxygen from the negative to the positive pole. Iu this scheme the chain of successive decompositions imagined by Grothuss does not take place, the only molecules decomposed being those adjacent to the poles. The appearance of the products of decomposition at the separate poles could be explained either in Grothuss' fashion by assuming dissociations throughout the mass of liquid, or in De La Rive's by supposing particular dissociated atoms to travel considerable distances. Perhaps a preconceived idea of economy in Nature deterred the workers of that time from accepting the two assumptions together, when either of them separately would meet the case. Yet it is to this apparent redundancy that later researches have pointed as the truth. Nature is what she is, and not what we would make her. De La Rive was one of the most thoroughgoing opponents of Volta's contact theory of the pile; even in the case when two metals are in contact in air only, without the intervention 3,Bl,ZEdhyG00gle 80 Galvanism, from Galvani to Ohm. of any liquid, he attributed the electric effect wholly to the chemical affinity of the air for the metals. During the long interval between the publication of the rival hypotheses of Grothuss and De La Rive, little real progress ': was made with the special problems of the cell; but mean- while electric theory was developing in other directions. One of these, to which our attention will first be turned, was the electro-chemical theory of the celebrated Swedish chemist, Jons Jacob Berzelins (6. 1779, d. 1848). Berzelius founded hiB theory,* which had been in one or two of its features anticipated by Davy,t on inferences drawn from Volta's contact effects. " Two bodies," he remarked, " which have affinity for each other, and which have been brought into mutual contact, are found upon separation to be in opposite electrical stateB. That which has the greatest affinity for oxygen usually becomes positively electrified, and the other negatively." This seemed to him to indicate that chemical affinity arises from the play of electric forces, which in turn spring from electric charges within the atoms of matter. To be precise, * he supposed each atom to possess two poles, which are the seat of opposite electrifications, and whose electrostatic field is the cause of chemical affinity. By aid of this conception Berzelius drew a simple and vivid picture of chemical combination. Two atoms, which are about to unite, dispose themselves bo that the positive pole of one touches the negative pole of the other ; the electricities of these two poles then discharge each other, giving rise to the heat and light which are observed to accompany the act of combination.^ The disappearance of these leaves the compound molecule with the two remaining poles ; and it cannot be dissociated into its constituent atoms again until some means is found of restoring to the vanished poles their charges. Such a means is afforded • Memoira of the Acad, of Stockholm. 1812 ; Nicool»on'» Journal of Nat. PhE, xxiir (1813), 142, 1C3, 240, 319; ixxv, 38, 118, 169. + Pail. Trim*., 1807. I Tbw idea wu Davy's. 3,Bl,ZEdhyG00gle Galvanism, from Gaivani to Ohm. 81 by the action of the galvanie pile in electrolysis : the opposite electricities of the current invade the molecules of the electrolyte, and restore the atoms to their original state of polarization. If, as Berzelius taught, all chemical compounds are formed by the mutual neutralization of pairs of atoms, it is evident / that they must have a binary character. Thus he conceived a salt to be compounded of an acid and an oxide, and each of these to be compounded of two other constituents. Moreover, in any compound the electropositive member would be replace- able only by another electropositive member, and the electro- negative member only by another member also electronegative ; so that the substitution of, e.g., chlorine for hydrogen in a compound would be impossible — an inference which was overthrown by subsequent discoveries in chemistry. Berzelius succeeded in bringing the most curiously diverse facte within the scope of bis theory. Thus " the combination of polarized atoms requires a motion to turn the opposite poles to each other ; and to this, circumstance is owing the facility with which combination takes place when one of the two bodies is in the liquid state, or when both are in that state ; and the extreme difficulty, or nearly impossibility, of effecting an union between bodies, both of which are solid. And again, since each polarized particle must have an electric atmosphere, and as this atmosphere is the predisposing cause of combination, as we have seen, it follows, that the particles cannot act but at certain distances, proportioned to the intensity of their polarity ; and hence it is that bodies, which have affinity for each other, always combine nearly on the instant when mixed in the liquid state, but less easily in the gaseous state, and the union ceases to be possible under a certain degree of dilatation of the gases ; as we know by the experiments of Grothuss, that a mixture of oxygen and hydrogen in due proportions, when rarefied to a certain degree, cannot be set on fire at any temperature whatever." , And again : " Many bodies require an elevation of temperature to G 3,Bl,ZEdhyG00gle 82 Gatvanism,from Galvani to Ohm. enable them to act upon each other. It appears, therefore, that heat possesses the property of augmenting the polarity of these bodies." Berzelius accounted for Volta's electromotive series by assuming the electrification at one pole of an atom to be some- what more or somewhat less than what would be required to neutralize the charge at the other pola Thus each atom would possess a certain net or residual charge, which might be of either sign ; and the order of the elements in Volta's series could be interpreted simply as the order in which they would stand when ranged according to the magnitude of this residual charge. As we shall see, this conception was afterwards overthrown by Faraday. Berzelius permitted himself to publish some speculations on the nature of heat and electricity, which bring vividly before us the outlook of an able thinker in the first quarter of the nineteenth century. The great question, he says, is whether v the electricities and caloric are matter or merely phenomena. If the title of matter is to be granted only to such things as are ponderable, then these problematic entities are certainly not matter; but thus to narrow the application of the term is, he believes, a mistake; and he inclines to the opinion that caloric is truly matter, possessing chemical affinities without obeying the law of gravitation, and that light and all radiations consist in modes of propagating Buoh matter. This conclusion makes it easier to decide regarding electricity. " From the relation which exists between caloric and the electricities," he remarks, " it is clear that what may be true with regard to the materiality of one of them must also be true with regard to that of the other. There are, however, a quantity of phenomena produced by electricity which do not admit of explanation without admitting at the same time that electricity is matter. Electricity, for instance, very often detacheB everything which covers the surface of those bodies which conduct it. It, indeed, passes through conductors without leaving any trace of its passage ; but it penetrates non-con- 3,Bl,ZEdhyG00gle Galvanism, from Galvani to Ohm. £& doctors which •ffWB its course, and makes a perforation jpeMnety of the same description as would have been made by something which had need of place for its passage. We often observe this when electric jars are broken by an over- charge, or when the electric shock is passed through a number of cards, etc. We may therefore, at least with some proba- J bility, imagine caloric and the electricities to be matter, destitute of gravitation, but possessing affinity to gravitating bodies. When they are not confined by these affinities, they tend to place themselves in equilibrium in the universe. The J suns destroy at every moment this equilibrium, and they send the re-united electricities in the form of luminous rays towards the planetary bodies, upon the surface of which the rays, being arrested, manifest themselves as caloric ; and this last in its turn, during the time required to replace it in equilibrium in the universe, supports the chemical activity of organic and inorganic nature." It was scarcely to be expected that anything so speculative as Berzelius' electric conception of chemical combination would be confirmed in all particulars by subsequent discovery ; and, as a matter of fact, it did not as a coherent theory survive the lifetime of its author. But some of its ideas have persisted, and among them the conviction which lies at its foundation, that chemical affinities are, in the last resort, of * electrical origin. While the attention of chemists was for long directed to the theory of Berzelius, the interest of electricians was diverted from it by a discovery of the first magnitude in a different region. That a relation of some kind subsists between electricity and magnetism had been suspected by the philosophers of the eighteenth century. The suspicion was based in part on some curious effects produced by lightning, of a kind which may be illustrated by a paper published in the Philosophical Transactions in 1735." A tradesman of Wakefield, we are told, "having put •Phil. Twos, mix (1T3S), p. 74. G 2 3,Bl,ZEdhyG00gle 84 Galvanism, from Galvani to Ohm. up a great number of knives and1 forks in a large box, and having placed the box in the corner of a large room, there- happen'd in July, 1731, a sudden storm of thunder, lightning, etc., by which the corner of the room was damaged, the Box split, and a good many knives and forks melted, the sheaths being untouched. The owner emptying the box upon a Counter where some NailB lay, the Persons who took up the knives, that lay upon the Nails, observed that the knives took up the Nails." Lightning thus came to be credited with the power of magnetizing steel ; and it was doubtless this which led Franklin* in 1751 to attempt to magnetize a sewing-needle by means of the discharge of Leyden jars. The attempt was indeed success- ful ; but, as Van Marum afterwards showed, it was doubtful whether the magnetism was due directly to the current. More experiments followed.t In 1805 Jean Nicholas Pierre Hachette (6. 1769, d. 1834) and Charles Bernard Desonnes (6. 1777, d. 1862) attempted to determine whether an insulated voltaic pile, freely suspended, is oriented by terrestrial mag- netism ; but without positive result. In 1807 Hans Christian Oersted (b. 1777, d. 1851), Professor of Natural Philosophy in Copenhagen, announced his intention of examining the action of electricity on the magnetic needle ; but it was not for some years that his hopes were realized. If one of his pupils ia to he believed,! he was " a man of genius, but a very unhappy experi- menter ; he could not manipulate instruments. He must always have an assistant, or one of his auditors who had easy hands, to arrange the experiment." During a course of lectures which he delivered in the winter of 1819-20 ou " Electricity, Galvanism, and Magnetism," the idea occurred to him that the changes observed with the compasB-needle during a thunderstorm might give the clue to the effect of which he was in search ; and this led him to think that the experiment should be tried with the galvanic circuit * Latter vi from Franklin to Collinaon. tin 1774 the Electoral Academy of Bavaria proponed the question, " Is there a roil and physical analogy between electric and magnetic forces ? " as the subject of a prue. iCf. a letter from Hanateen inserted in Hence Jones' Life of Faraday, ii, p. 395. 3,Bl,ZEdhyG00gle Galvanism, from Gaivani to Ohm. '86 closed iastead of open, and to inquire whether any effect is produced on a magnetic needle when an electric Current is passed through a neighbouring wire. At first he placed the wire *t right angles to the needle, but observed no result After the end of a lecture in which this negative experiment had been shown, the idea occurred to him to place the wire parallel to the needle : on trying it, a pronounced deflexion was observed, and the relation between magnetism and the electric current was discovered. Alter confirmatory experiments with more powerful apparatus, the public announcement was made , in July, 1820.' / Oersted did not determine the quantitative laws of the action, but contented himself with a statement of the qualita- ■ live effect and some remarks on its cause, which recall the magnetic speculations of Descartes : indeed, Oersted's concep- tions may be regarded as linking those of the Cartesian school to those which were introduced subsequently by Faraday. " To the effect which takes place in the conductor and in the sur- rounding space," be wrote, " we shall give the name of the ■conflict of electricity." " The electric conflict acts only on the magnetic particles of matter. All non-magnetic bodies appear penetrable by the electric conflict, while magnetic bodies, or rather their magnetic particles, resist the passage of this conflict Hence they can be moved by the impetus of the contending powers. " It is sufficiently evident from the preceding facts that the ■electric conflict is not confined to the conductor, hut dispersed pretty widely in the circumjacent space. " From the preceding facts we may likewise collect, that -this conflict performs circles ; for without this condition, it seems impossible that the -one part of the uniting wire, when placed below the magnetic pole, should drive it toward the east, and when placed above it toward the west; for it is the nature of a • ScliTOggerN Journal fur Chemia and Phyiik, xxix (1820), p. 2TS ; Thomwn'a Annali of Philosophy, itj (1820), p. 273; OttwaU'a Xhmktr *r uw««a ■WitMuctyti*, Nr. 83. 3,Bl,ZEdhyG00gle '86 Galvanism, from Galvani to Okm. circle that the motions in opposite parts should have an opposite direction." Oersted's discovery was described at the meeting of the French Academy on September 11th, 1820, by an academician. (Arago) who had just returned from abroad. Several investi- gators in France repeated and extended his experiments ; and the first precise analysis of the effect was published by two of these, Jean-Baptiste Biot (b. 1774, d. 1862) and Felix Savart (6. 1791, d. 1841), who, at a meeting of the Academy of Sciences on October 30th, 1820, announced* that the action experienced by a pole of austral or boreal magnetism, when placed at any distance from a straight wire carrying a voltaic current, may be thus expressed : " Draw from the pole a perpendicular to the wire ; the force on the pole is at right angles to this line and to the wire, and its intensity is proportional to the reciprocal of the distance." This result was soon further analysed, the attractive force being divided into constituents, each of which was supposed to be due to some particular element of the current ; in its new form the law may be stated thus : the magnetic force due to an element Am of a circuit, in which a current i is flowing, at a point whose vector distance from di m r, is (in suitable units) -[di,r]t or curl— .J It was now recognized that a magnetic field may be produced as readily by an electric current as by a magnet ; and, as Arago soon showed,j$ this, like any other magnetic field, is capable of • AhuIm de Chimie, it (1820), p. 222; Journal de Phjs., ili,p. 61. t If * and b denote two rectors, the vector whoso component* are (a,*, — b.A,. «.l. — /••&•, "•*» - 'yM !■ called the vtetvr product of a and b , and ia denoted by (a, b]. Ita direction is at right angles to those of a and b, and its magnitude is represented by twice the area of the triangle formed by them. ' J If a denote! any vector, the vector whoee oomponenU are s-* - ^, %~~ *?j jrj* - v— ia denoted by curl a. i Annalea de Chimie, it (1820), p. 93. 3,Bl,ZEdhyG00gle Galvanism, from Galvani to Ohm. 87 inducing magnetization in iron. The question naturally sug- gested itself as to whether the similarity of properties between 1 currents and magnets extended still further, e.g. whether conductors carrying currents would, like magnets, experience ponderomotive forces when placed in a magnetic field, and whether such conductors would consequently, like magnets, exert ponderomotive forces on each other. The first step towards answering these inquiries was taken by Oersted* himself. " As," he said, " a body cannot put another in motion without being moved in its turn, when it possesses the requisite mobility, it is easy to foresee that the galvanic arc must be moved by the magnet " ; and this he verified experimentally. The next Btep came from Andre* Marie Ampere \b. 1775, d. 1836), who at the meeting of the Academy on September 18th, exactly a week after the news of Oersted's first discovery had arrived, showed that two parallel wires carrying currents y^ attract each other if the currents are in the same direction, and repel each other if the currents are in opposite directions. During the next three years Ampere continued to prosecute the researches thus inaugurated, and in 1825 published his y collected results in one of the most celebrated memoirsf in the' history of natural philosophy. Ampere introduces his work by proclaiming himself a follower of that school which explained all physical phenomena in terms of equal and oppositely directed forces between pairs ; of particles ; and he renounces the attempt to seek more speculative, though possibly more fundamental, explanations in terms of the motions of ultimate fluids and aethers. Never- theless, he indicates two conceptions of this latter character, on which such explanations might be founded. In the first* he suggests that the ponderomotive forces * Bcfaweigger'i Journal Mr Chain, u. F07S., uii (1820), p. 364; Thornton's Annals of Philosophy, xri (1820), p. ST6. t Mem. de 1' Acad., ti, p. 175. % Stetuil tFofatrtationt ilttfre-dynamiquti, p. 215 ; and the memoir just cited, pp. 285, 370. • 3,Bl,ZEdhyG00gle 88 Gaiva?iism, from Galvani to Okm. between circuits carrying electric currents may be due to " the - reaction of the elastic fluid which extends throughout all space, whose vibrations produce the phenomena of light," and which is " put in motion by electric currents." This fluid or aether can, he says, " be no other than that which results from the combination of the two electricities." In the second conception* Ampere suggests that the interspaces between the metallic molecules of a wire which carries a current may be occupied by a fluid composed of the two electricities, not in the proportions which form the neutral fluid, but with an excess of that one of them which is opposite to the electricity peculiar to the molecules of the metal, and which consequently masks this latter electricity. In this inter- molecular fluid the opposite electricities are continually being dissociated and recombined ; a dissociation of the fluid within one inter- molecular interval having taken place, the positive electricity thus produced unites with the negative electricity of the interval next to it in the direction of the current, while the negative electricity of the first interval unites with the positive electricity of the next interval in the other direction. Such interchanges, according to this hypothesis, constitute the electric current Ampere's memoir is, however, hut little occupied with the more speculative side of the subject. His first aim was to investigate thoroughly by experiment the ponderomotive forces on electric currents. " When," he remarks, " M. Oersted discovered the action which a current exercises on a magnet, one might certainly have suspected the existence of a mutual action between two circuits carrying currents; but this was not a necessary consequence ; for a bar of soft iron also acts on a magnetized needle, although there is no mutual action between two bars of soft iron." , Ampere, therefore, submitted the matter to the test of the laboratory, and discovered that circuits carrying electric currents exert ponderomotive forces on each other, and that * Stencil tFebttrvaliom Usttrc-dytwmiijiwi, pp. 297, 300, 371. 3,Bl,ZEdhyG00gle Galvanism, from Gatvani to Ohm. 89 ponderomotive forces are exerted on such currents by magnets. To the science which deals with the mutual action of currents he gave the name electro-dynamics ;* and he showed that the action obeys the following laws : — (1) The effect of a current is reversed when the direction of the current is reversed. (2) The effect of a current flowing in a circuit twisted into small sinuosities is the same as if the circuit were smoothed out (3) The force exerted by a closed, circuit on an element of Another circuit is at right angles to the latter. (4) The force between two elements of circuits is unaffected when all linear dimensions are increased proportionately, the ■current-strengths remaining unaltered. From these data, together with his assumption that the force between two elements of circuits acts along the line joining them. Ampere obtained an expression of this force : the deduction may be made in the following way : — Let ds, dV be the elements, r the line joining them, and i, i' the current-strengths. From (2) we see that the effect of d» on di* is the vector sum of the effects of dx, dy, dz on ds', where these are the three components of di: so the required force must he of the form — r x a scalar quantity which is linear and homogeneous in ds ; ■and it must similarly be linear and homogeneous in ds' ; so using (1), we see that the force must be of the form F-«"r|(di.aO*<»-) + (di.r) and $ denote undetermined functions of ft From (4) it follows that when ds, ds', r are all multiplied by the same number, F is unaffected : this shows that ♦ _ I * Loo. cit, p. 3tt. 90 Galvanism, from Galvani to Ohm. Now, by (3), the resolved part of F along di' must vanish when integrated round the circuit s, i.e. it must be a complete differential when r* must be a complete differential ; or -^i.(t.dO' + f(i» ')•(»•*')■ must be a complete differentia] ; and therefore <■£- > r), or 3A ; >■ or B - -iA. Thus finally we have ; Thus we see that the pouderomotive force on a current-element di' in a magnetic field B is i' [di'. B]. Ampere developed to a considerable extent the theory of the equivalence of magnets with circuits carrying currents ; and showed that an electric current is equivalent, in its magnetic effects, to a distribution of magnetism on any surface terminated by the circuit, the axes of the magnetic molecules being everywhere normal to this surface :f such a magnetized surface is called a magnetic shell. He preferred, however, to regard the current rather than the magnetic fluid as the fundamental entity, and considered magnetism to be really an electrical phenomenon : each magnetic molecule owes its properties, according to this view, to the presence within it of a small closed circuit in which an electric current is perpetually flowing. The impression produced by Ampere's memoir was great and lasting. Writing half a century afterwards, Maxwell speaks of it as "one of the most brilliant achievements in science." " The whole," he says, " theory and experiment, seems as if it had leaped, full-grown and full-armed, from the brain of the ' Newton of electricity.' It is perfect in form and unassailable in accuracy ; and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electrodynamics." Not long after the discovery by Oersted of the connexion between galvanism and magnetism, a connexion was discovered between galvanism and heat. \ In 1822 Thomas Johann Seebeck * See ante, p. 86. t Loc- oil., p. M7. 3,Bl,ZEdhyG00gle Galvanism, from Galvani to Ohm. 93 (l. 1770, d. 1831), of Berlin discovered* that an electric current can be set up in a circuit of metals, without the interposition of any liquid, merely by disturbing the equilibrium of temperature. Let a ring be formed of copper aud bismuth soldered together at the two extremities; to establish a current it is only necessary to heat the ring at one of these junctions. To this new class of circuits the name thermo- electric was given. It was found that the metals can be arranged as a thermo-electric aeries, in the order of their power of generating currents when thus paired, and that this order is quite different fromVolta's order of electromotive potency. Indeed antimony and bismath, which are near each other in the latter series, are at opposite extremities of the former. The currents generated by thermo-electric means are generally feeble : and the mention of this fact brings us to the question, which was about this time engaging attention, of the efficacy of different voltaic arrangements. Comparisons of a rough kind had been instituted soon after the discovery of the pile- The French chemists Antoine Francois de Fourcroy (b. 1755, d. 1809), Louis Nicolas Vauquelin (6. 1763, d. 1829), and Louis Jacques Thenard (S. 1777, d. 1857) foundf in 1801, on varying the size of the metallic disks constituting the pile, that the sensations produced on the human frame were unaffected so long as the number of disks remained the same ; but that the power "of burning finely drawn wire was altered; and that the latter power was proportional to the total surface of the disks employed, whether this were distributed among a small number of large disks, or a large number of small ones. This was •Abhandl. d. Berlin Akad. 1821-3 ; Ann. d. Phys. Ixxiii (1823), pp. 116, 430; Ti (1826), pp. 1, 133, 263. Vulta hud prerioiuily noticed Out a silver pints whose end* were at different temperatures appeared to act like a voltaic cell . Further experiment* vera performed by Jamei dimming (>. 1777, &. 1861), Profeaeorof Chemiatrr at Cambridge, Tram. Curob. Phil. Soc. ii (1823), p. 47, and by Antoine Cesar Becquerel \b. 1788, d. 1878), Aanalea de Chimie, ixii (1820), p. 371. t Ann. de Chimie, xxxix (1801), p. 103. D,Bl,ZEdhyG00gle 04 Galvanism, from Gatvani to Ohm. explained by supposing that small plates give a small quantity of the electric fluid with a high velocity, while large plates give a larger quantity with no greater velocity. Shocks, which were supposed to depend on the velocity of the fluid alone, would therefore not be intensified by increasing the size of the plates. The effect of varying the conductors which connect the terminals of the pile was also studied. Nicolas Crautherot (b. 1753, d. 1803) observed* that water contained in tubes which have a narrow opening does not conduct voltaic currents so well as when the opening is more considerable. This experi- ment is evidently very similar to that which Beccaria had performed half a century previously! with electrostatic discharges. As we have already seen, Cavendish investigated very completely the power of metals to conduct electrostatic discharges; their power of conducting voltaic currents was now examined by Davy. J His method was to connect the terminals of a voltaic battery by a path containing water {which it decomposed), and also by an alternative path consisting of the metallic wire under examination. When the length of the wire was less than a certain quantity, the water ■ceased to be decomposed ; Davy measured the lengths and weights of wires of different materials and cross-sections under these limiting circumstances ; and, by comparing them, showed that the conducting power of a wire formed of any one metal I is inversely proportional to its length and directly proportional / " to its sectional area, but independent of the shape of the cross- section.§ The latter fact, as he remarked, showed that voltaic currents pass through the substance of the conductor and not along its surface. Davy, in the same memoir, compared the conductivities of various metals, and studied the effect of temperature : he found * Annate da Chim., xxxix (1801), p. 203. t See p. 63. X Phil. Trent,, 1821, p. 433. Eii reiulta ware confirmed afterwards by Becquere], Annalei da Chimie, izzii (1826], p. 423. $ These resulta had been known to Ca»endi«h. dhyGoogle Galvanism, from Galvani to Ohm. 95 that the conductivity varied with the temperature, being " lower in some inverse ratio ae the temperature was higher." He also observed that the same magnetic power is exhibited by every part of the same circuit, even though it be formed of wires of different conducting powers pieced into a chain, so that " the magnetism seems directly as the quantity of electricity which they transmit." The current which Aowb in a given voltaic circuit evidently depends not only on the conductors which form the circuit, but also on the driving-power of the battery. In order to form a complete theory of voltaic circuits, it was therefore necessary to extend Davy's laws by taking the driving-power into account. This advance was effected in 1826 by Georg Simon> J Ohm* (ft. 1787, d. 1854). J Ohm had already carried out a considerable amount of experimental work on the subject, and had, e.g., discovered that if a number of voltaic cells are placed in series in a circuit, the current is proportional to their number if the external resistance is very large, but is independent of their number if the external resistance is small. He now essayed the task of combining all the known results into a consistent theory. For this purpose he adopted the idea of comparing the flow of electricity in a current to the flow of heat along a wire, the theory of which had been familiar to all physicists since the publication of Fourier's ThSorie analytiquc de la ekaieur in 1822. " I have proceeded," be says, " from the supposition that the communication of the electricity from one particle takes place directly only to the one next to it, so that no immediate transition from that particle to any other situate at a greater distance occurs. The magnitude of the flow between two adjacent particles, under otherwise exactly similar circum- stances, I have assumed to be proportional to the difference of •Aim. d. rhye. ti (1826), p. 459 ; vii, pp. 46, 117; Bit Qalvaniuhe Kettt matJUmatuch btarbtiitt : Berlin, 182' ; translated in Taylor's Seimtyfie Memoir*, ii (18*1], p. 401. Cf. alio subsequent papers by Ohm in Eutner'i Archiv fir 4. git. Naturlthre, and SchweiggerU Jahrbuch. 3,Bl,ZEdhyG00gle 96 • Galvanism, front Galvani to Ohm. the electric forces existing in the two particles ; juBt as, in the theory of heat, the flow of caloric between two particles ia regarded as proportional to the difference of their temperatures." yf The comparison between the flow of electricity and the flow of heat suggested the propriety of introducing a quantity whose behaviour in electrical problems should resemble that of temperature in the theory of heat. The differences in the values of such a quantity at two points of a circuit would provide what waa so much needed, namely, a measure of the " driving-power " acting on the electricity between these points. To carry out this idea, Ohm recurred to Volta's theory of the electrostatic condition of the open pile. It was cus- tomary to measure the " tension " of a pile by connecting one terminal to earth and testing the other terminal by an electroscope. Accordingly Ohm says : " In order to investigate the changes which occur in the electric condition of a body A in a perfectly definite manner, the body is each time brought, under similar circumstances, into relation with a second moveable body of invariable electrical condition, called the electroscope ; and the force with which the electroscope is repelled or attracted by the body is determined. This force is termed the electrotcopic force of the body A." " The same body A may also serve to determine the electro- acopic force in various parts of the same body. For this purpose take the body A of very small dimensions, so that when we bring it into contact with the part to be tested of any third body, it may from its smallness be regarded as a substitute for this part : then its electroacopic force, measured in the way described, will, when it happens to be different at the various places, make known the relative differences with regard to electricity between these places." Ohm assumed, as was customary at that period, that when two metals are placed in contact, " they constantly maintain at the point of contact the same difference between their electro- acopic forces." He accordingly supposed that each voltaic cell possesses a definite tension, or discontinuity of electroacopic 3,Bl,ZEdhyG00gle Galvanism, from Ga/vani to Ohm. 97 force, which is to be regarded as its contribution to the driving- force of any circuit in which it may be placed. This assumption confers a definite meaning on his use of the term " electroecopic force " ; the force in question is identical with the electrostatic potential. But Ohm and his contemporaries did not correctly 1 understand the relation of galvanic conceptions to the ] electrostatic functions of Foisson. The electroecopic force in the open pile was generally identified with the thickness of the electrical stratum at the place tested ; while Ohm, recognizing that electric currents are not confined to the surface of the conductors, but penetrate their substance, seems to have thought of the electroBcopic force at a place in - a circuit as being proportional to the volume-density of electricity there — an idea in which he was confirmed by the relation which, in an analogous case, exists between the temperature of a body and the volume-density of heat supposed to be contained in it. Denoting, then, by S the current which flows in a wire of conductivity y, when the difference of the electroecopic forces at the terminals is E, Ohm writes S-yE. From this formula it is easy to deduce the Iawa already given by Davy. Thus, if the area of the cross-section of a wire is A, we can by placing n such wires side by side construct a wire of cross-section nA. If the quantity E is the same for each, equal currents will flow in the wires ; and therefore the current in the compound wire will be n times that in the single wire ; so when the quantity E is unchanged, the current is proportional to the cross-section ; that is, the conductivity of a wire is directly proportional to its cross-section, which is one of Davy's laws. In spite of the confusion which was attached to the idea of electroecopic force, and which was not dispelled for some years, the publication of Ohm's memoir marked a great advance in electrical philosophy. It was now clearly understood that the current flowing in any conductor depends only on the D,Bl,ZEdhyG00gle 98 Galvanism, frotn-Galvani to Ohm. .conductivity inherent in the conductor and on another variable which bears to electricity the same relation that temperature bears to heat ; and, moreover, it was realized that this latter ** variable is the link connecting the theory of currents with the older theory of electrostatics. These principles were a sufficient foundation for future progress; and much of the work which was published in the second quarter of the century was no more than the natural development of the principles laid down by Ohm* It ia painful to relate that the discoverer had long to wait before the merits of his great achievement were officially recognized. Twenty-two years after the publication of the memoir on the galvanic circuit, he was promoted to a university professorship ; this he held for the five years which remained until his death in 1854. * Ohm's theory in confirmed experimentally by teveral inveitigatore, among whom may be mentioned Guitar Theodor Fephner (i. 1801, i. 1887] {Manitkutm- utimgm tihr dit Qalvaniiahi Ksltt, Leipzig, 1831], and Cbarlei Wlieattbme (b. 1802, i. 1875) (Phil. Tram,, 18*3, p. 303). .yGoogle CHAPTER IV. THI LUMINIFEBOUS MEDIUM, FKOM BRADLKY TO FREBNEL. Although Newton, as we have seen, refrained from committing himself to any doctrine regarding the ultimate nature of light, the writers of the next generation interpreted his criticism of the wave-theory as equivalent to an acceptance of the corpuscular hypothesis. As it happened, the chief optical discovery of this period tended to support the latter theory, by which it was first and most readily explained. In 1728 James Bradley (b. 1692, d. 1762), at that time Savilian Professor of Astronomy at Oxford, sent to the Astronomer Royal (Halley) an " Account of a new discovered motion of the Fix'd Stars."* In observing the star y in the head of the Dragon, he had found that during the winter of 1725-6 the transit across the meridian was continually more southerly, while during the following summer its original position was restored by a motion northwards. Such an effect could not be explained as a result of parallax ; and eventually Bradley guessed it to be due to the gradual propagation of light. t Thus, let CA denote a ray of light, falling on the line BA \ and suppose that the eye of the observer is travelling along BA, with a velocity which is to the velocity of light as BA is to CA. Then the corpuscle of light, by which the object is discernible to the eye at A, would have been at C when the eye was at B. The tube of a telescope must therefore be pointed in the direction BO, in order to receive the rays from an object whose light is really propagated in the direction CA. The angle BCA measures the difference between the real and apparent positions .,„ of the object ; and it is evident from the figure that the sine of •Phfl. Timn». xmt (1728), p. 637. tBoemer, in a letter to Huygena of date 30th Dec., 1677, mention* a lutpeeted diijilaceinent of the apparent petition of a Mar, due to the motion of the earth at right angle* to the line of right. Of . Cumtfondmm I ■'")'','' or ct + - — j— w, as before. /* This formula was experimentally confirmed in 1851 by H. Fizeau,t who measured the displacement of interference- fringes formed by light which had passed through a column of moving water. * Phil. Mag. uriii (1846) p. 76. t Annalei de Ciiimio, Mi (18GB), p. 386. A!»o by A. A. Michel*™ and E. W. Hurler, Am- Journ. Science, xxxi (1886), p. 377. ihyGoogle 118 The Lumini/erotts Medium, The same result may easily be deduced from an experiment performed by Hoek * In this a beam of light was divided into two portions, one of which was made to pass through a tube of water AB and was then reflected at a mirror C, the light being afterwards allowed to return to A without passing through the water : while the other portion of the bifurcated beam was '" made to describe the same path in the reverse order, Le. passing through the water on its return (. journey from C instead of on the outward journey. On causing the two portions of the beam to inter- fere, Hoek found that no difference of phase was produced between them when the apparatus was oriented in the direction of the terrestrial motion. Let w denote the velocity of the earth, supposed to be directed from the tube towards the mirror. Let cju denote the velocity of light in the water at rest, and c/p +

- re, A„A, - rift If we write C,A,S, - t, and denote the total deviation of the wave-front by {„ we have AtI> - AJ3 - A^ cos 8, = re - rw cos S„ 3,Bl,ZEdhyG00gle 120 The Lnminiferous Medium, and therefore (neglecting second-order terms in w/c) ain A,S,D c - wcoaSi c w w B Denoting by 8 the value of S. when w is zero, we have Bin (i-B) c sin t e. Subtracting this equation from the preceding, we have 8 - 8, w sin 8 c Now the telescope by which the emergent wave-front B, D is received is itself being carried forward by the earth's motion; and we must therefore apply the usual correction for aberration in order to find the apparent direction of the emergent ray. But this correction is w sin B/c, and precisely counteracts the effect which has been calculated as due to the motion of the prism. So finally we see that the motion of the earth has no first-order " influence on the refraction of light from the stars. Fresnel inferred from his formula that if observations were made with a telescope filled with water, the aberration would be unaffected by the presence of the water — a result which was verified by Airy* in 1871. He showed, moreover, that the apparent positions of terrestrial objects, carried along with the observer, are not displaced by the earth's motion ; that experi- ments in refraction and interference are not influenced by any motion which is common to the source, apparatus, and observer ; and that light travels between given points of a moving material system by the path of least time. These predictions have also been confirmed by observation: Eespighifin 1861,andHoekJin 1868, experimenting with a telescope filled with water and a terrestrial source of light, found that no effect was produced on the phenomena of reflexion and refraction by altering the orienta- * Proo. Bo;. Soc., iz, p. 35. t Mem. Aocad. Sci. Bologna, ii, p. 279. J Art. Niwb., Ixxiii, p. IBS. 3,Bl,ZEdhyG00gk from Bradley to FresneL 121 tion of the apparatus relative to the direction of the earth's motion. E. Maacart* in 1872 discussed experimentally the question of the effect of motion of the Bource or recipient of light in all its bearings, and showed that the light of the sun and that derived from artificial sources are alike incapable of revealing by diffraction-phenomena the translatory motion of the earth. The greatest problem now confronting the investigators of light was to reconcile the facts of polarization with the principles of the wave-theory. Young had long been pondering over this, but had hitherto been baffled by it In 1816 he received a visit from Arago, who told him of a new experimental result which he and Fresnel had lately obtained! — namely, that two pencils of light, polarized in planes at right angles, do not interfere with each other under circumstances in which ordinary light shows interference-phenomena, but always give by their reunion the same intensity of light, whatever he their difference of path. Arago had not long left him when Young, reflecting on the new experiment, discovered the long-sought key to the myBtery : it consisted in the very alternative which Bernoulli had rejected eighty years before, of supposing that the vibrations of light are executed at right angles to the direction of propagation. Young's ideas were first embodied in a letter to AragoJ dated Jan. 12, 1817. " I have been reflecting," he wrote, " on the possibility of giving an imperfect explanation of the affection of light which constitutes polarization, without departing from the genuine doctrine of undulations. It is a principle in this theory, that all undulations are simply propagated through homogeneous mediums in concentric spherical surfaces like the • Ann. de l'Jicole Noeraale, (2) i, p. 167. I fit waanot publiibed until IH19, in Anniles d« Chimin, x; Freanel'e (Eurret, i, p. 609. By meuns of this result, Fresnel was able to give a complete explana- tion Of a clan of phenomena which Arngo had discovered in 1811, viz. that when polarised light El transmitted through thin plates of sulphate of lime or mica, and afterwards analysed by a prism of Iceland spar, beautiful complementary colours ire displayed. Young had shown that these effect! are due essentially to inter- ference, but had not made dear the part played by polarization. J Young's fFerit, i., p. 380. 3,Bl,ZEdhyG00gle 122 The Lumini/erous Medium, undulations of sound, consisting simply in the direct and retro- grade motions of the particles in the direction of the radius, with their concomitant condensation and rarefactions. And yet it is possible to explain in this theory a transverse vibration, propagated also in the direction of the radius, and with equal velocity, the motions of the particles being in a certain constant direction with respect to that radius ; and this is a polarization.." In an article on " Chromatics," which was written in September of the same year* for the supplement to the Encyclopaedia Britannica, he says :f " If we assume as a mathe- matical postulate, on the undulating theory, without attempting to demonstrate its physical foundation, that a transverse motion may be propagated in a direct line, we may derive from this assumption a tolerable illustration of the subdivision of polarized light by reflexion in an oblique plane," by " supposing the polar motion to be resolved " into two constituents, which fare differently at reflexion. In a further letter to Arago, dated April 29th, 1818, Young recurred to the subject of transverse vibrationB, comparing light to the undulations of a cord agitated by one of its extremities-! This letter was Bhown by Arago to Fresnel, who at once saw that it presented the true explanation of the non-interference of beams polarized in perpendicular planes, and that the latter effect could even be made the basis of a proof of the correctness of Young's hypothesis : for if the vibration of each beam be supposed resolved into three components, one along the ray and the other two at right angles to it, it is obvious from the Arago- Fresnel experiment that the components in the direction of the ray must vanish : in other words, that the vibrations which constitute light are executed in the wave-front. It must be remembered that the theory of the propagation of waves in an elastic solid was as yet unknown, and light was * Peacock's Lift of Young, p. 391. t Young's Work; i., p. 379. J This (oology bad been given by Huoke in a communication to the Royal Society on Feb. 16, 1671-2. But tbere seem* no nawm to suppose that Hook* appreciated the point now advanced by Young. 3,Bl,ZEdhyG00gle from Bradley to Fresnel. 123 BtSll always interpreted by the analogy with the vibrations of sound in air, for which the direction of vibration is the same as that of propagation. It was therefore necessary to give some justification for the new departure. With wonderful insight Fresnel indicated* the precise direction in which the theory of vibrations in ponderable bodies needed to be extended in order to allow of waves similar to those of light : " the geometers," he wrote, " who have discussed the vibrations of elastic fluids hitherto have taken account of no accelerating forces except those arising from the difference of condensation or dilatation between conse- cutive layers." He pointed out that if we alao suppose the medium to possess a rigidity, or power of resisting distortion, such as is manifested by all actual solid bodies, it will be capable of transverse vibration. The absence of longitudinal waves in the aether he accounted for by supposing that the forces which oppose condensation are far more powerful than those which oppose distortion, and that the velocity with which condensations are propagated is so great compared with the speed of the oscillations of light, that a practical equilibrium of pressure is maintained perpetually. The nature of ordinary non-polarized light was next discussed. " If then," Fresnel wrote,t " the polarization of a ray of light consists in this, that all its vibrations are executed in the same direction, it results from any hypothesis on the generation of light-waves, that a ray emanating from a single centre of dis- turbance will always be polarized in a definite plane at any instant. But an instant afterwards, the direction of the motion changes, and with it the plane of polarization ; and these variations follow each other as quickly as the perturbations of the vibrations of the luminous particle : so that even if we could •Annalee de Chimie, ivii (1821), p. ISO; (Event, i, p. 629. Young had already drawn attention to this point. " It U difficult," ha eay* in hi* Ltetun* on A'ttvrul Philosophy, ed. 1807, toI. i, p. 138, "to compare the lateral adheiion, or the force which restate the detriurfon of the parta of a solid, with any form of direct tobeiion. Thia force constitutes the rigidity or hardneaa of a aolid body, and ia wholly abaent from liquida." t lac. dt., p. 186. 3,Bl,ZEdhyG00gle 124 The Lumini/erous Medium, isolate the light of this particular particle from that of other Luminous particles, we should doubtless not recognize in it any appearance of polarization. If we consider now the effect pro- duced by the union of all the waves which emanate from the different points of a luminous body, we see that at each instant, at a definite point of the aether, the general resultant of all the motions which commingle there will have a determinate direction, but this direction will vary from one instant to the next. So direct light can be considered as the union, or more exactly as the rapid succession, of systems of waves polarized in all directions. According to this way of looking at the matter, the act of polarization consists not in creating these transverse motions, but in decomposing them in two invariable directions, and separating the components from each other; for then, in each of them, the oscillatory motions take place always in the same plane." He then proceeded to consider the relation of the direction of vibration to the plane of polarization. " Apply these ideas to double refraction, and regard a uniaxal crystal as an elastic medium in which the accelerating force which results from the displacement of a row of molecules perpendicular to the axis, relative to contiguous rows, is the same all round the axis ; while the displacements parallel to the axis produce accelerating forces of a different intensity, stronger if the crystal is " repulsive," and weaker if it is " attractive." The distinctive character of the rays which are ordinarily refracted being that of propagating themselves with the same velocity in all directions, we must admit that their oscillatory motions are executed at right angles to the plane drawn through these rays and the axis of the crystal; for then the displacements which they occasion, always taking place along directions perpendicular to the axis, will, by hypothesis, always give rise to the same accelerating forces. But, with the conventional meaning which is attached to the expression -plane of polarization, the plane of polarization of the ordinary rays is the plane through the axis: thus, t» q^pepsil- of polarized lights the 3,Bl,ZEdhyG00gle from Bradley to Fresnel. 125 oscillatory motion is executed at right angles to the plane of polarizatvm." This result afforded Fresnel a foothold in dealing with the problem which occupied the rest of hie life : henceforth hie aim was to base the theory of light on the dynamical properties of the luminiferous medium. The first topic which he attacked from this point of view was the propagation of light in crystalline bodies. Since Brewster's discovery that many crystals do not conform to the type to which Huygens' construction is applicable, the wave theory had to some extent lost credit in this region. Fresnel, now, by what was perhaps the most brilliant of all his efforts,* not only reconquered the lost territory, but added a new domain to science. He had, as he tells us himself, never believed the doctrine that in crystals there are two different luminiferous media, one to transmit the ordinary, and the other the extraordinary waves. The alternative to which he inclined was that the two velocities of propagation were really the two rootB of a quadratic equation, derivable in some way from the theory of a single aether. Could this equation be obtained, he was confident of finding the explanation, not only of double refraction, but also of the polarization by which it is always accompanied. The first step was to take the case of uniaxal crystals, which had been discussed by Huygens, and to see whether Huygens' sphere and spheroid could be replaced by, or made to depend on, a single surface.t Now a wave propagated in any direction through a uniaxal ■His flrat memoir on Doable Refraction m presented to the Aoademy on Not. 19th, 1821, but ha* not bean published except in liii collected work*: 4 , " 1. and if {I, m, n) denote the direction-cosines of the normal to the plane of the wave, we have therefore »»* = a*{m* + »*) + 5*P. But the quantities 1/v, and l/vt, as given by theae equations, are easily seen to be the lengths of the semi-axes of the ellipse in which the spheroid J*(y' 4 a1) + a'as* - 1 iB intersected by the plane Ix 4 -my + nz « 0 ; and thus the construction in terms of Huygens' sphere and spheroid can be replaced by one which depends only on a single surface, namely the spheroid Vif + 2s) 4 will therefore have the property that the square of its radius vector in any direction is proportional to the component in that direction of the elastic force due to a unit displacement in that direction : it is called the surface of elasticity. Consider now a displacement along one of the axes of the section on which the surface of elasticity ie intersected by the plane of the wave. It is easily seen that in this case the com- ponent of the elastic force at right angles to the displacement acts along the normal to the wave-front ; and Fresnel assumes that it will be without influence on the propagation of the vibrations, on the ground of hiB fundamental hypothesis that the vibrations of light are performed solely in the wave-front. This step is evidently open to criticism ; for in a dynamical theory everything should be deduced from the laws of motion without special assumptions. But granting his contention, it follows that such a displacement will retain its direction, and will be propagated as a plane-polarized wave with a definite velocity. Now, in order that a stretched cord may vibrate with unchanged period, when its tension is varied, its length must be increased proportionally to the square root of its tension ; and similarly the wave-length of a luminous vibration of given period is proportional to the Bquare root of the elastic force (per unit 3,Bl,ZEdhyG00gle from Bradley to Fresnel. 129 displacement), which urges the molecules of the medium parallel to the wave-front. Hence the velocity of propagation of a wave, measured at right angles to its front, is proportional to the square root of the component, along the direction of dis- placement, of the elastic force per unit displacement ; and the velocity of propagation of such a plane-polarized wave as we have considered is proportional to the radius vector of the surface of elasticity in the direction of displacement. Moreover, any displacement in the given wave-front can be resolved into two, which are respectively parallel to the two axes of the diametral section of the surface of elasticity by a plane parallel to this wave:front ; and it follows from what has been said that each of these component displacements will be propagated as an independent plane-polarized wave, the velocities of propagation being proportional to the axes of the section* and therefore inversely proportional to the axes of the section of the inverse surface of this with respect to the origin, which is the ellipsoid Cl fl *1 But this is precisely the result to which, as we have seen, Fresnel had been led by purely geometrical considerations ; and thus his geometrical conjecture could now be regarded as substantiated by a study of the dynamics of the medium. It is easy to determine the wave-Burface or locus at any instant— say, t - 1 — of a disturbance originated at some previous instant — say,(-0— at some particular point — say, the origin. For this wave-surface will evidently be the envelope of plane waves emitted from the origin at the instant t = 0 — that is, it will be the envelope of planes Ix + my + nz-v - 0, where the constants /, m, n, v are connected by the identical equation P + m* + n* - 1, * It is evident from Out that the opiie azu, or lines of tingle wave -velocity, along which then is no double refrution, will be perpendicular to the two circular section! of the surface of obuvtici t j . K 3,Bl,ZEdhyG00gle 130 The Luminiferom Medium, and by the relation previously found — namely, By the usual procedure for determining envelopes, it may be shown that the locus in question is the surface of the fourth degree _*_+_£_ + _*_-o «,r»-l h!*-1 w*-l ' which is called FresneFs wave-surface* It is a two-sheeted surface, as must evidently be the case from physical considerations. In uniaxal crystals, for which t, and c, are equal, it degenerates into the sphere f* = I/*». and the spheroid (rC + ti [f + S*J = 1. It is to these two surfaces that tangent-planeB are drawn in the construction given by Huygens for the ordinary and extraordinary refracted rays in Iceland spar. As Fresnel observed, exactly the same construction applies to biaxal crystals, when the two sheets of the wave-surface are substi- tuted for Huygens' sphere and spheroid. " The theory which I have adopted," says Fresnel at the end of this memorable paper, " and the simple constructions which I have deduced from it, have this remarkable character, that all the unknown quantities are determined together by the solution of the problem. We find at the Bame time the velocities of the ordinary ray and of the extraordinary ray, and their planes of polarization. Physicists who have studied attentively the laws of nature will feel that such simplicity and • Another construe lion for tie wavc-surfacs ii the following, which is due to MaeCullagb, Coll. Work; p. 1. Let the ellipsoid e^ + V' + V*-' be intersected by a plan take length! equal to th is the wave-surf aue . 3,Bl,ZEdhyG00gle from Bradtey to Fresnel. 131 such close relations between the different elements of the phenomenon ore conclusive in favour of the hypothesis on which they are based" The question as to the correctness of Fresnel's construction was discussed for many years afterwards. A striking conse- quence of it was pointed out in 1832 by William Rowan Hamilton (b. 1805, d. 1865), Royal Astronomer of Ireland, who remarked* that the surface defined by Fresnel's equation has four conical points, at each of which there is an infinite number of tangent planes ; consequently, a single ray, proceeding from a point within the crystal in the direction of one of these points, must be divided on emergence into an infinite number of rays, constituting a conical surface. Hamilton also showed that there are four planes, each of which touches the wave- surface in an infinite number of points, constituting a circle of contact : so that a corresponding ray incident externally should be divided within the crystal into an infinite number of refracted rays, again constituting a conical surface. These singular and unexpected consequences of the theory were shortly afterwards verified experimentally by Humphrey Lloyd,t and helped greatly to confirm belief in Fresnel's theory. It should, however, be observed that conical refraction only shows his form of the wave-surface to be correct in its general features, and is no test of its accuracy in all details. But it was shown experimentally by Stokes in 1872,+ Glazebrook in 1879.S and Hastings in 1887,1 1 that the construction of Huygens and Fresnel is certainly correct to a very high degree of approximation; and Fresnel's final formulae have since been regarded as unassailable. The dynamical substructure on which he based them is, as we have seen, open to objection ; • Tnna. Boy. Irish Acad., ivii (1888), p. 1. t Tram. Eoy. Irish Acad., xvii (1833), p. 146. Strictly speaking, tlie bright ecus which U usually observed ariset from rsya adjrumt to the singular ray : the Utter can, however, be observed, it* entablement by dispersion into the conical form causing it to appear dark. J Proo. B. 8., xx, p. **8. f Phil, Tram., olxxi, p. 431, f An. Jour. Sei. (3), xirr, p. 80. K 2 3,Bl,ZEdhyG00gle 132 The I.uminiferous Medium, but, as Stokes observed*: "If we reflect on the state of the subject as Fresnel found it, and as he left it, the wonder is, not that he failed to give a rigorous dynamical theory, but that a single mind was capable of effecting so much." In a second supplement to his first memoir on Double Refraction, presented to the Academy on November 26th, 1821,f Fresnel indicated the lines on which his theory might be extended so as to take account of dispersion. " The molecular groups, or the particles of bodies," he wrote, " may be separated by intervals which, though small, are certainly not altogether insensible relatively to the length of a wave." Such a coarse- grainedness of the medium would, as he foresaw, introduce into the equations terms by which dispersion might be explained ; indeed, the theory of dispersion which was afterwards given by Cauchy was actually based on this principle. It seems likely that, towards the clone of his life, Fresnel was contemplating a great memoir on dispersion^ which was never completed. Fresnel had reason at first to be pleased with the reception of his work on the optics of crystals : for in August, 1822, Laplace spoke highly of it in public j and when at the end of the year a seat in the Academy became vacant, he was encouraged to hope that the choice would fall on him. In this he was disappointed! Meanwhile his researches were steadily continued ; and in January, 1823, the very month of his rejection, he presented to the Academy a theory in which reflexion and refraotion|| are referred to the dynamical properties of the luminiferous media. • Brit. Assoc. Rep., 1862, p. 254. t aSuvrt; ii, p. *38. J Cf. the biography in (Euvrtt it Frttnal, i, p. xcvi. § Writing to Young in the spring of 1823, he says : " Toua oca memoires, que demieraniBnt j'ai preaentea coup aur coup a l'Academis des Sciences, ns in* en out p» Dependant ouvert la porta. Cast M. Dulong qui ■ fits noinmS pour lemplii la place vaoante dans la section de physique. . . Yous Toyet, Monsieur, que la theoria dec ondulationa ne m'a point poite bonheur : mail eela ne m'en d£goute pat : et je ma console de oe malheur en in' occupant d'optique avee une nouvells ardour." | The Mae. ma for some time believed to be lost, but waa ultimately found among the papers of Fourier, and printed in Mem, de 1'Acad. xi (1832), p. 393 : Of «*!•»», i, p. T6T. 3,Bl,ZEdhyG00gle from BradUy to Frestul. 133 As in his previous investigations, he assumes that the vibrations which constitute light are executed at right angles to the plane of polarization. He adopts Young's principle, that reflexion and refraction are doe to differences in the inertia of the aether in different material bodies, and supposes (as in hie memoir on Aberration) that the inertia is proportional to the inverse square of the velocity of propagation of light in the medium. The conditions which he proposes to satisfy at the interface between two media are that the displacements of the adjacent molecules, resolved parallel to this interface, shall be equal in the two media ; and that the energy of the reflected and refracted waves together shall be equal to that of the incident wave. On these assumptions the intensity of the reflected and refracted light may be obtained in the following way : — Consider first the case in which the incident light is polarized in the plane of incidence, so that the displacement is at right angles to the plane of incidence; let the amplitude of the displacement at a given point of the interface be / for the incident ray, g for the reflected ray, and h for the refracted ray. The quantities of energy propagated per second across unit cross-section of the incident, reflected, and refracted beams are proportional respectively to eipif> ctpi9*, ttpth1, where c,, c?, denote the velocities of light, and p,,p, the densities of aether, in the two media ; and the cross-sections of the beams which meet the interface in unit area are cos i, cos i, cos r respectively. The principle of conservation of energy therefore gives C,pi COS i ./* = dpi COS i . BizBdb¥Google from Bradley to Fresnel. 135 perpendicular to each other, the reflected light will be wholly polarized in the plane of reflexion. Frefenel'a investigation can scarcely be called a dynamical theory in the strict sense, as the qualities of the medium are not denned. His method was to work backwards from the known properties of light, in the hope of arriving at a mechanism to which they could be attributed ; he succeeded in accounting for the phenomena in terms of a few simple principles, but was not able to specify an aether which would in turn account for these principles. The " displacement " of Fresnel could not be a displacement in an elastic solid of the usual type, since its normal component is not continuous across the interface between two media.* The theory of ordinary reflexion was completed by a dis- cussion of the case in which light is reflected totally. This had formed the subject of some of Fresnel 's experimental researches several years before ; and in two papersf presented to the Academy in November, 1817, and January, 1818, he had shown that light polarized in any plane inclined to the plane of reflexion is partly "depolarized" by total reflexion, and that this is dne to differences of phase which are introduced between the components polarized in and perpendicular to the plane of reflexion, " When the reflexion is total," he said, " rays polarized in the plane of reflexion are reflected nearer the surface of the glass than those polarized at right angles to the same plane, so that there is a difference in the paths described." This change of phase he now deduced from the formulae already obtained for ordinary reflexion. Considering light polarized in the plane of reflexion, the ratio of the amplitudes of the reflected and incident light is, as we have seen, »in(f-r), sin (i + r) ' when the sine of the angle of incidence is greater than jii/fii, * Fresnel'* theory of reflexion can, however, he reconciled with the electro- magnetic theory of light, by identifying his " displacement " with the electric force. t (Euvra dt Fruntl, i., pp. 441, 487. 3,Bl,ZEdhyG00gle 136 The Lumini/erous Medium. so that total reflexion takes place, this ratio may be written in the form where 0 denotes a real quantity denned by the equation .("''8i°','-.'"')'. fl, COB I Freenel interpreted this expression to mean that the amplitude of the reflected light is equal to that of the incident, but that the two waves differ in phase by an amount 8. The case of light polarized at right angles to the plane of reflexion may be treated in the same way, and the resulting formulae are completely confirmed by experiment. A few months after the memoir on reflexion had been presented, Fresnel was elected to a seat in the Academy ; and during the rest of his short life honours came to him both from France and abroad. In 1827 the Royal Society awarded him the Bumford medal; but Arago, to whom Young had confided the mission of conveying the medal, found him dying ; and eight days afterwards he breathed bis last By the genius of Young and Fresnel the wave-theory of light was established in a position which has since remained unquestioned ; and it seemed almost a work of supererogation when, in 1850, Foucault* and Fizeau,t carrying out a plan long before imagined by Arago, directly measured the velocity of light in air and in water, and found that on the question so long debated between the rival schools the adherents of the undulatory theory had been in the right. * Compte* Rendu*, xxx (I860), p. 551. t Ibid., p. 562. 3,Bl,ZEdhyG00gle ( 137 ) CHAPTER V. THB ASTHEB AS AN ELASTIC SOLID. "Whek Young and Freenel put forward the view that the vibrations of light are performed at right angles to its direction of propagation, they at the same time pointed out that this peculiarity might be explained by making a new hypothesis regarding the nature of the luminiferous medium ; namely, that it possesses the power of resisting attempts to distort its shape. It is by the possession of such a power that solid bodies are distinguished from fluids, which offer no resistance to distortion; the idea of Young and Fresnel may therefore be expressed by the simple statement that the aether behaves as an elastic solid. ~ After the death of Fresnel this conception was developed in a brilliant series of memoirs to which our attention must now be directed. The elastic-solid theory meets with one obvious difficulty at the outset If the aether has the qualities of a solid, how is it that the planets in their orbital motions are able to journey through it at immense speeds without encountering any perceptible resistance ? This objection was first satisfactorily answered by Sir George Gabriel Stokes* (b. 1819, d. 1903), who remarked that such substances as pitch and shoemaker's wax, though so rigid as to be capable of elastic vibration, are yet sufficiently plastic to permit other bodies to pass slowly through them. The aether, he suggested, may have this combination of qualities in an extreme degree, behaving like an elastic solid for vibrations so rapid as those of light, but yielding like a fluid to the much slower progressive motions of the planets. Stokes's explanation harmonizes in a curious way with Fresnel's hypothesis that the velocity of longitudinal waves in • Tram. Camb. Phil. Soc, viii, p. 287 (1845). 3,Bl,ZEdhyG00gle 138 The Aether as an Elastic Solid. the aether is indefinitely great compared with that of the transverse waves ; for it is found by experiment with actual substances that the ratio of the velocity of propagation of longitudinal waves to that of transverse waves increases rapidly as the medium becomes softer and more plastic. In attempting to set forth a parallel between light and the vibrations of an elastic substance, the investigator is compelled more than once to make a choice between alternatives. He may, for instance, suppose that the vibrations of the aether are executed either parallel to the plane of polarization of the light or at right angles to it ; and he may suppose that the different refractive powers of different media are due either to differences in the inertia of the aether within the media, or to differences in its power of resisting distortion, or to both these causes combined. There are, moreover, several distinct methods for avoiding the difficulties caused by the presence of longitudinal vibrations ; and as, alas ! we shall see, a further source of diversity is to be found in that liability to error from which no man is free. It is therefore not surprising that the list of - elastic-solid theories is a long one. At the time when the transversality of light was dis- covered, no general method had 'been developed for investi- gating mathematically the properties of elastic bodies; but under the stimulus of Fresnel's discoveries, some of the best intellects of the age were attracted to the subject. The volume of Memoirs of the Academy which contains Fresnel's theory of crystal -optics contains also a memoir by Claud Louis Marie Henri Navier" (b. 1785, d. 1836), at tliat time Professor of Mechanics in Paris, in which the correct equations of vibratory motion for a particular type of elastic solid were for the first time given. Navier supposed the medium to be ultimately constituted of an immense numl>er of particles, which act on each other with forces directed along the lines joining them, and depending on their distances apart ; and showed that if e denote . 376. The memoir was presented in 1821, vA 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. 139 the (vector) displacement of the particle whose undisturbed position is {x, y, z), and if p denote the density of the medium, the equation of motion is P 5^ " ~ ***' ^ca^ *^v e ~ n cur* cur' *' where n denotes a constant which measures the rigidity, 01 power of resisting distortion, of the medium. All such elastic properties of the body as the velocity of propagation of waves in it must evidently depend on the ratio njp. Among the referees of one of Navier's papers was Augustine Louis Cauohy (J. 1789, d. 1857), one of the greatest analysts of the nineteenth century* who, becoming interested in the question, published in 1828f a discussion of it from an entirely different point of view. Instead of assuming, as Navier had done, that the medium is an aggregate of point-centres of force, and thus involving himself in doubtful molecular hypotheses, he devised a method of directly studying the elastic properties of matter in bulk, and by its means showed that the vibrations of an isotropic solid are determined by the equation p — - - ( k + ^Til'grad div e - n curl curl •; here n denotes, as before, the constant of rigidity; and the constant k, which is called the modulits of compression,^ denotes the ratio of a pressure to the cubical compression produced by it Cauchy's equation evidently differs from Navier's in that " Hamiltou's opinion, written in 1S33, is worth repenting : " The principal theories of algebraical analysis (under which I include Calculi) require to be entire]}- remodelled; and Cauchy has done much already for this great object. Poiawn also ha* done much ; but he doe* not seem to me to have nearly so logical a mind as Cauchy, great as his talents and clearness arc ; and both are in my judgment very fat inferior to Fourier, whom I place at the head of the French School of Mathematical Philosophy, even abose Lagrnnge and Laplace, tbough I rank their talents above those of Caucby and Poisson." (Life of Sir W. S. BMtailtox, ii, p. 68.) t Cauchy, Extreutt dt Mathimatiquit hi, p. ISO (1828). I This notation was introduced at a later period, but is used here in order to iroid subsequent changes. dhyGoogle 140 The Aether as an Elastic Solid. two constants, k and n, appear instead of one. The reason for this is that a body constituted from point-centres of force in Navier's fashion has its moduli of rigidity and compression connected by the relation* Actual bodies do not necessarily obey this condition ; eg. for india-rubber, k is much larger than ^ n ;t and there seems to be no reason why we should impose it on the aether. In the same year PoissonJ succeeded in solving the diffe- rential equation which had thus been shown to determine the wave-motions possible in an elastic solid. The solution, which is both simple and elegant, may be derived as follows : — Let the displacement vector • be resolved into two components, of which one e is circuital, or satisfies the condition div c - 0, while the other b is irrotational, or satisfies the condition curl b - 0. The equation takes the form * In order to construct a body whose elastic properties are not limited by thii equation, William John Macquorn Bankine (*. 1820, d. 1872) considered a con- tinuous fluid in which a number of point-centres of force are situated : the fluid is supposed to be partially condensed round these centre*, the elastic atmosphere of each nucleus being retained round it by attraction. An additional volume-elasticity due to the fluid it thus acquired ; and no relation between i and n it now necessary. Cf. Bankine'a MitetUaneoui Seimtifii Paptri, pp. SI *qq. Sir William Thomson (Lord Kelvin), in 1889, formed a solid not obeying Navier's condition by using pairs or dissimilar atoms. Cf. Thomson'! Poprrt, iii, p. 395. Cf. also Baitimore Lecturti, pp. 133 sqq. t It may, however, be objected that india-rubber and other bodies which fail to fulfil Navier's relation are not true solids. On this historic controversy, cf. Todhunter and Pearson's Hitters of ElaitiHly, i, p. 490. i Mem. de l'Aced., viii (1828), p. 623. Poiseon takes the equation in the restricted form given by Navier ; but this does not affect the question of wave- propagation. 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. 141 The terms which involve b and those which involve o muat be separately zero, since they represent respectively the irrota- tioiial and the circuital parts of the equation. Thus, o satisfies the pair of equations 3*0 while b is to be determined from /,P"(*+i")7'b' curlb"°- A particular solution of the equations for o is easily seen to be ex - A sinAfa- t /-), cr ~ Ban.\(z- t /-), c, - 0, which represents a transverse plane wave propagated with velocity \/(n/p). It can be shown that the general solution of the differential equations for c is formed of such waves as this, travelling in all directions, superposed on each other A particular solution of the equations for b is bx =0, b, - 0, ft. - Ctan\(* - *J*±i*Y which represents a longitudinal wave propagated with velocity the general solution of the differential equation for b is formed by the superposition of euch waves as this, travelling in all directions. Poisson thus discovered that the waves in an elastic solid are of two kinds : those in c are transverse, and are propagated with velocity («//>)* ; while those in b are longitudinal, and are propagated with velocity \(k+$n)fp\K The latter are* waves of dilatation and condensation, like sound-waves ; in the o-waves, on the other hand, the medium is not dilated or condensed, but. •Cf. Stokee, "On the Dyuamiual Problem of Diffraction," Cunb. Phil. Tim, U (1MB). 3,Bl,ZEdhyG00gle 142 The Aether as an Elastic Solid. only distorted in a manner consistent with the preservation of a constant density.* The researches which have been mentioned hitherto have all been concerned with isotropic bodies. Cauchy in 1828f extended the equations to the case of crystalline substances. This, however, he accomplished only by reverting to Navier*a plan of conceiving an elastic body as a cluster of particles which attract each other with forces depending on their distances apart ; the aelotropy he accounted for by supposing the particles to be packed more closely in some directions than in others. The general equations thus obtained for the vibrations of an elastic solid contain twenty-one constants ; six of these depend on the initial Btress, bo that if the body is initially without stress, only fifteen constants are involved. If, retaining the initial stress, the medium is supposed to be symmetrical with respect to three mutually orthogonal planes, the twenty-one constants reduce to nine, and the equations which determine the vibrations may be written in the form* g-^+fl^ + fr + J^+to + i)^ + 2 and two similar equations. The three constants 0, S, I re- present the stresses across planes parallel to the coordinate planes in the undisturbed state of the aether. § * It may easily be shown that any disturbance, in either isotropic or crystalline media, for which the direction of vibration of the molecules lies in the wave-front or surface of constant phase, must, satisfy the equation div • — 0, where e denotes the displacement ; if, on the other hand, the direction of vibration of the molecules is perpendicular to the wave-front, the disturbance must satisfy the equation curl e - 0. These results were proved by M. O'Brien, Trans. Camh. Phil. 800., 18*2. t Exercicen di Math., iii (1828), p. 1S8. 1 These are substantially equations (68) on page 208 of the third volume of the Sxtreiat. $ O, H, I are tensions when they are positive, and pressures when they ue negative. dhyGoogle The Aether as an Elastic Solid. 1 43 On the basis of these equations, Cauchy worked out a theory of light, of which an instalment relating to crystal-optics was presented to the Academy in 1830* Its characteristic features will now be sketched. By substitution in the equations last given, it is found that when the wave-front of the vibration is parallel to the plane of yz, the velocity of propagation must be (A + (?)i if the vibration takes place parallel to the axis of y, and (g-\-G)l if it takes place parallel to the axis of z. Similarly when the wave-front is parallel to the plane of zx, the velocity must be (A + E~fi if the vibration is parallel to the axis of x, and (/+ J7)l if it is parallel €o the axis of z; and when the wave-front is parallel to the plane of xy, the velocity must be {g + /)* if the vibration is parallel to the axis of x, and (/+/)* if it is parallel to the axis of y. Now it is known from experiment that the velocity of a ray polarized parallel to one of the planes in question is the same, whether its direction of propagation is along one or the other of the axes in that plane : so, if we assume that the vibrations which constitute light are executed parallel to the plane of polarization, we must have /+J3"=/+7, 5 + 7 = ^+6?, A + if-A + G; or, G-S-I. This is the assumption made in the memoir of 1830 : the theory based on it is generally known as Caucky's FHrst Theory ;f the equilibrium pressures G, B, I, being all equal, are taken to be zero. If, on the other hand, we make the alternative assumption that the vibrations of the aether are executed at right angles to the plane of polarization, we must have h + B=g + I, f+r-k+G, o + G-f+H; • Mem. de 1' Acad., z, p. 293. In the previnua year (Mem. de l'Acad., ix, p. 114) Cauchy had iUted that tlio equations of elasticity lead in the caM of uniaxal cryitala to a wave-wirface, of which two sheets are a sphere and spheroid aa in Huygena' theory. f The equations and remit* of Cauchy'e First Theory of crystal -optic* were independently obtained shortly afterward* by Franz Ernst Neumann (*. 1798, d. 1895) : of. Ann. d. Fbys. ni (1632), p. 41B, reprinted at No. 76 of Oitwald'i Klmilur dtr eiaklen Wintmehafttn, with notea by A. Wangerin. 3,Bl,ZEdhyG00gle 144 The Aether as an Elastic Solid. the theory baaed on this supposition is known as Cauehy's Second Theory : it was published in 1836* In both theories, Cauchy imposes the condition that the section of two of the sheets of the wave-surface made by any one of the coordinate planes is to be formed of a circle and an ellipse, as in Freanel's theory ; this yields the three conditions 36c -f(b + c +/) ; 3ca - g(c + a + g) ; 3ab = h(a + b + h). Thus in the first theory we have these together with the equations (3-0, H-Q, 7-0, which express the condition that the undisturbed state of the aether is unstressed ; and the aethereal vibrations are executed parallel to the plane of polarization. In the second theory we have the three first equations, together with f-G-h-I-g-ff; and the plane of polarization is interpreted to be the plane at right angles to the direction of vibration of the aether. Either of Cauehy's theories accounts tolerably well for the phenomena of crystal-optics ; but the wave-surface (or rather the two sheets of it which correspond to nearly transverse waves) is not exactly Fresnel's. In both theories the existence of a third wave, formed of nearly longitudinal vibrations, is a formidable difficulty. Cauchy himself anticipated that the existence of these vibrations would ultimately be demonstrated by experiment, and in one placef oonjectured that they might be of a calorific nature. A further objection to Cauehy's theories ia that the relations between the constants do not appear to admit of any simple physical interpretation, being evidently assumed for the sole purpose of forcing the formulae into some degree of conformity with the results of experiment. And further difficulties will appear when we proceed subse- quently to compare the properties which are assigned to the aether in crystal-optics with those which must be postulated in order to account for reflexion and refraction. * Comptt* Eendua, ii (18S6), p. 311 : Mfin. da l'Acad. iviii (1839), p. 15S. t Mem. do 1'Acsd. Xfiii, p. 161. 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. 145 To the latter problem Cauchy soon addressed himself, his investigations being in fact published* in the same year (1830) as the first of his theories of cryBtal-optics, At the outset of any work on refraction, it is necessary to assign a cause for the existence of refractive indices, i.e. for the variation in the velocity of light from one body to another. Huygens, as we have seen, suggested that transparent bodies consist of hard particles which interact with the aethereal matter, modifying its elasticity. Cauchy in his earlier papersf followed this lead more or less closely, assuming that the density p of the aether is the same in all media, but that its rigidity n varies from one medium to another. Let the axis of x be taken at right angles to the surface of separation of the media, and the axis of 2 parallel to the inter- section of this interface with the incident wave-front; and suppose, first, that the incident vibration is executed at right angles to the plane of incidence, so that it may be represented bv - /(- x cos i - y sin i + P t\ where i denotes the angle of incidence ; the reflected wave may be represented by i + ^' and the refracted wave by e, = /, ( - x cos r - y sin r + I— t\ where r denotes the angle of refraction, and n' the rigidity of the second medium. To obtain the conditions satisfied at the reflecting surface, Cauchy assumed (without assigning reasons) that the x- and ^-components of the stress across the $y-plane are equal in • Bull, dee Sciences Math. xir. (1830), p. S. t As will appear, his views on this subject subsequently changed. 3,Bl,ZEdhyG00gle 146 The Aether as an Elastic Solid. the media on either side the interface. This implies in the present case that the quantities de, , 8e, n — and n — etc 3y are to be continuous acrosB the interface : so we have naosi'. (f-F') = n'cosr.f,; ttsini .(f + F )-»' Binr./,. Eliminating /'i, we have F sin (r - t) / " Bin (r + i) Now this is Fresnel'B sine-law for the ratio of the intensity of the reflected ray to that of the incident ray ; and it is known that the light to which it applies ia that which is polarized parallel to the plane of incidence. Thus Cauchy was driven to the conclusion that, in order to satisfy the known facte of reflexion and refraction, the vibrations of the aether must be supposed executed at right angles to the plane of polarization of the light. The case of a vibration performed in the plane of incidence he discussed in the same way. It was found that Fresnel'B tangent-law could be obtained by assuming that e, and the normal pressure across the interface have equal values in the two contiguous media. The theory thus advanced waa encumbered with many diffi- culties. In the first place, the identification of the plane of polarization with the plane at right angles to the direction of vibration was contrary to the only theory of crystal-optics which Cauchy had as yet published. In the second place, no reasons were given for the choice of the conditions at the interface. Cauchy's motive in selecting these particular conditions was evidently to secure the fulfilment of Fresnel's Bine-law and tangent-law ; but the results are inconsistent with the true boundary-conditionB, which were given later by Green. It is probable that the results of the theory of reflexion had much to do with the decision, which Cauchy now made,* to •CompteB Bendua, ii. (1830), p. 341. 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. 147 reject the first theory of crystal-optics in favour of the second. After 1836 he consistently adhered to the view that the vibra- tions of the aether are performed at right angles to the plane of polarization. In that year he made another attempt to frame a satisfactory theory of reflexion * based on the assumption just mentioned, and on the following boundary-conditions: — At the interface between two media curl e is to be continuous, and (taking the axis of x normal to the interface) 3cz/dx is also to be continuous. Again we find no very satisfactory reasons assigned for the choice of the boundary-conditions ; and as the continuity of e itself across the interface is not included amongst the conditions chosen/they are obviously open to criticism ; but they lead to Fresnel's sine- and tangent-equations, which correctly express the actual behaviour of light, t Cauchy remarkB that in order to justify them it is necessary to abandon the assumption of his earlier theory, that the density of the aether is the same in all material bodies. It may be remarked that neither in this nor in Cauchy's earlier theory of reflexion is any trouble caused by the appear- ance of longitudinal waves when a transverse wave is reflected, for the simple reason that he assumes the boundary-conditions to lx! only four in number ; and these can all be satisfied without the necessity for introducing any but transverse vibrations. These features bring out the weakness of Cauchy's method of - attacking the problem. His object was to derive the properties of light from a theory of the vibrations of elastic solids. At the * outset he had already in his possession the differential equations of motion of the Bolid, which were to be his Btarting-point, and the equations of Fresnel, which were to be his goaL It only "Coinptea Bendns, ii. (1836), p. 341 : " Memoire «ur ladiapertion delalumilre " (AWwbki tstrtitn ■ matter of fact, satisfied by the electric force in toe electro- magnetic theory of light. The continuity of ■curt e if eqniTaJent to the continuity of the magnetic vector acroai the interface, and the continuity of ~de,ftiz leadj to the ma equation aa the continuity of the component of electric force in the direction of the intersection of the interface with the plane of incidence. D,Bl,ZEdhyG00gle 1 48 The Aether as an Elastic Solid. remained to supply the boundary -co nditione at an interface, which are required in the discussion of reflexion, and the relations between the elastic constants of the solid, which are required in the optics of crystals. Cauchy seems to have con- sidered the question from the purely analytical point of view. Given certain differential equations, what supplementary con- ditions must be adjoined to them in order to produce a given analytical result ? The problem when stated in this form admits of more than one solution ; and hence it is not surprising that within the space of ten years the great French mathe- matician produced two distinct theories of crystal-optics and three distinct theories of reflexion* almost all yielding correct or nearly correct final formulae, and yet mostly irreconcilable with each other, and involving incorrect boundary-conditions and improbable relations between elastic constants. Cauchy's theories, then, resemble Fresuel s in postulating types of elastic Bolid which do not exist, and for whose assumed properties no dynamical justification is offered. The same objection applies, though in a loss degree, to the original form of a theory of reflexion and refraction which was discovered about this timet almost simultaneously by James MacCullagh (b. 1809, d. 1847), of Trinity College, Dublin, and Franz Neumann (J. 1798, d. 1895), of Kbnigsberg. To these authors is due the merit of having extended the laws of reflexion to crystalline media; but the principles of the theory were originally derived in connexion with the simpler case of isotropic media, to which our attention will for the present he confined. ' One yet remains to be mentioned. + The outlinns of the theory were published by MacCullagb, in Brit. Ateoc Bop. 1835 ; and hie results were given in Phil. Mug. i (Jan., 1837), and in Proc. Royal Irish Acad, iviii. (Jan., 1837). Neumann's memoir waa presented to the Berlin Academy towards the end of 1836, and published in 1837 in Abh. Berl. Ak. Buadem Jahre 1835, Math. Klaate, p. 1. So far aa publication ie concerned, the priority would seem to belong to MacCuUagh; but there are reoaona for believing that (he priority of discovery really reata with Neumann, who bad arrived at his equations a year before they were communicated to the Berlin Academy. dhyGoogle . The Aether as an Elastic Solid. 140 MacCullagh and Neumann felt that the great objection to Fresnel's theory of reflexion was its failure to provide for the continuity of the normal component of displacement, at the interface between two media ; it is obvious that a discontinuity in this component could not exist in any true elastic-solid theory, since it would imply that the two media do not remain in contact. Accordingly, they made it a fundamental con- dition that all three componente of the displacement must be coutinuous- at the interface, and found that the sine-law and tangent-law can be reconciled with this condition only by supposing that the aether-vibrations are parallel to the plane of polarization : which supposition they accordingly adopted. In place of the remaining three true boundary- conditions, however, they used only a single equation, derived by assuming that transverse incident waves give rise only to transverse reflected and refracted waves, and that the conservation of energy holds for these — i.e. that the masses of aether put in motion, multiplied by the squares of the amplitudes of vibration, are the same before and after incidence. This is, of course, the same device as had been used previously by Fresnel; it must, however, be remarked that the principle is unsound as applied to an ordinary elastic solid; for in such a body the refracted and reflected energy would in part be carried away by longitudinal waves. In order to obtain the sine and tangent laws, MacCullagh and Neumann found it necessary to assume that the inertia of the luminiferous medium is everywhere the same, and that the differences in behaviour of this medium in different substances are due to differences in its elasticity. The two laws may then be deduced in much the same way as in the previous investigations of Fresnel and Gauchy. Although to insist on continuity of displacement at the interface was a decided advance, the theory of MacCullagh and Neumann scarcely showed as yet much superiority over the quasi mechanical theories of their predecessors. Indeed, MacCullagh himself expressly disavowed any claim to regard 3,Bl,ZEdhyG00gle 150 The Aether as an Elastic Solid. his theory, in the form to which it had then been brought, as a final explanation of the properties of light. " If we are asked," he wrote, " what reasons can be assigned for the hypotheses on Which the preceding theory is founded, we are far from being able to give a satisfactory answer. We are obliged to confess that, with the exception of the law of vis viva, the hypotheses are nothing more than fortunate conjectures. These conjectures are very probably right, since they have led to elegant laws which are fully borne out by experiments ; but this is all we can assert respecting them. We cannot attempt to deduce them from first principles ; because, in the theory of light, such principles are still to be sought for. It is certain, indeed, that light is produced by undulations, propagated, with transversal vibrations, through a highly elastic aether ; but the constitution of this aether, and the laws of its connexion (if it has any connexion) with the particles of bodies, are utterly unknown." The needful reformation of the elastic-solid theory of reflexion was effected by Green, in a paper* read to the Cambridge Philosophical Society in December, 1837. Green, though inferior to Cauchy as an analyst, was his superior in physical insight ; instead of designing boundary-equations for the express purpose of yielding Freenel's sine and tangent formulae, he set to work to determine the conditions which are actually satisfied at the interfaces of real elastic solids. These he obtained by means of general dynamical principles. In an isotropic medium which is strained, the potential energy per unit volume due to the state of stress is _ If, 4 \ /3e, fc„ &,\' 1 l/de. fe-V fdem 3e,\* where e denotes the displacement, and k and n denote the two * Tram. C»mb. Phil. Soo., 1838 ; Green's Math. Paperi, p. its, 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. 151 elastic constants already introduced; by substituting this value of

S--l£ + -n] grad div e - n curl curl e ; ' (where p denotes the density), the equation of motion may be deduced. But this method does more than merely furnish the equation of motion K^Tijgrad div e + »Ve, which had already been obtained by Cauchy ; for it also yields the boundary-conditions which must be satisfied at the interface between two elastic media in contact ; these are, as might be guessed by physical intuition, that the three components of the displacement* and the three components of stress across the interface are to be equal in the two media. If the axis of x be taken normal to the interface, the latter three quantities are (.-J.)*.-*. ,(M), -a „£♦*). The correct boundary-conditions being thus obtained, it was a simple matter to discuss the reflexion and refraction of an incident wave by the procedure of Fresnel and Cauchy. The result found by Green was that if the vibration of the aethereal molecules is executed at right angles to the plane of incidence, the intensity of the reflected light obeys Fresnel 'b sine-law, pro- vided the rigidity n is assumed to be the same for all media, but the inertia p to vary from one medium to another. Since the sine-law is known to be true for light polarized in the plane of incidence, Green's conclusion confirmed the hypotheses of * The** Aral three conditions ire of courae not dytifcmical but geometric*!. 3,Bl,ZEdhyG00gle 152 The Aether as an Elastic Solid. Fresnel, that the vibrations are executed at right angles to the plane of polarization, and that the optical differences between media are due to the different densities of aether within them. It now remained for Green to discuss the case in which the incident light is polarized at right angles to the plane of inci- dence, so that the motion of the aethereal particles is parallel to the intersection of the plane of incidence with the front of the wave. In this case it ia impossible to satisfy all the six boundary-conditions without assuming that longitudinal vibra- tions are generated by the act of reflexion. Taking the plane of incidence to be the plane of yz, and the interface to be the plane of xy, the incident wave may be represented by the equations 5 3 Cy - A £■/(< + h + my) ; e„ - - A —f(t + lz + my) ; oz oy where, if i denote the angle of incidence, we have I = J— cos i, m = - ./£- sin i. There will be a transverse reflected wave, and a transverse refracted wave, where, since the velocity of transverse waves in the second medium is \/n//>i> we can determine /i from the equation (,- + »■-£; ft there will also be a longitudinal reflected wave, ey - D g-/(( - A* + my); «, - D ^f(t - Xz + my), 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid: 153 where A is determined by the equation kt + ;7l and a longitudinal refracted wave, t, - E -f(t + X& + my), where A, ie determined by *,■ + »•- 5^-- it, -/(M)'**(M)"^-&)'- The usual variational equation I' ^^ + ^i^ + Wt^dxdydzm-\\\S*dxd'A' 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. 163 then yields the differential equations of motion, namely : and two similar equations. These differ from C&uchy's fundamental equations in having greater generality: for Cauchy's medium was supposed to be built up of point-centres of force attracting each other according to some function of the distance ; and, as we have seen, there are limitations in this method of construction, which render it incompetent to represent the most general type of elastic solid. Cauchy's equations for crystalline media are, in fact, exactly analogous to the equations originally found by Navier for isotropic media, which contain only one elastic constant instead of two. The number of constants in the above equations still exceeds the three which are required to specify the properties of a biaxal crystal : and Green now proceeds to consider how the number may be reduced. The condition which he imposes for this purpose is that for two of the three waves whose front is parallel to a given plane, the vibration of the aethereal molecules shall be accurately in the plane of the wave : in other words, that two of the three waves shall be purely distortional, the remaining one being consequently a normal vibration. This condition gives five relations* which may be written : — /' - fi - 2/ / - ft - 2g; V - ? - 2A; where p denotes a new constant^ * A* Green showed, the hypotheaia of tianivnnality really involve* the exiitence of plane* of lymmatry, eothat it alone ii Capable of giving 14 relatione between the 21 conatant* : and 3 of the remaining 7 constant* may be removed bj change of axet, leaving only four. t It wm afterward* ghown by Barre de Saint- Tenant (*. 1797, d. 1883), Journal dt Math., vii (1868), p. 399, that if the initial atreuea be auppoaed to vanish, the conditiona which must be aitie&ed among the remaining nine conatant* M 2 3,Bl,ZEdhyG00gle 164 The Aether as an Elastic Solid. Thus the potential energy per unit volume may be written -)'■ At this point Green's two theoriofl of crystal-optics diverge from each other. According to the first theory, the initial j G, H, I are zero, so that . , lie, 9a, fr.\ ^m+w-'m^i^w-'m >H%+m- .3e. dfj) ". *i »>/> 9> k>ftfi *'i 'n urdei' >hat the wave-surface may be Frwnel't, an ths folio wing : — (3i -/)(&•-/) -{/+/•}' {3«-j){M -*) = [> + •)' 1 (Sa - A) (3* - A) = (A + A")1 l(3a - j) (3d- A) (3» -/) + (3— A) (3» -/) (3« - f ) - 2(/+/) (? +j0 (A + A'). There reduce to Green's relation.! when the additional equation b — t ii inrame-i . Saint-Tenant disputed the validity of Green's relations, assailing that they?ere compatible only with ieotropy. On this controversy cf. E. T. G-laiebrook, Brit. Aino. Report, 1886, p. 171, and Karl Pearson in Todhunter and Pearson's Sultry of EUiticity, ii, } 147. 3,Bl,zEdhyG00gle The Aether as an Elastic Solid. 165 This expression contains the correct number of constants, namely, four: three of them represent the optical constants of a biaxal crystal, and one (namely, p) represents the square of the velocity of propagation of longitudinal waves. It is found that the two sheets of the wave-surface which correspond to the two distortions! waves form a Fresnel's wave-surface, the third sheet, which corresponds to the longitudinal wave, being an ellipsoid. The directions of polarization and the wave-velocities of the distortional waves are identical with those assigned by FresneL provided it is assumed that the direction of vibration of the aether-particles is parallel to the plane of polarization ; but this last assumption is of course inconsistent with Green's theory of reflexion and refraction. In his Second Theory, Green, like Cauchy, used the condition that for the waves whose fronts are parallel to the coordinate planes, the wave- velocity depends only on the plane of polariza- tion, and not on the direction of propagation. He thus obtained the equations already found by Cauchy — G-f- H-g = I-k. The wave-surface in this case also is Fresnel's, provided it is assumed that the vibrations of the aether are executed at right angles to the plane of polarization. The principle which underlies the Second Theories of Green aud Cauchy is that the aether in a. crystal resembles an elastic solid which is unequally pressed or pulled in different directions by the unmoved ponderable matter. This idea appealed strongly to W. Thomson (Kelvin), who long afterwards developed it further* arriving at the following interesting result :— Let an incompressible solid, isotropic when unstrained, be such that its potential energy per unit volume is where q denotes its modulus of rigidity when unstrained, and •Proc.B.S.Edin.XT(18S7),p.21: Phil. Mag. xxy (1888) p. 116: Baltimer* Laturt, (ed. 1904), pp. 228-259. 3,Bl,ZEdhyG00gle 1 66 The Aether as an Elastic Solid. o*. 0*> 7*. denote the proportions in which lines parallel to the axes of strain are altered ; then if the solid be initially strained in a way defined by given values of a, /3, y, by forces applied to its surface, and if waves of distortion be superposed on this initial strain, the transmission of these waves will follow exactly the laws of Fresnel's theory of crystal-optics, the wave-surface being V-l £^_i ?*>.! 2 9$ There is some difficulty in picturing the manner in which the molecules of ponderable matter act upon the aether so as to produce the initial strain required by this theory. Lord Kelvin utilized* the suggestion to which we have already referred, namely, that the aether may pervade the atoms of matter so as to occupy space jointly with them, and that its interaction with them may consist in attractions and repulsions exercised throughout the regions interior to the atoms. These forces may be supposed to be so large in comparison with those called into play in free aether that the resistance to compres- sion may be overcome, and the aether may be (Bay) condensed in the central region of an isolated atom, and rarefied in its outer parts. A crystal may be supposed to consist of a group of spherical atoms in which neighbouring spheres overlap each other ; in the central regions of the spheres the aether will be condensed, and within the lens-shaped regions of overlapping it will he still more rarefied than in the outer parts of a solitary atom, while in the interstices between the atoms its density will be unaffected. In consequence of these rarefactions and condensations, the reaction of the aether on the atoms tends to draw inwards the outermost atoms of the group, which, however, will be maintained in position by repulsions between the atoms themselves; and thus we can account for the pull which, according to the present hypothesis, is exerted on the aether by the ponderable molecules of crystals. * Baltimore Liuturei (*d. 1904), p. 263. 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. 167 Analysis similar to that of Cauchy's and Green's Second Theory of crystal-optics may be applied to explain the doubly refracting property which is possessed by strained glass ; but in this case the formulae derived are found to conflict with the results of experiment. The discordance led Kelvin to doubt the truth of the whole theory. "After earnest and hopeful consideration of the stress theory of double refraction during fourteen years," he said,* "lam unable to see how it can give the true explanation either of the double refraction of natural crystals, or of double refraction induced in isotropic solids by the application of unequal pressures in different directions." It is impossible to avoid noticing throughout all Kelvin's work evidences of the deep impression which was made upon him by the writings of Green. The same may be said of Kelvin's friend and contemporary Stokes; and, indeed, it is no exaggeration to describe Green as the real founder of that " Cambridge school " of natural philosophers, of which Kelvin, Stokes, Lord Bayleigh, and Clerk Maxwell were the most illustrious members in the latter half of the nineteenth century, and which is now led by Sir Joseph Thomson and Sir Joseph Larmor. In order to understand the peculiar position occupied by Green, it is necessary to recall some- thing of the history of mathematical studies at Cambridge. The century which elapsed between the death of Newton and the scientific activity of Green was the darkest in the history of the University. It is true that Cavendish and Young were educated at Cambridge; but they, after taking undergraduate courses, removed to London. In the entire period the only natural philosopher of distinction who lived and taught at Cambridge was Miehell ; and for some reason which at this distance of time it is difficult to understand fully, MichelTs researches seem to have attracted little or no attention among his collegiate contemporaries and successors, * Baitimori Ledum (fit. 1904), p. 268. 3,Bl,ZEdhyG00gle 168 The Aether as an Elastic Solid. who silently acquiesced when his discoveries were attributed to others, and allowed his name to perish entirely from Cambridge tradition. A few years before Green published his first paper, a notable revival of mathematical learning swept over the University ; the fluxional symbolism, which Bince the time of Newton had isolated Cambridge from the continental schools, was abandoned in favour of the differential notation, and the works of the groat French analysts were introduced and eagerly read. Green undoubtedly received his own early inspiration from this source ; but in clearness of physical insight and conciseness of exposition he far excelled his masters ; and the slight volume of his collected papers has to this day a charm which is wanting to the voluminous writings of Cauchy and Poisson. It was natural that such an example should powerfully influence the youthful intellects of Stokes — who was an undergraduate when Green read his memoir on double refraction to the Cambridge Philosophical Society' — - and of William Thomson (Kelvin), who came into residence two years afterwards.* In Bpite of the advances which were made in the great memoirs of the year 1839, the fundamental question as to whether the aether-particlea vibrate parallel or at right angles to the plane of polarization was still unanswered. More light was thrown on this problem ten years later by Stokes's inves- tigation of Diffraction. t Stokes showed that on almost any conceivable hypothesis regarding the aether, a disturbance in which the vibrations are executed at right angles to the plane of diffraction must be transmitted round the edge of an opaque body with less diminution of intensity than a disturbance whose vibrations are executed parallel to that plane. It follows that when light, of which the vibrations are oblique to the plane of * It iu in the year Thomson took hit degree (1816) that he bought, and read with delight, the electrical memoir -which Given had published at Nottingham in 1828. t Tram. Camb. Phil. Soc., ix (1849), p. 1. Stokei'a Melk. and Phy: Ftptr; U, p. 218. 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. 169 diffraction, is so transmitted, the plane of vibration will be more nearly at right angles to the plane of diffraction in the diffracted than in the incident light. Stokes himself performed experi- ments to test the matter, using a grating in order to obtain strong light diffracted at a large angle, and found that when the plane of polarization of the incident light was oblique to the plane of diffraction, the plane of polarization of the diffracted light was more nearly parallel to the plane of diffraction. This result, which was afterwards confirmed by L. Lorenz * appeared to confirm decisively the hypothesis of Fresnel, that the vibra- tions of the aethereal particles are executed at right angles to the plane of polarization. Three years afterwards Stokes indicated! a second line of proof leading to the same conclusion. It had long been known that the blue light of the sky, which is due to the scattering of the sun's direct rays by small particles or molecules in the atmosphere, is partly polarized. The polarization is most marked when the light comes from a part of the sky distant 90° from the sun, in which case it must have been scattered in a direction perpendicular to that of the direct sunlight incident on the small particles; and the polarization is in the plane through the Bun. If, then, the axis of y by taken parallel to the light incident on a small particle at the origin, and the scattered light be observed along the axis of x, this scattered light is found to be polarized in the plane xy. Considering the matter from the dynamical point of view, we may suppose the material particla to possess so much inertia (compared to the aether) that it is " - practically at rest. Its motion relative to the aether, which is the cause of the disturbance it creates in the aether, will there- fore be in the same line as the incident aethereal vibration, but in the opposite direction. The disturbance must be transversal, and must therefore be zero in a polar direction and * Ana. d. Ph ji. rati (1880), p. 316. Phil. Mag. ui (1861), p. 321. t Phil. Tram., 1852, p. 463. Stokw'i Math, and Fhyi. Paptrt, iii, p. 287. Cf. the foot-note addad on p. 361 ot the Math, and Phyi. Papert. 3,Bl,ZEdhyG00gle 170 The Aether as an Elastic Solia. a maximum in an equatorial direction, its amplitude being, in fact, proportional to the Bine of the polar distance. The polar line must, by considerations of symmetry, be the line of the incident vibration. Thus we see that none of the light scattered in the ^-direction can come from that constituent of the incident- light which vibrates parallel to the x axis ; so the light observed in this direction must consist of vibrations parallel to the z-axis. But we have seen that the plane of polarization of the scattered light is the plane of xy ; and therefore the vibration is at right angles to the plane of polarization." The phenomena of diffraction and of polarization by scatter- ing thus agreed in confirming the result arrived at in Fresnel's and Green's theory of reflexion. The chief difficulty in accepting it arose in connexion with the optics of crystals. As we have seen, Green and Cauchy were unable to reconcile the hypothesis of aethereal vibrations at right angles to the plane of polariza- tion with the correct formulae of crystal-optics, at any rate bo long as the aether within crystals was supposed to be free from initial stress. The underlying reason for this can be readily seen. In a crystal, where the elasticity is different in different directions, the resistance to distortion depends solely on the orientation of the plane of distortion, which in the case of light is the plane through the directions of propagation and vibration. Now it is known that for light propagated parallel to one of the axes of elasticity of a crystal, the velocity of propagation depends only on the plane of polarization of the light, being the same whichever of the two axes lying in that plane is the direction of propagation. Comparing these results, we see that the plane of polarization must be the plane of distortion, and therefore the vibrations of the aether-particles must be executed parallel to the plane of polarization, t * The theory of polarization by snmil particles wnl afterwards investigated by Lord Bayleigh, Phil. Meg. xli (1871). t In Freansl'B theory of crystal -optica, in which the aether -Tib ration* are at right angles to the plane of polarization, the velocity of propagation depend! only on the direction of vibration, not on the plane through thin and the direction of tranemiasion. 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. 171 A way of escape from this conclusion suggested itself to Stokes,* and later to Eankinet and Lord Kayleigh.J What if the aether in a crystal, instead of having its elasticity different in > different directions, were to have its rigidity invariable and its inertia different in different directions ? This would bring the \ theory of crystal-optics into complete agreement with Fresnel's and Green's theory of reflexion, in which the optical differences between media are attributed to differences of inertia of the aether contained within them. The only difficulty lies in conceiving bow aelotropy of inertia can exist ; and all three writers overcame this obstacle by pointing out that a solid which is immersed in a fluid may have its effective inertia different in different directions. For instance, a coin immersed in water moves much more readily in its own plane than in the direction at right angles to this. Suppose then that twice the kinetic energy per uuit volume of the aether within a crystal is represented by the expression fW /&,Y /fc.V and that the potential energy per unit volume has the same value as in space void of ordinary matter. The aether is assumed to be incompressible, so that div e is zero : the potential energy per unit volume is therefore *"** \\dy + dzj + \dz + to) + [ox + dy) ~ 3y & _ . fade, .Sfj&vl 3* 3a; ~ &c dy)' where n denotes as usual the rigidity. * Stokes, in a Utter to Lord Hay! sigh, inserted in his Mimoir and Scitntijie CorrapondiHct, ii, p. 09, explains that the idea presented itself to him while he n> writing the paper on Fluid Motion which appeared in Trans. Gamh. Phil. See., nil (1843), p. 10S. He loggeated the a-are-iiuface to which thia theorj lead* in Brit. Aeeoe. Sep., 188!, p. 269. t Phil. Meg. (4), i (1861), p. 441. I Phil. Hag. (4), xli (1871), p. 619. dhyGoOgk 172 The Aether as an Elastic Solid. The variational equation of motion is -fllMlM*!)}^ where y denotes an undetermined function of (x, y, z) : the term in p being introduced on account of the kinematics! constraint expressed by the equation div e - 0. The equations of motion which result from this variational equation are &t, dp _. r #* , 1792, d. 1871) in 1820 to be associated with differences in the crystalline form of the specimens, the two types bearing the same relation to each other as a right-handed and left-handed helix respectively. FresnelJ and W. Thomson § proposed the term helical to denote the property of rotating the plane of polarization, exhibited by such bodies as quartz : the less appropriate term natural rotatory polarization ie, however, generally used.[| Biot showed that many liquid organic bodies, ag. turpentine and sugar solutions, possess the natural rotatory property : we might be led to infer the presence of a helical structure in the molecules of such substances ; and this inference is sap- ported by the study of their chemical constitution ; for they are invariably of the " mirror-image " or "enantiomorphous" type, in which one of the atoms (generally carbon) is asym- metrically linked to other atoms. The next advance in the subject was due to Fresnel,1F who showed that in naturally active bodies the velocity of propa- gation of circularly polarized light is different according as the polarization is right-handed or left-handed. From this property the rotation of the plane of polarization of a plane- polarized ray may be immediately deduced ; for the plane- polarized ray may be resolved into two rays circularly polarized in opposite senses, and these advance in phase by different * Mem. de l'lnatitut, 18 12, Part I, p. 218, iqq. ; Annals* do Chun., u (1818), p. 372; I {1819), p. 63. t Camb. Phil. StW. Twni. i, p. 43. J Mem. de l'lntt. vii, p. 73. $ Baltimore Ltctwrtt (ed. 1804}, p. 31. || The term rotatory may be applied with propriety to the property duoorefed .by Faraday, which will be discussed later. 1 Annalea de Chim. xxriii (1824), p. 147. 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. 175 amountB in passing through a given thickness of the substan.ce : I at any stage they may be recompounded into a plane-polarized ray, the azimuth of whose plane of polarization varies with the length of path traversed. It is readily seen from this that a ray of light incident on a crystal of quartz will in general bifurcate into two refracted rays, each of which will be elliptically polarized, i.e. will be capable of resolution into two plane-polarized components which differ in phase by a definite amount The directions of these refracted rays may be determined by Huygens' con- struction, provided the wave-surface is supposed to consist of a sphere and spheroid which do not touch. The first attempt to frame a theory of naturally active bodies was made by MacCullagh in 1836.* Suppose a plane wave of light to be propagated within a crystal of quartz. Let (x, y, z) denote the coordinates of a vibrating molecule, when the axis of a; is taken at right angles to the plane of the wave, and the axis of z at right angles to the axis of the crystal. Using Y and Z to denote the displacements parallel to the axes of y and z respectively at any time t, MacCullagh assumed that the differential equations which determine Y and ^are ffY .&Y #Z -&=* d*+,l& tfZ t&2 &Y bp " * cw* ~ p ac where ,< denotes a constant on which the natural rotatory property of the crystal depends. In order to avoid compli- cations arising from the ordinary crystalline properties of quartz, we shall suppose that the light is propagated parallel to the optic axis, so that we can take c, equal to e,. Assuming first that the beam is circularly polarized, let it be represented by Y= Asva—(lx-t), Z - i A cos — (to - t), • Tram. Royal Irish Acad., >rii. ; MacCullagh 'a Co!!, Werkt, p. 63. 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. mgmenon was studied Bhortly afterwards by Biot, poIarizuLiuii ■kV^alteration consists in a rotation of the p) Substituting ins*tes direction of propagation: the an^" "N^o the tbicknesB of the plate * 1 - d't*yyx§ot the wave-length. * -otation is from left Since l/l denotes the velocity of propagation, it is evident that the reciprocals of the velocities of propagation of a right-handed and left-handed beam differ by the quantity re,*' from which it is easily shown that the angle through which the plane of polarization of a plane-polarized beam rotates in unit length of path is If we neglect the variation of c, with the period of the light, this expression satisfies Biot'B law that the angle of rotation in unit length of path is proportional to the inverse Bquare of the wave-length. MacCullagh's investigation can be scarcely called a theory, for it amounts only to a reduction of the phenomena to empirical, though mathematical, laws ; but it was on this foundation that later workers built the theory which is now * The later developments of this theory will be discussed in a subsequent chapter; but mention may here be made of an attempt which was made in 1S66 by Carl Neumann, then a very young man, to provide a rational basis for MacCullagh's equations. Neumann showed that the equations may be derived from the hypothesis that the relative displacement of one aethereal particle with respect to another act* on tbe latter according to the same law as an element of an electric current acta on a magnetic pole. Cf. the preface to C. Neumann's Jtis magnitiiehi Drihimg drr Polaritatinmibiru dti lie/itii, Halle, 1863. 3,Bl,ZEdhyG00gle TkAemmmZma. k T tic Solid. 177 snwiuitemfwK aswBtr^atn* - -j? « -^< eloped the theory of light a: mv s^t tSnr bb * *-«ntM- = . - -evoted considerable attention to m.tlieimK^«k»»k>->^T*u' metals. Their researches in this hethoffHtamK. be reviewed. itL=r»i2Ti>n.~Er>-.^ing properties of metals are the power of acrTsuJofl.iE!i*:iefiecting light at all angles of incidence, which is rn' so well shown by the mirrors of reflecting telescopes, and the opacity, which causes a train of waves to be extinguished before it has proceeded many wave-lengths into a metallic medium. That these two attributes are connected appears probable from the fact that certain non-metallic bodies — e.g., aniline dyes — which strongly absorb the rays in certain parte of the spectrum, reflect those rays with almost metallic brilliance. A third quality in which metalB differ from transparent bodies, and which, as we shall see, is again closely related to the other two, is in regard to the polarization of the light reflected from them. This was first noticed by Mains ; and in 1830 Sir David Brewster* showed that plane-polarized light incident on a metallic surface remains polarized in the same plane after reflexion if its polarization is either parallel or perpendicular to the plane of reflexion, but that in other cases the reflected light is polarized elliptical! v. It was this discovery of Brewster's which suggested to the mathematicians a theory of metallic reflexion. For, as we have seen, elliptic polarization is obtained when plane-polarized light is totally reflected at the surface of a transparent body ; and this analogy between the effects of total reflexion and metallic reflexion led to the surmise that the latter pheno- menon might be treated in the same way as Fresnel had treated the former, namely, by introducing imaginary quantities into the formulae of ordinary reflexion. On these principles mathe- matical formulae were devised by MacCullaght and Cauchy^ 'Phil. Tram., 1830. + Proc. Boy. Iriih Acad., i (1836), p. 1; ii (1843), p. 378 : Tram. Roy. Irith Aod., XTiii (1837), p. 71 : MacCuUagh'e Cstt. Worki, pp. fi8, 132, 330. J Coincles Rendu*, til (183S), p. 953 ; viii (1639), pp. S63, 668, Stl ; xxvi (1848), p. 86. N 3,Bl,ZEdhyG00gle 178 The Aether as an Elastic Solid. To explain their method, we shall suppose the incident light to be polarized in the plane of incidence. According to Fresnel's sine-law, the amplitude of the light (polarized in this way) reflected from a transparent body is to the amplitude of the incident light in the ratio -. sin (i - t) sin (i + r)' where i denotes the angle of incidence and r is determined from the equation sin i = n ain r. MacGullagh and Cauchy assumed that these equations hold good also for reflexion at a metallic surface, provided the refractive index ft is replaced by a complex quantity ft = vfl -Ky/ - 1) say, where v and k are to be regarded as two constants characteristic of the metal. We have therefore tan t - tan r (p* - ain' t)* - cos t tan t + tan r (ji* - sin' t)i + cos t If then we write m» (1 - k y/^Vf - sin" i - IFfS^1, so that equations defining U and v are obtained by equating separately the real and the imaginary parts of this equation, we have fTe V - l + cos i and this may be written in the form j.«rri where y U* + co8*i - 2 U cos v c U* + eoa'i +2U cos v c _ 2l7co8tsinu 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. 17ft The quantities J and 8 are interpreted in the same way as in Fresnel's theory of total reflexion : that is, we take J to mean the ratio of the intensities of the reflected and incident light, while $ measures the change of phase experienced by the light in reflexion. The case of light polarized at right angles to the plane of incidence may be treated in the same way. When the incidence is perpendicular, U evidently reduces to v [ 1 + k1)*, and v reduces to - tan " * «. For silver at perpen- dicular incidence almost all the light is reflected, so J is nearly unity : this requires cos v to be small, and k to be very large. The extreme case in which k is indefinitely great but v indefinitely small, bo that the quasi-index of refraction is a pure imaginary, is generally known as the case of ideal silver. The physical significance of the two constants v and k was more or less distinctly indicated by Cauchy ; in fact, as the difference between metals and transparent bodies depends on the constant «, it is evident that k must in some way measure the opacity of the substance. This will be more clearly seen if we inquire how the elastic-solid theory of light can be extended so as to provide a physical basis for the formulae of MacCullagh and Cauchy* The sine-formula of Fresnel, which was the starting-point of our investigation of metallic reflexion, is a consequence of Green's elastic-solid theory : and the differences between Green's results and those which we have derived arise solely from the complex value which we have assumed for /a. We have therefore to modify Green's theory in such a way as to obtain a. complex value for the index of refraction. Take the plane of incidence as plane of xy, and the metallic surface as plane of yz. If the light is polarized in the plane of incidence, bo that the light-vector is parallel to the axis of z, the incident light may be taken to be a function of the argument ax + by + ct, • TbU ni done by Lord R*yleigh, Phil. May. iliii (1872), p. 821. N 2 3,Bl,ZEdhyG00gle 180 The Aether as an Elastic Solid. where !--(?y-* ;--ey- here i denotes the angle of incidence, /> the inertia of the aether, and n ita rigidity. Let the reflected light be a function of the argument axx + by + ct, where, in order to secure continuity at the boundary, b and c must have the same values as before. Since Green's formulae are to be still applicable, we must have where ain i - p Bin r, but ft has now a complex value. This equation may be written in the form «,* + j» . EC, n Let the complex value of ft* be written P the real part being written pijp in order to exhibit the analogy with Green's theory of transparent media : then we have „,. + »■- e^-^-'.iv/rT. n n But an equation of this kind must (as in Green's theory) represent the condition to be satisfied in order that the quantity may satisfy the differential equation of motion of the aether ; from which we see that the equation of motion of the aether in the metallic medium is probably of the form tfe, , de, /3V, ?e.\ This equation of motion differs from that of a Greenian 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. 181 elastic solid by reason of the occurrence of the term in detfdt. But this is evidently a " viscous " term, representing something like a frictional dissipation of the energy of luminous vibra- tions : a dissipation which, in fact, occasions the opacity of the metal. Thus the term which expresses opacity in the equation of motion of the luminif erous medium appears as the origin of the peculiarities of metallic reflexion.* It is curious to notice how closely this accords with the idea of Huygens, that metals are characterized by the presence of soft particles which damp the vibrations of light. There is, however, one great difficulty attending this explanation of metallic reflexion, which was first pointed out by Lord Rayleigh-f We have seen that for ideal silver ff is real and negative: and therefore A must be zero and pi negative; that is to say, the inertia of the luminiferous medium in the metal most be negative. This seems to destroy entirely the physical intelligibility of the theory as applied to the case of ideal silver. The difficulty 1b a deep-seated one, and was not overcome for many years. The direction in which the true solution lies will suggest itself when we consider the resemblance which has already been noticed between metals and those substances which show "surface colour" — e.g. the aniline dyes. In the case of the latter substances, the light which is so copiously reflected from them lies within a restricted part of the spectrum ; and it therefore seems probable that the phenomenon is not to be attributed to the existence of dissipative terms, but that it belongs rather to the same class of effects as dispersion, and is to be referred to the same causes. In fact, dispersion means that the value of the refractive index of a substance with respect to any kind of light depends on the period of the light ; and we have only to suppose that the physical causes which operate in dispersion cause the refractive index ■ It ii saiily »een that the amplitude it reduced bj- the factor r™* when light Innli oau Wave-length in the metal: * is generally called the eo*Jieu*t nf ainrption. t Loc. cit. 3,Bl,ZEdhyG00gle 182 The Aether as an Elastic Solid. to become imaginary for certain kinds of light, in order to explain satisfactorily both the surface colours of the aniline dyes and the strong reflecting powers of the metals. Dispersion was the subject of several memoirs by the founders of the elastic-solid theory. So early as 1830 Cauchy's attention was directed* to the possibility of constructing a mathematical theory of this phenomenon on the basis of Fresnel's " Hypothesis of Finite Impacts "f — Le. the assumption that the radius of action of one particle of the luminiferoue medium on its' neighbours is so large as to be comparable with the wave-length of light. Cauchy supposed the medium to be formed, as in Navier's theory of elastic solids, of a system of point-centres of force : the force between two of these point-centres, m at (x, y, z), and ^ at (x + Ax, y + Ay, a + Az), may be denoted by mn/{r), where r denotes the distance between m and ;i. When this medium is disturbed by light-waves pro- pagated parallel to the s-axis, the displacement being parallel to the K-axis, the equation of motion of m is evidently ^..S */<*■ + ,)-— , where £ denotes the displacement of m, (£ + A£) the displace* moot of ft, and (r + p) the new value of r. Substituting for p its value, and retaining only terms of the first degree in A£, this equation becomes dt' r ^rfr| r ) c Now, by Taylor's theorem, since £ depends only on t, we have Substituting, and remembering that summations which involve odd powers of Ai must vanish when taken over all * Bull, dea Sc. Math, sir (1830), p. 8 : " Sur la duporriou i» U lumrtre," JTvln. Butrntti U Math., 1836. t Cf. p. 132. 3,Bl,ZEdhyG00gle The Aether as an Elastic Solid. ' 183 the point-centres within the sphere of influence of m, we obtain an equation of the form where o, /3, 7 . . . denote constants. Each successive term on the right-hand Bide of this equation involves an additional factor (As)W as compared with the pre- ceding term, where \ denotes the wave-length of the light : bo if the radii of influence of the point-centres were indefinitely small in comparison with the wave-length of the light, the equation would reduce to which is the ordinary equation of wave-propagation in one dimension in non-dispersive media. But if the medium is so coarse-grained that A is not large compared with the radii of influence, we must retain the higher derivates of £. Substi- tuting e-e* in the differential equation with these higher derivates retained, we have •.--Kx)'"(?r)'"- which shows that c„ the velocity of the light in the medium, depends on the wave-length A ; as it should do in order to explain dispersion. Dispersion is, then, according to the view of Fresnel and Cauchy, a consequence of the coarse-grainednesa of the medium. Since the luminiferous medium was found to be dispersive only within material bodies, it seemed natural to suppose that in these bodies the aether is loaded by the molecules of matter, and that dispersion depends essentially on the ratio of the wave-length to the distance between adjacent material molecules. D,Bl,ZEdhyG00gle 184 Thi Aether as an Elastic Solid. This theory, in one modification or another, held its ground until forty years later it was overthrown by the facts of anomalous dispersion. The distinction between aether and ponderable matter was more definitely drawn in memoirs which were published independently in 1841-2 by F. E. Neumann* and Matthew O'Brien. f These authors supposed the ponderable particles to remain sensibly at rest while the aether surges round them, and is acted on by them with forces which are proportional to its displacement. ThuBj the equation of motion of the aether P jj3 - - (A + $■«) grad div e - n curl curl e - Co, where C denotes a constant on which the phenomena of dis- persion depend. For polarized plane waves propagated parallel to the axis of x, this equation becomes _ d't _ _ &9 and substituting Co; 2*y~ '(-«■ where r denotes the period and V the velocity of the light, we have % C an equation which expresses the dependence of the velocity on the period. The attempt to represent the properties of the aether by those of an elastic solid loat some of its interest after the rise of the electromagnetic theory of light But in 1867, * Berlin Abhandlnngen am dem J>hrB 1841, ZweiterTcil, p. 1 ; Berlin, 1843. t Timni. C«nb. Phil. Soc. tu (1842), p. 367. X O'Brien, loc. ciL, \\ 19, 26. dhyGooglc The Aether as an Elastic Solid. 185 before the electromagnetic hypothesis had attracted much attention, an elastic-solid theory in many respects preferable to its predecessors was presented to the French Academy* by Joseph Bousainesq (b. 1842). Until this time, as we have seen, investigators had been divided into two parties, according as they attributed the optical properties of different bodies to variations in the inertia of the luminiferous medium, or to variations in its elastic properties. Boussinesq, taking up a position apart from both these schools, assumed that the aether is exactly the same in all material bodies as in interplanetary space, in regard both to inertia and to rigidity, and that the optical properties of matter are due to interaction between the aether and the material particles, as had been imagined more or less by Neumann and O'Brien. These material particles he supposed to be disseminated in the aether, in much the same way as dust-particles floating in the air. If a denote the displacement at the point (x, y, z) in the aether, and e' the displacement of the ponderable particles at the same place, the equation of motion of the aether is PoV "~ (*"•"**) graddiv* + WV*« - />i ^p (1) where p and pi denote the densities of the aether and matter respectively, and k and n denote as usual the elastic constants of the aether. This differs from the ordinary Cauehy-Green equation only in the presence of the term piSV/tK*, which represents the effect of the inertia of the matter. To this equation we must adjoin another expressing the connexion between the displacements of the matter and of the aether: if we assume that these are simply proportional to each other — say, e' = At, (2) * Journal de Math. (2) liii (1808], pp. S13, 125 : of. also Compte* Bendus, citii (1893), pp. 80, 189, 193. Equations kindred to «ome of those of Bonaiino»q were afterwards deduced by Earl Pennon, Proc. Lond. Math. 800 , xx (1889), p. 297, from the hypothesis that the strain-energy involves the velocities. 3,Bl,ZEdhyG00gle 186 The Aether as an Elastic Solid. where the constant A depends on the nature of the ponderable body— our equation becomes which is essentially the same equation as is obtained in those older theories which suppose the inertia of the lurainiferons medium to vary from one medium to another. So far there would Beem to be nothing very new in Boussinesq's work. But when we proceed to consider crystal-optics, dispersion, and rotatory polarization, the advantage of his method becomes evident: he retains equation (1) as a formula universally true — at any rate for bodies at rest — while equation (2) is varied to suit the circumstances of the case. Thus dispersion can be explained if, instead of equation (2), we take the relation e'--4e-i>V*e, where D is a constant which measures the dispersive power of the substance : the rotation of the plane of polarization of sugar solutions can be explained if we suppose that in these bodies equation (2) ie replaced by «' ■ At + B curl s, where B is a constant which measures the rotatory power; and the optical properties of crystals can be explained if we suppose that for them equation (2) is to be replaced by the equations «/ - Aitt, «,' = Ai?y, et' - Atfz When these values for the components of t are substituted in equation (1), we evidently obtain the same formulae as were derived from the Stokes-Rankine-R&yleigh hypothesis of inertia different in different directions in a crystal ; to which Boussinesq's theory of crystal-optics is practically equivalent. The optical properties of bodies in motion may be accounted for by modifying equation (1), so that it takes the form P^-(A+i«)graddiv + «V^-P,(- + K-,- + ^-+«,,-j., 3,Bl,ZEdhyG00gle 0V. Ve, .Id The Aether as an Elastic Solid, 187 where w denotes the velocity of the ponderable body. If the body is an ordinary isotropic one, and if we consider light propagated parallel to the axis of a, in a medium moving in that direction, the light-vector being parallel to the axis of x, the equation reduces to substituting '.-/{>- ri), where V denotes the velocity of propagation of light in the medium estimated with reference to the fixed aether, we obtain for V the value / n y + P>A w \ft + p,AJ p + fiXA The absolute velocity of light is therefore increased by the amount piAw/(p + piA) owing to the motion of the medium ; and this may be written {#' - 1) w//i*, where p denotes the refractive index ; so that BouBSinesq's theory leads to the same formula as had been given half a century previously by FresneL' It is Boussineaq's merit to have clearly asserted that all space, both within and without ponderable bodies, is occupied by one identical aether, the same everywhere both in inertia and elasticity; and that all aethereal processes are to be re- presented by two kinds of equations, of which one kind expresses the invariable equations of motion of the aether, while the other kind expresses the interaction between aether and matter. Many years afterwards these ideas were revived in connexion with the electromagnetic theory, in the modern forms of which they are indeed of fundamental importance. *Cf. p. 116 iqq. 3,Bl,ZEdhyG00gle 188 ) CHAPTER VI. FARADAY. Towards the end of the year 1812, Davy received a letter in which the writer, a bookbinder's journeyman named Michael Faraday, expressed a desire to escape from trade, and obtain employment in a scientific laboratory. With the letter was enclosed a neatly written copy of notes which the young man ■ — he was twenty-one years of age — had made of Davy's own public lectures. The great chemist replied courteously, and arranged an interview ; at which he learnt that his correspon- dent had educated himself by reading the volumes which came into his hands for binding. "There were two," Faraday wrote later, "that especially helped me, the 'Encyclopaedia Britannica,' from which I gained my first notions of electricity, and Mrs. Marcet's ' Conversations on Chemistry,' which gave me my foundation in that science." Already, before his applica- tion to Davy, he had performed a number of chemical experiments, and had made for himself a voltaic pile, with which he had decomposed several compound bodies. At Davy's recommendation Faraday was in the following spring appointed to a post in the laboratory of the Royal Institution, which had been established at the close of the eighteenth century under the auspices of Count Rumford ; and here he remained for the whole of his active life, first as assistant, then as director of the laboratory, and from 1833 onwards as the occupant of a chair of chemistry which was founded for his benefit. For many years Faraday was directly under Davy's influence, and was occupied chiefly in chemical investigations. But in 1821, when the new field of inquiry opened by Oersted's 3,Bl,ZEdhyG00gle Faraday. 189 discovery waB attracting attention, he wrote an Historical Sketch of Electro- Magnetism,,* as a preparation for which he carefully repeated the experiments described by the writers he was reviewing ; and this seems to have been the beginning of the researches to which his fame is chiefly due. The memoir which stands firat in the published volumes of Faraday's electrical workt was communicated to the Royal Society on November 24th, 1831. The investigation was inspired, as he tells us, by the hope of discovering analogies between the behaviour of electricity as observed in motion in currents, and the behaviour of electricity at rest on conductors. Static electricity was known to possess the power of " induction " — La, of causing an opposite electrical state on bodies in its neighbourhood ; was it not possible that electric currents might show a similar property ? The idea at first was that if in any circuit a current were made to flow, any adjacent circuit would be traversed by an induced current, which would persist exactly as long as the inducing current. Faraday found that this was not the case ; a current was indeed induced, but it lasted only for an instant, being in fact perceived only when the primary current was started or stopped. It depended, as he soon convinced himself, not on the mere existence of the inducing current, but on its variation. Faraday now set himself to determine the laws of induction of currents, and for this purpose devised a new way of repre- senting the state of a magnetic field. Philosophers had been long accustomed? to illustrate magnetic power by strewing iron filings on a sheet of paper, and observing the curves in which they dispose themselves when a magnet is brought underneath. •Publkhed in Annalt of FAHeiophy, ii (1821), pp. 135. 2T* ; iii (1822)-, p. 107. t Experimental JUit&rehtt in EUeiriaty, bj Michael Faraday : 3 Tola. J The practice goea back at leaat ai tar M Niocolo Cabeo ; indeed the curve* traced by Petrua Peragrinua on hi* globular lodeatone (of. p. S) were projection! of lines of force. Among eighteenth -century writere La Hire mentions the uee of iron filings. Hint, del' Acad., 1717. JanuUr had referred to them in hia electro- magnetic paper of 1821, Eip. Set. ii, p. 127. 3,Bl,ZEdhyG00gle 190 Faraday, These curves suggested to Faraday* the idea of lines of magnetic force, or curves whose direction at every point coincides with the direction of the magnetic intensity at that point; the curves in which the iron filings arrange themselves on the paper resemble these curves so far as is possible subject to the condition of not leaving the plane of the paper. With these lines of magnetic force Faraday conceived all space to be filled. Every line of force is a closed curve, which in some part of its course passes through the magnet to which it belongs.! Hence if any small closed curve be taken in space, the lines of force which intersect this curve must form a tubular surface returning into itself ; such a surface is called a tube of force. From a tube of force we may derive information not only regarding the direction of the magnetic intensity, bat also regarding its magnitude; for the product of this magnitude} and the cross-section of any tube is constant along the entire length of the tube.§ On the basis of this result, Faraday conceived the idea of partitioning all space into compartments by tubes, each tube being such that this product has the same definite value. For simplicity, each of these tubes may be called a " unit line of force " ; the strength of the field is then indicated by the separation or concentration of the unit lines of force, [| so that the number of them which intersect a unit area placed at right angles to their direction * They wen first defined in Eip. Si:, §114: "By magnetic curves, I mean the linei of magnetic forces, however modified by the juxtaposition of pole*, which could be depicted by iron filing! ; or those to which a very small magnetic needle would form a tangent." t Exp. Rtt. ill, p. 406. X Within the substance of magnetized bodies we mutt in thia connexion under- stand the magnetic intensity to be that experienced in a crevice whose aides •» perpendicular to the lines of magnetisation : in other word*, we must take it to be what since Maxwell'* time has been called the magnetic induction. } Exp. lie:, $ 3073. This theorem waa first proved by tie French geometer Michel Chasles, in bis memoir on the attraction of an ellipsoidal sheet, Journal de l'E'cole Polyt. xv (1837), p. 266. || Ibid., , 3122. "The relative amount of force, or of lines of force, in a given apace is indicated by their concentration or separation— i.e., by their number io that apace." 3,Bl,ZEdhyG00gle Faraday. 191 at any point measures the intensity of the magnetic field at that point. Faraday constantly thought in terms of lines of force. " I cannot refrain," he wrote, in 1851,* " from again expressing my conviction of the truthfulness of the representation, which the idea of lines of force affords in regard to magnetic action. All the points which are experimentally established in regard to that action — La all that is not hypothetical — appear to be well and truly represented by £t."f Faraday found that a current is induced in a circuit either when the strength of an adjacent current is altered, or when a magnet is brought near to the circuit, or when the circuit itself is moved about in presence of another current or a magnet. He saw from the first; that in all cases the induction depends on the relative motion of the circuit and the lines of magnetic force in its vicinity. The precise nature of this dependence was the subject of long-continued further experiments. In 1832 he foundjj that the currents produced by induction under the same circumstances in different wires are proportional to the conducting powers of the wires — a result which showed that the induction consists in the production of a definite electromotive force, independent of the nature of the wire, and dependent only on the intersections of the wire and the magnetic curves. This electromotive force is produced whether the wire forms a closed circuit (so that a current flows) or is open (so that electric tension results). All that now remained was to inquire in what way the electromotive force depends on the relative motion of the wire and the lines of force. The answer to this inquiry is, in t Some of Faraday's most distinguished contemporaries »ere far from sharing this conviction. " I declare," wrote Sir George Airy in 1855, " that I can hardly imagine anyone who practically and numerically knows thii agreement " between observation ud the results of calculation based on action at a distance, "to hesitate an instant in the tlioice between this simple and precise action, on the one hand, and anything so vague and varying as lines of force, on the Other hand." Cf. Bcncc Jones's Life of Faratbiy, ii, p. 363. J.-Eio, Re:, } 116. } Ibid., j 213. 3,Bl,ZEdhyG00gle 192 Faraday. Faraday's own words," that "whether the wire moves directly or obliquely across the lines of force, in one direction or another, it sums up the amount of the forces represented by the lines it has crossed," so that " the quantity of electricity thrown into a current is directly as the number of curves intersected, "t The induced electromotive force is, in fact, simply proportional to the number of the unit lines of magnetic force intersected by the wire per second. This is the fundamental principle of the induction of currents. Faraday is undoubtedly entitled to the full honour of its discovery ; but for a right understanding of the progress of electrical theory at this period, it is necessary to remember that many years elapsed before all the conceptions involved in Faraday's principle became clear and familiar to his contem- poraries ; and that in the meantime the problem of formulating the laws of induced currents was approached with success from other points of view. There were indeed many obstacles to the direct appropriation of Faraday's work by the mathematical physicists of his own generation ; not being himself a mathe- matician, he was unable to address them in their own language ; and his favourite mode of representation by moving lines of force repelled analysts who had been trained in the school of Laplace and Poisson. Moreover, the idea of electromotive force itself, which had been applied to currents a few years previously in Ohm's memoir, was, as we have Been, still involved in obBcurity and misapprehension. A curious question which arose out of Faraday's theory was whether a bar-magnet which is rotated on its own axis carries its lines of magnetic force in rotation with it. Faraday himself believed that the lines of force do not rotate *-. on this view a revolving magnet like the earth is to be regarded as moving through its own lines of force, so that it must become charged at the equator and poles with electricity of opposite signs ; and if a wire not partaking in the earth's rotation were to have sliding contact with the earth at a pole and at the • Exp. Si:, $ 3082. t Ibid., } 3116. % Hid., §3090. 3,Bl,ZEdhyG00gle Faraday. 193 equator, a current would steadily flow through it. Experiments confirmatory of these views were made by Faraday himself ;* but they do not strictly prove his hypothesis that the lines of force remain at rest ; for it is easily seenf that, if they were to rotate, that part of the electromotive force which would be produced by. their rotation would be derivable from a potential, and so would produce no effect in closed circuits such as Faraday used. Three years after the commencement of Faraday's researches on induced currents he was led to an important extension of them by an observation which was communicated to him by another worker. William Jenkin had noticed that an electric shock may be obtained with no more powerful source of electricity than a single cell, provided the wire through which the current passes is long and coiled ; the shock being felt when contact is broken.J As Jenkin did not choose to investigate the matter further, Faraday took it up, and showedg that the powerful momentary current, which was observed when the circuit was interrupted, was really an induced current governed by the same laws as all other induced currents, but with this peculiarity, that the induced and inducing currents now flowed in the same circuit In fact, the current in its steady state establishes in the surrounding region a magnetic field, whose lines of force are linked with the circuit ; and the removal of these lines of force when the circuit is broken originates an induced current, which greatly reinforces the primary current just before its final extinction. To this phenomenon the name of self-induction has been given. The circumstances attending the discovery of self-induction • Rep. Stt., j) 218, 3109, 1ms. t Cf. W. Weber, Ann. d. Phyi. lii (1341) ; 8. Tolver PiwtoD, Phil. Maa-. *ii (1BBG), p. 131. In 1891 8. T. Preston, Phil. Mag. nii, p. 100, deaigned a crucial experiment to test the question ; hut it irai not tried for want of a sufficient! y delicate electrometer. J A similar observation had been made by Henry, and published in the Amer. Jour. Sti. xxii (1832), p. 403. The spark at the rupture of a spirally- wound circuit had been often obeerwd, e.g., by Pouillet and Hobili. f Exp. R»., i 1048. DslzEdhyGoOgle 194 Faraday. occasioned a comment from Faraday on the number of sugges- tions which were continually being laid before him. He re- marked that although at different times a large number of authors had presented him with their ideas, this case of Jenkin was the only one in which any result had followed. " The volunteers are Berious embarrassments generally to the experienced philosopher."* The discoveries of Oersted, Ampere, and Faraday had shown the close connexion of magnetic with electric science. But the connexion of the different branches of electric science with each other was still not altogether clear. Although Wollaston's experiments of 1801 had in effect proved the identity in kind of the currents derived from fractional and voltaic sources, the question was still regarded as open thirty years afterwards,f no satisfactory explanation being forthcoming of the fact that frietional electricity appeared to be a surface-phenomenon, whereas voltaic electricity was conducted within the interior substance of bodies. To this question Faraday now applied him- self ; and in 1833 he succeeded* in showing that every known effect of electricity — physiological, magnetic, luminous, calorific, chemical, and mechanical — may be obtained indifferently either with the electricity which is obtained by friction or with that obtained from a voltaic battery. Henceforth the identity of the two was beyond dispute. Some misapprehension, however, has existed among later writers as to the conclusions which may be drawn from this identification. What Faraday proved is that the process which goes on hi a wire connecting the terminals of a voltaic cell is of the same nature as the process which for a short time goes on in a wire by whieh a condenser is discharged. He did not prove, 'Bonce Jones's Lift of Faraday, ii, p. 4G. t Cf. John Davy, Phil. Train., 1832, p. 259 ; W. Ritchie, Hid., p. 279. Daty suggested Out the electrical power, " according to the analogy of the solar raj," might be " not a simple power, but a combination of powers, which may occur variously associated, and produce all the varieties of electricity with which we are acquainted." t Bxp. St., Series in. 3,Bl,ZEdhyG00gle Faraday. 195 and did not profess to have proved, that this process consists in the actual movement of a quasi-substance, electricity, from one plate of the condenser to the other, or of two quasi-substances, the resinous and vitreous electricities, in opposite directions. The process had been pictured in this way by many of his predecessors, notably by Volta; and it has since been so pictured by most of his successors : but from such assumptions Faraday himself carefully abstained. What is common to all theories, and is universally conceded, is that the rate of increase in the total quantity of electrostatic charge within any volume-element is equal to the excess of the hiflux over the efflux of current from it. This statement may be represented by the equation |+divi.O, (,) where p denotes the volume-density of electrostatic charge, and i the current, at the place (a;, y, z) at the time t. Volte's assumption is really one way of interpreting this equation physically : it presents itself when we compare equation (1) with the equation dt which is the equation of continuity for a fluid of density p and velocity v : we may identify the two equations by supposing i to be of the same physical nature as the product pv; and this is precisely what is done by those who accept Volte's assumption. But other assumptions might be made which would equally well furnish physical interpretations to equation (1). For instance, if we suppose p to be the convergence of any vector of which i is the time-flux* equation (1) ia satisfied automatically ; * In sy inboli, div • - - p, I-. where ( denote* the Teotor in question. 02 3,Bl,ZEdhyG00gle 196 Faraday. we can picture this vector ae being of the nature of a displace- ment. By such an assumption we should avoid altogether the necessity for regarding the conduction-current as an actual flow of electric charges, or for speculating whether the drifting charges are positive or negative ; and there would be no longer anything surprising in the production of a null effect by the coalescence of electric charges of opposite signs,. Faraday himself wished to leave the matter open, and to avoid any definite assumption." Perhaps the best indication of his views is afforded by a laboratory notef of date 1837 : — "After much consideration of the manner in which the electric forces are arranged in the various phenomena generally*. I have come to certain conclusions which I will endeavour to note down without committing myself to any opinion as to the cause of electricity, Le., as to the nature of the power. If electricity exist independently of matter, then I think that the hypothesis of one fluid will not stand against that of two fluids. There are, I think, evidently what I may call two elements of power, of equal force and acting toward each other. But these powers may be distinguished only by direction, and may be no more separate than the north and south forces in the elements of a magnetic needle. They may be the polar points of the forces originally placed in the particles of matter." It may be remarked that since the rise of the mathematical theory of electrostatics, the controversy between the supporters of the one-fluid and the two-fluid theories had become manifestly barren. The analytical equations, in which interest was now largely centred, could be interpreted equally well on either hypothesis; and there seemed to be little prospect of discriminating between them by any new experi- mental discovery. But a problem does not lose its fascination *"HU principal urn," said Helmholta in the Fnrsday Lecture of 1881, " »-aa to express in bit new conceptions only (acta, with the least possible usa of hypothetical substances and fumes. This was really a progress in general scientific: method, destined to purify science from the last remains of meta- physics." t Bence Jones's Lift of Fnraduy, ti, p. 77. DslzEahyGoOgIC Faraday. X9T because it appears insoluble. " I said once to Faraday," wrote Stokes to his father-in-law in 1879, " as I sat beside him at a British Association dinner, that I thought a great step would be made when we should be able to say of electricity that which we say of light, in saying that it consists of undula- tions. He said to me he thought we were a long way off that yet"* For his next series of researches,t Faraday reverted to subjects which had been among the first to attract him as an apprentice attending Davy's lectures : the voltaic pile, and the relations of electricity to chemistry. It was at thiB time generally supposed that the decomposi- tion of a solution, through which an electric current is passed, is due primarily to attractive and repellent forces exercised on its molecules by the metallic terminals at which the current enters and leaves the solution. Such forces had been assumed both in the hypothesis of Grothuss and Davy, and in the rival hypothesis of De La Rive ;J the chief difference between these being that whereas Grothuss and Davy supposed a chain of decompositions and recompositions in the liquid, De La Rive supposed the molecules adjacent to the terminals to be the only ones decomposed, and attributed to their fragments the power of travelling through the liquid from one terminal to the other. To test this doctrine of the influence of terminals, Faraday moistened a piece of paper in a saline solution, and supported it in the air on wax, so as to occupy part of the interval between two needle-points which were connected with an electric machine. When the machine was worked, the current was conveyed between the needle-points by way of the moistened paper and the two air-intervals on either side of it ; and under these circumstances it was found that the salt under- went decomposition. ' Since in this case no metallic terminals of any kind were in contact with the solution, it was evident that •Stokei's Scientific Cormpondtnct, vol. i, p. 363. t Sip. So., § 450 (1833). t CI. pp. 78- -9. 3,Bl,ZEdhyG00gle 198 Faraday. all hypotheses which attributed decomposition to the action of the terminals were untenable. The ground being thus cleared by the demolition of previous theories, Faraday was at liberty to construct a theory of his own. He retained one of the ideas of Grothuss' and Davy's doctrine, namely, that a chain of decompositions and recombi- nations takes place in the liquid ; but these molecular processes he attributed not to any action of the terminals, but to a power possessed by the electric current itself, at all places in its course through the solution. If as an example we consider neighbouring molecules A, B, C, D, ... of the compound — say water, which was at that time believed to be directly decom- posed by the current — Faraday supposed that before the passage of the current the hydrogen of A would be in close union with the oxygen of A, and also in a less close relation with the oxygen atoms of B, C, D, . . . : these latter relations being conjectured to be the cause of the attraction of aggregation in solids and fluids.* When an electric current is sent through the liquid, the affinity of the hydrogen of A for the oxygen of B is strengthened, if A and B lie along the direction of the current ; while the hydrogen of A withdraws some of its bonds from the oxygen of A, with which it is at the moment combined. So long as the hydrogen and oxygen of A remain in association, the state thus induced is merely one of polarization ; but the compound molecule is unable to stand the strain thus imposed on it, and the hydrogen and oxygen of A part company from each other. Thus decompositions take place, followed by recombinations : with the result that after each exchange an oxygen atom associates itself with a partner nearer to the positive terminal, while a hydrogen atom associates with a partner nearer to the negative terminal, This theory explains why, in all ordinary cases, the evolved substances appear only at the terminals ; for the terminals are the limiting surfaces of the decomposing substance ; and, except at them, every particle finds other particles having a contrary *Exp. Rm., J 623. 3,Bl,ZEdhyG00gle Faraday. 199 tendency with which it can combine. It also explains why, in numerous caseB, the atoms of the evolved substances are not retained by the terminals (an obvioiw difficulty in the way of all theories which suppose the terminals to attract the atoms) : for the evolved substances . are expelled from the liquid, not drawn out by an attraction. Many of the perplexities which had harassed the older theories were at once removed when the phenomena were re- garded from Faraday's point of view. Thus, mere mixtures (as opposed to chemical compounds) are not separated into their constituents by the electric current ; although there would seem to be no reason why the Grothuss-Davy polar attraction should not operate as well on elements contained in mixtures as on elements contained in compounds. In the latter part of the same year (1833) Faraday took up the subject again." It was at this time that he introduced the terms which have ever since been generally used to describe the phenomena of electro-chemical decomposition. To the terminals by which the electric current passes into or out of the decomposing body he gave the name electrodes. The electrode of high potential, at which oxygen, chlorine, acids, &c, are evolved, he called the anode, and the electrode of low potential, at which metals, alkalis, and bases are evolved, the cathode. Those bodies which are decomposed directly by the current he named electrolytes ; the parts into which they are decomposed. ions ; the acid ions, which travel to the anode, he named anions ; and the metallic ions, which pass to the cathode, cations. Faraday now proceeded to test the truth of a supposition which he had published rather more than a year previously ,f and which indeed had apparently been suspected by Gay-Lussac and Thenard* so early as 1811; namely, that the rate at which an electrolyte is decomposed depends solely on the intensity of the electric current passing through it, and not at all on the size of the electrodes or the strength of the solution. Having • Exp. -R»., { 661. f Hid., i 377 (Dec. 1332). J Siehertkn phyiiso-thimiqvu faitti tur U pU$ ', Puis, 181 1, p. 12. 3,Bl,ZEdhyG00gle 200 Faraday. established the accuracy of this law,* he found by a comparison of different electrolytes that the mass of any ion liberated by a given quantity of electricity is proportional to its chemical equivalent, i.e. to the amount of it required to combine with some standard mass of some standard element. If an element is m-valent, so that one of its atoms can hold in combination n atoms of hydrogen, the chemical equivalent of this element may be taken to be l/» of its atomic weight; and therefore Faraday's result may be expressed by saying that an electric current will liberate exactly one atom of the element in question in the time which it would take to liberate » atoms of hydrogen.f The quantitative law seemed to FaradayJ to indicate that " the atoms of matter are in some way endowed or associated with electrical powers, to which they owe their most striking qualities, and amongst them their mutual chemical affinity." Looking at the facts of electrolytic decomposition from this point of view, he showed how natural it is to suppose that the electricity which passes through the electrolyte is the exact equivalent of that which is possessed by the atoms separated at the electrodes ; which implies that there is a certain absolute quantity of the electric power associated with eacJi atom of matter. The claims of this splendid speculation he advocated with conviction. " The harmony," he wrote,§ " which it introduces into the associated theories of definite proportions and electro- chemical affinity is very great. According to it, the equivalent weights of bodies are simply those quantities of them which contain equal quantities of electricity, or have naturally equal electric powers ; it being the electricity which determines the equivalent number, because it determines the combining force. Or, if we adopt the atomic theory or phraseology, then the * Exp. Ret., H 713-821. t In die modem unit*, 96680 coulombs of electricity must put round the circuit in order to liberate of each ion a number of gram* equal to the quotient of the iitomio weight by the valency. t Kxp. St,., f 852, i Ibid., { 869. 3,Bl,ZEdhyG00gle Faraday. 201 atoms of bodies which are equivalent to each other in their ordinary chemical action, have equal quantities of electricity naturally associated with them. " But," he added, " I must confess I am jealous of the term atom : for though it is very easy to talk of atoms, it is very difficult to form a clear idea of their nature, especially when compound bodies are under consideration." These discoveries and ideas tended to confirm Faraday in preferring, among the rival theories of the voltaic cell, that one to which all his antecedents and connexions predisposed him. The controversy between the supporters of Volta's contact hypothesis on the one hand, and the chemical hypothesis of Davy and "Wollaston on the other, had now been carried on for a generation without any very decisive result. In Germany and Italy the contact explanation was generally accepted, under the influence of Christian Heinrich Pfaff, of Kiel (6. 1773, d. 1852), and of Ohm, and, among the younger men, of Gustav Theodor Fechner (b. 1801, d. 1887), of Leipzig," and Stefano Marianini (6. 1790, d. 1866), of Modena. Among French writers De La Rive, of Geneva, was, as we have seen, active in support of the chemical hypothesis ; and this side in the dispute had always been favoured by the English philosophers. There is no doubt that when two different metals are put in contact, a difference of potential is set up between them without any apparent chemical action ; but while the contact party regarded this as a direct manifestation of a "contact- force " distinct in kind from all other known forces of nature, * Johann Cliriitiim Poggendorff {». 1796, d. ISTT), of Berlin, for long the editor of the Aiinalen der Phyaik, leaned originally to the chemical tide, but in 183S became convinced of the truth of the contact theory, which he afterwards actively defended. Horiti Hermann Jaeobi (». 1801, d. 1874), of Dorpat, is also to be mentioned among it* advocate). Faraday'* first series of investigations on this subject were made in 1834 : E*p. &-., series viii. In 1810 De L* Hive followed on the same tide with his aWasrato iur U Gssim aV t'Elttf. Vntlaiqu*. The views of Faraday and De La Rive were criticised by Pfaff, Stvitiut dtr Lt\rt vain Qalvimumut, Kiel, 1337, an* by Fechner, Ann. d. Phys., zlii (1337), p. 481, and xliii (1338), p. 4S3 : translated Phil. Hag., aiii (1833), pp. 206, 367. Faraday returned to tbe question in 1840, Exp. Stt., series xvi and ivii. 3,Bl,ZEdhyG00gle 202 Faraday. the chemical party explained it as a consequence of chemical affinity or incipient chemical action between the metals and the surrounding air or moisture. There is also no doubt that the continued activity of a voltaic cell is always accompanied by chemical unions or decompositions ; but while the chemical party asserted that these constitute the efficient source of the current, the contact party regarded them as secondary actions, and attributed the continual circulation of electricity to the perpetual tendency of the electromotive force of contact to- transfer charge from one substance to another. One of the most active supporters of the chemical theory among the English physicists immediately preceding Faraday was Peter Mark Koget (£. 1779, d. 1869), to whom are due two of the strongest arguments in its favour. In the first place, carefully distinguishing between the quantity of electricity put into circulation by a cell and the tension at which this electricity is furnished, he showed that the latter quantity depends on the "energy of the chemical action'"* — a fact which, when taken together with Faraday's discovery that the quantity of electricity put into circulation depends on the amount of chemicals con- sumed, places the origin of voltaic activity beyond all question. Eoget's principle was afterwards verified by Faradayt and by De La RiveJ; " the electricity of the voltaic pile is proportionate in its intensity to the intensity of the affinities concerned in its production," said the former in 1834; while De La Rive ■wrote in 1836, " The intensity of the currents developed in combinations and in decompositions is exactly proportional to the degree of affinity which subsists between the atoms whose combination or separation has given rise to these currents." * " The absolute quantity of electricity which, ii thui developed, and made to circulate, will depend upon a variety of circumatancee, aueh aa the extent of the aurfacee in chemical action, the futilities afforded to its tranemiaetoii, Ac. But it* degree of intensity, or tuition, aa it is often termed, will be regulated by other causea, and more especially by the energy of the chemical action. " Roget'a Oaltanitm (1832), j 70. t Exp. Hit., f j 90S, 908, 916, 988, 19GS. J Antilles de Cbim., lxi (183S), p. 38. 3,Bl,ZEdhyG00gle Faraday. 203 Not resting here, however, Boget brought up another argu- ment of far-reaching significance. " If," he wrote,* " there could exiBt a power having the property ascribed to it by the [contact] hypothesis, namely, that of giving continual impulse to a fluid in one constant direction, without being exhausted by its own action, it would differ essentially from all the other known powers in nature. All the powers and sources of motion, with the operation of which we are acquainted, when producing their peculiar effects, are expended in the same proportion as those effects are produced ; and hence arises the impossibility of obtaining by their agency a perpetual effect ; or, in other words, a perpetual motion. But the electro-motive force ascribed by Volta to the metals when in contact is a force which, as long as a free course is allowed to the electricity it sets in motion, is never expended, and continues to be exerted with undi- minished power, in the production of a never-ceasing effect. Against the truth of such a supposition the probabilities are all but infinite." This principle, which is little less than the doctrine of conservation of energy applied to a voltaic cell, was reasserted by Faraday. The process imagined by the contact school " would," he wrote, " indeed be a creation of power, like no other force in nature." In all known cases energy is not generated, but only transformed. There is no such thing in the world as "a pure creation of force; a production of power without a corresponding exhaustion of something to supply iff As time went on, each of the rival theories of the cell became modified in the direction of the other. The contact party admitted the importance of the surfaces at which the metals are in contact with the liquid, where of course the chief chemical action takes place ; and the chemical party confessed their inability to explain the state of tension which subsists before the circuit is closed, without introducing hypotheses just as uncertain as that of contact force, •Boget1. ffofrwiutiJ (1832), % 113. f Xxp. Xtt., i 2071 (1810). 3,Bl,ZEdhyG00gle 204 Faraday. Faraday's own view on this point* was that a plate of amalgamated zinc, when placed in dilute sulphuric acid, " has power so far to act, by its attraction for the oxygen of the particles^in contact with it, as to place the similar forces already active between these and the other particles of oxygen and the particles of hydrogen in the water, in a peculiar state of tension or polarity, and probably alBO at the same time to throw those of its own particles which are in contact with the water into a similar but opposed state. Whilst this state is retained, no further change occurs : but when it iH relieved by completion of the circuit, in which case the forces determined in opposite directions, with respect to the zinc and the electro- lyte, are found exactly competent to neutralize each other, then a series of decompositions and recompositions takes place amongst the particles of oxygen and hydrogen which constitute the water, between the place of contact with the platina and the place where the zinc is active : these intervening particles being evidently in close dependence upon and relation to each other. The zinc forms a direct compound with those particles of oxygen which were, previously, in divided relation to both it and the hydrogen : the oxide is removed by the acid, and a fresh surface of zinc is presented to the water, to renew and repeat the action." These ideas were developed further by the later adherents of the chemical theory, especially by Faraday's friend Christian Friedrich Schonbein.f of Basle (6. 1799, d. 1868), the discoverer of ozone. Schonbein made the hypothesis more definite by assuming that when the circuit is open, the molecules of water adjacent to the ziuc plate are electrically polarized, the oxygen side of each molecule being turned towards the zinc and being negatively charged, while the hydrogen side is turned away from the zinc and is positively charged. In the third quarter * Sap. Set., j 918. t Ann. d. Phyi., Ixiviii (18*3), p. 289, translated Archive* de* re. phy*., iiii (I860), p. 192. Faraday and Suhdnbain for many yours carried on a correapondnnm, which dm been edited by 0. W. A. Knlilbaum and F. V. DarbUUre : London, William* and Nor gate. 3,Bl,ZEdhyG00gle Faraday. 205 of the nineteenth century, the general opinion was in favour of some such conception as this. Helmholtz" attempted to- grasp the molecular processes more intimately by assuming that the different chemical elements have different attractive- powers (exerted only at small distances) for the vitreous and resinous electricities : thus potassium and zinc have strong attractions for positive charges, while oxygen, chlorine, and bromine have strong attractions for negative electricity. This- differs from Volte's original hypothesis in little eke but. in assuming two electric fluids where Volta assumed only one. It is evident that the contact difference of potential between two metals may be at once explained by Helmholtz's hypothesis, as it was by Volte's ; and the activity of the voltaic cell may be referred to the same principles : for the two ions of which the liquid molecules are composed will also possess different attractive powers for the electricities, and may be supposed to be united respectively with vitreouB and resinous. charges. Thus when two metals are immersed in the liquid, the circuit being open, the positive ions are attracted to the negative metal and the negative ions to the positive metal,, thereby causing a polarized arrangement of the liquid molecules near the metals. When the circuit is closed, the positively charged surface of the positive metal is dissolved into the fluid ;. and as the atoms carry their charge with them, the positive- charge on the immersed surface of this metal must be per- petually renewed by a current flowing in the outer circuit. It will be seen that Helmholtz did not adhere to Davy's; doctrine of the electrical nature of chemical affinity quite as. simply or closely as Faraday, who preferred it in its most direct and uncompromising form, " All the facts show us," he wrote, t "that that power commonly called chemical affinity can he- communicated to a distance through the metals and certain forms of carbon ; that the electric current is only another form of the forces of chemical affinity ; that its power is in proportion . ■ In liia celebnteu memoir of 1847 on the ConiervMion.oC .Energy. tAp.*«.tf VM. 3,Bl,ZEdhyG00gle 206 Faraday. to the chemical affinities producing it ; that when it is deficient in force it may be helped by calling in chemical aid, the want in the former being made up by an equivalent of the latter ; that, in other words, the forces termed chemical affinity and electricity are one and the tame." In the interval between Faraday's earlier and later papers on the cell, some important results on the same subject were published by Frederic Daniell (b. 1790, d. 1845X Professor of Chemistry in King's College, London.* Daniell showed that when a current is passed through a solution of a salt in water, the ions which carry the current are those derived from the salt, and not the oxygen and hydrogen ions derived from the water ; this follows since a current divides itself between different mixed electrolytes according to the difficulty of decomposing each, and it is known that pure water can be electrolysed only with great difficulty. Daniell further showed that the ions arising from (say) sodium sulphate are not represented by Na,0 and SOi, but by Na and SO, ; and that in such a case as this, sulphuric acid is formed at the anode and soda at the cathode by secondary action, giving rise to the observed evolution of oxygen and hydrogen respectively at these terminals. The researches of Faraday on the decomposition of chemical compounds placed between electrodeB maintained at different potentials led him in 1837 to reflect on the behaviour of such substances as oil of turpentine or sulphur, when placed in the same situation. These bodies do not conduct electricity, and are not decomposed ; but if the metallic faces of a condenser are maintained at a definite potential difference, and if the space between them is occupied by one of these insulating substances, it is found that the charge on either face depends on the nature of the insulating substance. If for any particular insulator the charge has a value e times the value which it would have if the intervening body were air, the number i may be regarded as a measure of the influence which the insulator exerts on the propagation of electrostatic action • Phil. Tnuu., 18S9, p. »7. 3,Bl,ZEdhyG00gle Faraday. 207 through it : it was called by Faraday the specific inductive capacity of the insulator.* The discovery of this property of insulating substances or dielectrics raised the question as to whether it could be harmonized with the old ideas of electrostatic action. Consider, for example, the force of attraction or repulsion between two small electrically-charged bodies. So long as they are in air, the force is proportional to the inverse square of the distance ; but if the medium in which they are immersed be partly changed — e.g., if a globe of sulphur be inserted in the intervening space — this law is no longer valid : the change in the dielectric affecte the distribution of electric intensity throughout the entire field The problem could be satisfactorily solved only by forming a physical conception of the action of dielectrics : and such a conception Faraday now put forward. The original idea had been promulgated long before by bis master Davy. Davy.it will be remembered,t in his explanation of the voltaic pile, bad supposed that at first, before chemical decompositions take place, the liquid plays a part analogous to that of the glass in a Leyden jar, and that in this is involved an electric polarization of the liquid molecules.} This hypothesis was now developed by Faraday. Referring first to his own work on electrolysis, he asserted^ that the behaviour of a dielectric is exactly the same as that of an electrolyte, up to the point at which the electrolyte breaks down under the electric stress ; a dielectric being, in fact, a body which is capable of sustaining the stress without suffering decomposition. " For," he argued,! " let the electrolyte be water, a plate of ice being coated with platina foil on its two surfaces, and these * Exp. Bmm., $ 1252 (1837). CftTandiih had diaoorarod ipeciSc inductive capacity long before, ljut bit paper* were still unpublished. I Cf. p. 77. % Tbu is eipremly Hated in Davy's EUuunU of Chemical Fhiloiaphy (1812), Qiv. i, { 7, where he lay i it down that an MMntUl " property of non-conductor*" it " to receive electrical polarities." f &p. Xf,, H 1164, 1338, 1313, 1621. \B*f-B**., { 1164. 3,Bl,ZEdhyG00gle 208 Faraday. coatings connected with any continued source of the two electrical powers, the ice will charge like a Leyden arrangement, presenting a case of common induction, but no current will pass. If the ice be liquefied, the induction will now fall to a certain degree, because a current can now pass ; but its passing is dependent upon a peculiar molecular arrangement of the particles consistent with the transfer of the elements of the electrolyte in opposite directions . . . Ae, therefore, in the electrolytic action, induction, appeared to \»1Aib first step,and decomposition the second (the power of separating these steps from each other by giving the solid or fluid condition to the electrolyte being in our hands) ; as the induction was the same in its nature as that through air, glass, wax, &c, produced by any of the ordinary means ; and as the whole effect in the electrolyte appeared to be an action of the particles thrown into a peculiar or polarized state, I was glad to suspect that common induction itself was in all cases an action of contiguous particles, and that electrical action at a distance (i.e., ordinary inductive action) never occurred except through the influence of the intervening matter." Thus at the root of Faraday's conception of electrostatic induction lay Una idea that the whole of the insulating medium through which the action takes place is in a state of polarization similar to that which precedes decomposition in an electrolyte. " Insulators," he wrote," " may be said to be bodies whose particles can retain the polarized state, whilst conductors are those whose particles cannot be permanently polarized." The conception which he at this time entertained of the polarization may be reconstructed from what he had already written concerning electrolytes. He supposedf that in the ordinary or unpolarized condition of a body, the molecules con- sist of atoms which are bound to each other by the forces of chemical affinity, these forces being really electrical in their nature ; and that the same forces are exerted, though to a less * Exp. Set., i 1338. t Thie must not be taken to be more than an idea which Faraday mentioned aa present to hia mind. He declined aa yet to formulate a definite h ypotheeia. 3,Bl,ZEdhyG00gle Faraday. 209 degree, between atoms which belong to different molecules, thus producing the phenomena of cohesion. When an electric field is set up, a change takes place in the distribution of these forces; some are strengthened and some are weakened, the effect being symmetrical about the direction of the applied electric force. Such a polarized condition acquired by a dielectric when placed in an electric field presents an evident analogy to the condition of magnetic polarization which is acquired by a mass of soft iron when placed in a magnetic field ; and it was there- fore natural that in discussing the matter Faraday should introduce lines of cleetrU force, similar to the lines of magnetic force which he had employed so successfully in his previous researches. A line of electric force he defined to be a curve whose tangent at every point has the same direction as the electric intensity. The changes which take place in an electric field when the dielectric is varied may be very simply described in terms of lines of force. Thus if a mass of sulphur, or other substance of high specific inductive capacity, is introduced into the field, the effect is as if the lines of force tend to crowd into it : as W. Thomson (Kelvin) showed later, they are altered in the same way as the lines of flow of heat, in a case of steady con- duction of heat, would be altered by introducing a body of greater conducting power for heat. By studying the figures of the lines t>f force in a great number of individual cases, Faraday was led to notice that they always dispose themselves as if they were subject to a mutual repulsion, or as if the tubes of force had an inherent tendency to dilate." It is interesting to interpret by aid of these conceptions the law of Priestley and Coulomb regarding the attraction between two oppositely-charged spheres. In Faraday's view, the medium intervening between the spheres is the seat of a system of stresses, which may be represented by an attraction or tension along the lines of electric force at every point, together with a • Exp. St., Jj 1224, 1597 (1837). P D,Bl,ZEdhyG00gle 210 Faraday. mutual repulsion of these lines, or pressure laterally. Where a line of force ends on one of the spheres, its tension is exercised on the sphere: in this way, every surface-element of each sphere is pulled outwards. If the spheres were entirely removed from each other's influence, the state of stress would be uniform round each sphere, and the pulls on its surface-elements would balance, giving no resultant force on the sphere. But when the two spheres are brought into each other's presence, the unit lines of force become somewhat more crowded together on the sides of the spheres which face than on the remote sides, and thus the resultant pull on either sphere tends to draw it toward the other. When the spheres are at distances great compared with their radii, the attraction is nearly proportional to the inverse square of the distance, which is Priestley's law. In the following year (1838) Faraday amplified* his theory of electrostatic induction, by making further use of the analogy with the induction of magnetism. Fourteen years previously Foisson had imagined! an admirable model of the molecular processes which accompany magnetization ; and this was now applied with very little change by Faraday to the case of induc- tion in dielectrics. " The particles of an insulating dielectric," he suggested,* " whilst under induction may be compared to a series of small magnetic needles, or, more correctly still, to a series of small insulated conductors. If the space round a charged globe were filled with a mixture of an insulating dielectric, as oil of turpentine or air, and small globular conductors, as shot, the latter being at a little distance from each other so as to be insulated, then these would in their condition and action exactly resemble what I consider to be the condition and action of the particles of the insulating dielectric itself. If the globe were charged, these little con- ductors would all be polar ; if the globe were discharged, they would all return to their normal state, to be polarized again upon the recharging of the globe." That this explanation accounts for the phenomena of specific * Exy. BM., SerUa xiv. t Cf. p. 65. % Sip. &n„ § 1679. 3,Bl,ZEdhyG00gle Faraday. 211 inductive capacity may be seen by what follows, which is substantially a translation into electrostatieal language of Poisson's theory of induced magnetism.* Let p denote volume-density of electric charge. For each of Faraday's " small shot " the integral flj pdxdydx, integrated throughout the shot, will vanish, since the total charge of the shot is zero : but if r denote the vector (x, y, z), the integral fjj pt dx dy dz will not be zero, since it represents the electric polarization of the abut. ; if there are iV shot per unit volume, the quantity P - N ffj (> r dx dy dz will represent the total polarization per unit volume. If d denote the electric force, and £ the average value of A, P will be proportional to £, say P-(.-l)K By integration by parts, assuming all the quantities concerned to vary continuously and to vanish at infinity, we have where 9 denotes an arbitrary function, and tbe volume-integrals are taken throughout infinite space. This equation shows that the polar-distribution of electric charge on the shot is equivalent to a volume-distribution throughout space, of density p~ = - div P. Now the fundamental equation of electrostatics may in suitable units be written, div d - p ; * W. Thornton (Kelvin), Camb. and Dub. Hath. Journal, November, 18*5 ; W. Thomson's Paptn on Eleetroitatict and Magmtirm, § 43 aqq. ; P. 0. MoootU, Arch, dea ac. phys. (Geneva) ri (1847), p. 193 ; Hem. della Son. ltd. Modena, (2) rir (1850), p. 49. D,Bl,ZEdhyG00gle 212 Faraday. • and this gives on averaging div E - />i + p, where pi denotes the volume-density of free electric charge, i.e. excluding that in the doublets ; or div (E + P) - p., or div (f E) - p,. This is the fundamental equation of electrostatics, as modified in order to take into account the effect of the specific inductive capacity t. The conception of action propagated step by step through a medium by the influence of contiguous particles had a firm hold on Faraday's mind, and was applied by him in almost every part of physics. " It appears to me possible," he wrote in 1838* " and even probable, that magnetic action may be communicated to a distance by the action of the intervening particles, in a manner having a relation to the way in which the inductive forces of static electricity are transferred to a distance ; the intervening particles assuming for the time more or less of a peculiar condition, which (though with a very imperfect idea) I have several times expressed by the term electro-tonic state."-f The same set of ideas sufficed to explain electric currents. Conduction, Faraday suggested,? might be " an action of contiguous particles, dependent on the forces developed in electrical excitement ; these forces bring the particles into a state of tension or polarity ;§ and being hi this state the contiguous particles have a power or capability of communicating these forces, one to the other, by which they are lowered and discharge occurs." » Bxp Sm., } 1729. + This duds had baen devised in 1831 to expreaa the slate of matter lubJMt to magneto-electric induction ; cf. Exp.^Rn., $ 60. J Sip. En. iii, p. 513. { Ai in olectroBtatic induction in dielectric*. 3,Bl,ZEdhyG00gle Faraday. 213 After working strenuously for the ten years which followed the discovery of induced currents, Faraday found in 1841 that his health waa affected ; and for four years he rested. A second period of brilliant discoveries began in 1845. Many experiments had been made at different times by various investigators* with the purpose of discovering a connexion between magnetism and light. These had generally taken the form of attempts to magnetize bodies by exposure in particular ways to particular kinds of radiation ; and a successful issue had been more than once reported, only to be negatived on reexamination. The true path was first indicated by Sir John HerscheL After his discovery of the connexion between the outward form of quartz crystals and their property of rotating the plane of polarization of light, Herschel remarked that a rectilinear electric current, deflecting a needle to right and left all round it, possesses a heiicoidal dissymmetry similar to that displayed by the crystals. " Therefore," he wrote,t " induction led me to conclude that a similar connexion exists, and must turn up somehow or other, between the electric current and polarized light, and that the plane of polarization would be deflected by magneto-electricity." The nature of this connexion was discovered by Faraday, who so far back as 1834} had transmitted polarized light through an electrolytic solution during the passage of the current, in the hope of observing a change of polarization. This early attempt failed ; but in September, 1845, he varied the experiment by placing a piece of heavy glass between the poles of an excited electro-magnet ; and found that the plane of polarization of a beam of light was rotated when the beam travelled through the glass parallel to the lines of force of the magnetic field.§ •e.g. by Horichini, o( Borne, in 1813, Quart. Journ. Sci. xix, p. 338; by Samuel Hunter Chriltie, of Cambridge, in 182S, Phil. Tram., 1828, p. 219 ; and by Miry Somernlle in the same year, Phil. Trans., 1826, p. 132. tSir. J. 'EenehO. in Benceloaet'* Lift of Faraday, p. 206. XSrp. Itn., f Ml. If lb., §2152. 3,Bl,ZEdhyG00gle 214 Faraday. In the year following Faraday's discovery, Airy* suggested a way of representing the effect analytically ; as might have been expected, this was by modifying the equations which had been already introduced by Mac Cullagh for the case of naturally active bodies. In Mac Cullagh 's equations fffY , ffY &Z \ffZ t&Z $ 2837. 3,Bl,ZEdhyG00gle Faraday. '2'2l permit the exertion of the magnetic force with more facility in one direction than another ; and that direction would be the magnecryatallie axis. Hence, when in the magnetic field, the magnecryatallie axis would be urged into a position coincident with the magnetic axis, by a force correspondent to that difference, just as if two different bodies were taken, when the one with the greater conducting power displaces that which is weaker." This hypothesis led Faraday to predict the existence of another type of magnecrystallic effect, as yet unobserved. " If such a view were correct," he wrote,* " it would appear to follow that a diamagnetic body like bismuth ought to be less diamagnetic when its magnecryatallie axis is parallel to the magnetic axis than when it is perpendicular to it. In the two positions it should be equivalent to two substances having different conducting powers for magnetism, and therefore if submitted to the differential balance ought to present differential phaenomena." This expectation was realized when the matter was subjected to the test of experiment, f The aeries of Faraday's "Experimental Researches in Electricity " end in the year 1855. The closing period of his life was quietly spent at Hampton Court, in a house placed at his disposal by the kindness of the Queen ; and here on August 25th, 1867, be passed away. Among experimental philosophers Faraday holds by uni- versal consent the foremost place. The memoirs in which his discoveries are enshrined will never cease to be read with admiration and delight ; and future generations will preserve with an affection not less enduring the personal records and familiar letters, which recall the memory of his humble and unselfish spirit. •Exp. Mu., i 283B. t Hid., S 28*1. 3,Bl,ZEdhyG00gle The Mathematical Electricians of the CHAPTER VI I. TEE MATHEMATICAL ELECTRICIANS OF THE MIDDLE OF THE NINETEENTH CENTURY. While Faraday was engaged in discovering the laws of induced currents in his own way, by use of the conception of lines of force, his contemporary Franz Neumann was attacking the same problem from a different point of view. Neumann preferred to take Ampere as his model ; and in 1845 published a memoir* in which the laws of induction of currents were deduced by the help of Ampere's analysis. Among the assumptions on which Neumann based his work was a rule which had been formulated, not long after Faraday's original discovery, by Emil Lenz.t and which may be enunciated as follows : when a conducting circuit is moved in a magnetic field, the induced current flows in such a direction that the ponderomotive forces on it tend to oppose the motion. Let da denote an element of the circuit which is in motion, and let G da denote the component, taken in the direction of motion, of the ponderomotive force exerted by the inducing current on da, when the latter is carrying unit current ; so that the value of G is known from Ampere's theory. Then Lenz's rule requires that the product of 0 into the strength of the induced current should be negative. Neumann assumed that this is because it consists of a negative coefficient multiplying the square of G; that is, he assumed the induced electro- motive force to be proportional to 0. He further assumed it to be proportional to the velocity v of the motion; and thus obtained for the electromotive force induced in da the expression - tvGds, where * denotes a constant coefficient. By aid of this formula, ■ Berlin Abhandlungan, 18(5, p. I ; 1848, p. 1 ; reprinted u No. 10 and No. 36 of Oatwald'a KLuiiktr; translated Journal de Hath. liii (IS*8)- P- "3. t Ann. d. Phya. mi (IBM), p. 483. dhyGoOgk Middle of the Nineteenth Century, 223 in the earlier part* of the memoir, he calculated the induced currents in various particular cases. But having arrived at the formulae in this way, Neumann noticedf a peculiarity in them which suggested a totally different method of treating the subject. In fact, on examining the expression for the current induced in a circuit which is in motion in the field due to a magnet, it appeared that this induced current depends only on the alteration caused by the motion in the value of a certain function ; and, moreover, that this function is no other than the potential of the ponderomotive forces which, according to Ampere's theory, act between the circuit, supposed traversed by unit current, and the magnet. Accordingly, Neumann now proposed to reconstruct his theory by taking this potential function as the foundation. The nature of Neumann's potential, and its connexion with Faraday's theory, will be understood from the following considerations : — The potential energy of a magnetic molecule M in a field of magnetic intensity B is (B . K) ; and therefore the potential energy of a current i Sowing in a circuit s in this field is i|f(B.as), where S denotes a diaphragm bounded by the circuit s; as is Been at once on replacing the circuit by its equivalent magnetic shell S. If the field B be produced by a current i' flowing in a circuit s", we have, by the formula of Biot and Savart, ,.418*1 * {{ 1-8. It may be remarked that Neumann, in making use of Ohm'i law, »u (like everyone elm it this time) unaware of the identity uf eloctrowopic force with electrostatic potential. t j 9. 3,Bl,ZEdhyG00gle 224 The Mathematical Electricians of the Hence, the mutual potential energy of the two currents is which by Stokes's transformation may be written in the fo (di.diQ 'IL*. This expression represents the amount of mechanical work which must be performed against the electro-dynamic pondero- motive forces, in order to separate the two circuits to an infinite distance apart, when the current-strengths are maintained unaltered. The above potential function has been obtained by con- sidering the ponderomotive forces ; but it can now be connected with Faraday's theory of induction of currents. For by interpreting the expression jj(B.a») in terms of hues of force, we see that the potential function represents the product of * into the number of unit-lines of magnetic force due to s', which pass through the gap formed by the circuit s ; and since by Faraday's law the currents induced in s depend entirely on the variation in the number of these lines, it is evident that the potential function supplies all that is needed for the analytical treatment of the induced currents. This was Neumann's discovery. The electromotive force induced in a circuit s by the motion of other circuits $', carrying currents i', is thuB proportional to the time-rate of variation of the potential (di.dQ. •1,1^ so that if we denote by a the vector 'I.." dhyGoogle Middle of the Nineteenth Century. 225 which, of course, is a function of the position of the element ds from which t ie measured, then the electromotive force induced in any circuit-element ds by any alteration in the currents which rive rise to a is (i.a.). The induction of currents is therefore governed by the vector a ; this, which is generally known as the vector-potential, has from Neumann's time onwards played a great part in electrical theory. It may be readily interpreted in terms of Faraday's conceptions ; for (a. da) represents the total number of unit lines of magnetic force which have passed across the line-element ds prior to the instant t. The vector- potential may in fact be regarded as the analytical measure of Faraday's electTotonic state* While Neumann was endeavouring to comprehend the laws of induced currents in an extended form of Ampere's theory, another investigator was attempting a still more ambitions project : no less than that of uniting electrodynamics into a coherent whole with electrostatics. Wilhelm Weber (ft. 1804, d. 1890) was in the earlier part of Ms scientific career a friend and colleague of Gauss at Giittingen. In 1837, however, he became involved in political trouble. The union of Hanover with the British Empire, which had subsisted since the accession of the Hanoverian dynasty to the British throne, was in that year dissolved by the operation of the Salic law ; the Princess Victoria succeeded to the crown of England, and her uncle Ernest-Augustus to that of Hanover. The new king, who was a pronounced reactionary, revoked the free constitution which the Hanoverians had for some time enjoyed ; and Weber, who took a prominent part in opposing this action, was deprived of his professorship. From 1843 to 1849, when his principal theoretical researches in electricity were made, be occupied a chair in the University of Leipzig. The theory of Weber was in its origin closely connected with the work of another Leipzig professor, Fechner, who in 1845f introduced certain assumptions regarding the nature of • Cf. up. 212, 272. r Ann. d. Pbya. Uiv (1846), p. 337. 0 3,Bl,ZEdhyG00gle 226 The Mathematical Electricians of the electric currents. Fechner supposed every current to consist in a streaming of electric charges, the vitreous charges travelling in one direction, and the resinous charges, equal to them in magnitude and number, travelling in the opposite direction with equal velocity. He further supposed that like charges attract each other when they are moving parallel to the same direction, while unlike charges attract when they are moving in opposite directions. On these assumptions he succeeded in bringing Faraday's induction effectB into connexion with Ampere's laws of electrodynamics. In 1846 Weber * adopting the same assumptions aa Fechner, analysed the phenomena in the following way: — The formula of Ampere for the ponderomotive force between two elements ds, da' of currents *, i', may be written „ .., . ,,/2 * The unit* which haTe been adopted in the above investigation depend on the electrodTnajnia action* of curranta ; i.e. the; ere such that two unit currents flowing in parallel circular cirouite at a certain distance apart eiert unit poaderomutive tone on each other. The quantity of electricity conveyed in unit time by such a unit current is adopted ai the unitt&acge. This unit charge is not identical with the eleetrottatio unit charge, which is deSned-toka such that two unit charge* at unit distance apart repel each other with unit polnieiojcotive force. Hence lha necessity for introducing the factor e. D,Bl,ZEdhyG00gle 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 eUetrvn-theorus — 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), bat 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 dU d /BU Consider now two material particles at distance r apart, whose mechanical kinetic energy is T, and whose mechanical potential energy is V, and which carry charges t and e1. 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 w and the potential energy is V+— ■ To such a system the principle of conservation of energy may be applied : the equation of energy is, in fact, dhyGoogle 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 - w'r*/2r 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 p 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 J(/*-ep)»a+ F"- 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 where the integration is taken over the sphere, and where a denotes the surface-density ; p is independent of the position of the particle /* within the sphere. If now the electric charge on the sphere is so great that ep is greater-tharr/^ then v* and V must increase and diminish together; which ie 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 eeV b-E f r |(r.T-)-(r.T)|n T w J' where t and Y denote the velocities of the charges ; so that the * This example »«» gran by Helmholtt, Journal fiir Math. Ixxv {1873), p. 36 ; PML Mag. xlir (1873), p. 680. 3,Bl,ZEdhyG00gle 230 The Mathematical Electricians of the mutual energy of two current-elements containing charges e, e respectively of each kind of electricity, ib ^M(r-r>(r.T)}'+[(r.T>(p.T)!'t(-(r.vO-(r.T)}'-((-r.TX«)n, If ds, ds' denote the lengths of the elements, and i, % the currents in them, we have ids - 2#t, t'dV = 2eV ; bo the mutual energy of two current-elements is £(r.d<) .(r.*). The mutual energy of tdi with all the other currents is therefore t(di.a), where a denotes a vector-potential r ,(r.a.-).r By reasoning similar to Neumann's, it may be shown that the electromotive force induced in di by any alteration in the rest of the field is -(«■.•)! 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 3,Bl,ZEdhyG00gle 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 Been 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,t indeed, that the assumption of opposite electricities moving with equal and opposite veloci- ties in a circuit ia 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 Riemann (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. Riemann proposed as the electrokinetic energy of two electrons e(x, y, a) and e'(x', y', zf) 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 £ + Fx. * Cumptes Rendui, Inr {1872}, p. 760. Cf. alao Comptea Rendui, ox (1890), p. 813, and Holimiiller, Zeitachrift fur Hath. u. Phya., 1870, p. SB. + Thia bad been doae in in inaugural diaaertation by Beegen, Oiittingan, 1SS4. dhyGoogle 234 The Mathematical Electricians of the and, regarding Ft 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 h 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 Vender 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 # 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 Vender's own hypothesis of an intra-mercurial planet But it was found later* that j of the excess could be explained by substituting Hlemann's electrodynamic law for Weber's, and that a com binatioD of the laws of Riemann and Weber would give exactly the amount desired, t After the publication of his memoir on the law of force between electrons, Weber turned hie 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 ts quite natural to suppose that currents would be induced in these molecular ■ By Mauri™ Levy, Comptes Rendno, cz 11860), p. 645. t Tie consequences of adopting tie eleetrodynamic lav of Clausius {for which *ee later) were discussed by Oppenheim, Zur Fragi ttasA ibr PortpJUiuimft- gaohwindigkeit dtr Gravitation, Wien, 1886. ♦ Leipzig Berichte, i (1S4T), p. 348 ; Ann. d. Php. lxxiii (1818), p. 241 ; tranalated Taylor's Scirntifie Mtmoin, y, p. 477 J Abhandl. der K. Sochi. Go*, i (1862), p. 483; Ann. d. Fhyt. Ixxnil (1862), p. US; tram. Tyndall and Francia' Scientific Mmoir; p. 163. D,Bl,ZEdhyG00gle 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: , 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 * Annates de Chimie (7) t (1846), p. 289, D,Bl,ZEdhyG00gle 53© 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 Poiason.but is easilydeducible 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 ; hut 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. l'hys. lmrii(1862),p. U6 ; TyndnlUnd Francis' Bei. *«.., p. HIS. 3,Bl,ZEdhyG00gle Middle o/ the Nineteenth Century. 237 mutual action of the molecular magnets themselves. In the uii magnetized condition the molecules " arrange themselves so as to satisfy their mutual attraction hy 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£ 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 & Now let an increasing magnetic force B 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 3 attains the value 3m/e*, they become perpendicular to the line e, after which they remain in this position, when B is increased further. Thus they display the phenomena of induc- tion initially proportional to the magnetizing force, and of saturation. If the magnetizing force B 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 B acts in the opposite direction, the equilibrium will be stable only so long as R is less than m/c*; when H increases beyond this limit, the equilibrium becomes unstable, and the magnets tarn over so as to point in the direction of B; when H is gradually decreased to zero, they remain in their new posi- tions, thus illustrating the phenomenon of residual magnetism. * Proc. Roy. Soe. xxxr (1883), p. 17S. t Trtatiu M EUct. & Mag., \ 443. J Phil. Mag. «>i (I8B0),p. 205; Magnetic Induction in Iron and tlntr Mita.1*, mi. of Hath, zzvii (1897), p. 6. D,Bl,ZEdhyG00gle H38 The Mathematical Electricians of the m By taking a large number of such pairs of magnetic molecules, originally oriented iu 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 moss 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 Koget's and Faraday's defeneef 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.aos. 3,Bl,ZEdhyG00gle Middle of the Nineteenth Century. 239 and completed by James Prescott Joule, of Manchester, in 1841. Jonle, 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 waB 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 electrolysed 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 he+ 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 Roget and Faraday, thus perfected by Jonle, 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, ZnlZn SOJCu SO,|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. S8. t Phil. Mag. xix (1841), p. 260 ; Joule'* Scitntijtc Fapirs i, p. SO. I Phil. Mag. XX (18*1), p. 98: of. aUo Phil. Mag. xxil (IS43), p. '201. 3,Bl,ZEdhyG00gle 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 tiling) 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 (6. 1821, d. 1894), in a memoir which was published in 1847, and which will be more folly noticed presently, and by W. Thomson (Lord Kelvin) in 1851t; the equations have subsequently received only one important modification, which is due to Helmholtz. J Helmholtz pointed out that the electrical energy famished 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," ie., possible output of mechanical work, of a system maintained at the absolute temperature T, then a fraction TdE EdT of this work is obtained, not at the expense of the thermal or * The heat of formation of a gramme-molecule of ZnSOi ie greater than the boat of formation of a gramme- molecule of CuSOt by about 50,000 calories ; and with diialent metals, 46,000 calorie* per gramme- molecule corresponds to an e.ro.f. of oai volt; so the e.m.f. of a Daniell cell should be 60/46 volts, which it nearly the f Kelvin's Math, and PhtJI. Paptrt, t, pp. 473, 490. I Berlin Sitzuugsber., 18B2, pp. 22, 825: 1883, p. 647. } Quart. Jouni. Math., April, 1855; Kelvin's Mali, and Phf. Papm, i, p. 297, eqn. (7). 3,Bl,ZEdhyG00gle Middle oj 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 Boget, Faraday, and Joule ia expressed by the equation where E denotes the available or electrical energy, which is measured by the electromotive force of the cell, and where A denotes the heat of the chemical reaction which supplies this energy. In accordance with Thomson's principle, we must replace this equation by 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 TdEfdT; 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. Eelmholtz'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 Fhysik ; and it was eventually published as a separate treatise. f lu this memoir it was asserted? that the conservation of • Od July 23rd, 1847. t Berlin, G. A. Reimer. English Translation in Tyndall 4 Francis' ScUntiJU Iftmoiri, p. 114. The publisher, to HelmhohVs " great surprise," gave him an honorarium- CI. Htrmenn cm Stlmholtt, by Leo Koenigabeiger ; English translation by F. A. Welby. I Helmholtx had been partly anticipated by W. R. Grove, in his lectures on the Correlation of Phyticnl Fitrcii, which wers 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 i.C physical forces, is the establish moot of their equivalent of power, Or their measurable relation to a riven standard." 3,Bl,ZEdhyG00gle 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 Rumford, Joule, and Robert Mayor* ; and that any of these forms of energy may be converted into the chemical, electro- static, voltaic, and magnetic forma. 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 t 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 X V, where X denotes a proper fraction. At this stage let charges edX be brought from a great distance and added to the charges Xe. The work required for this is SedX.XV, so the total work required in order to bring the system from infinite dispersion to its final state is 2«r. [ XdX, or JS«F. 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 I peptic dydz, where p0 denotes the density of Foisson's equivalent magnetUa- ■ Julius Eobeit Mayer (*. 1811, rf. 1878), who m a medical man in Heilbronn, asserted the equivalence of heat and «-orV in 1842, Annul, d. Chemie, xlii, p. 233 ; hie memoir, like that uf Helmbolti, was first declined by the editors of the Annalen der Phyaik. An English Iranal&tton of one of Mayer's memoirs u printed in Phil. Mag. iit (1883), p. 493. 3,Bl,ZEdhyG00gle Middle of the Nineteenth Century. 243 tion, for the permanent magnets only, and at denotes the magnetic potential." Helmhoitz, 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, Helmhoitz 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. The correct theory of the energy of magnetic and electro- magnetic fields is due mainly to W. Thomson (Lord Kelvin). Thomson's researches on this subject commenced with one or two short investigations regarding the ponderomotive forces which act on temporary magnets. In 1847 he discussedf the case of a small iron sphere placed in a magnetic field, showing that it is acted on by a ponderomotive force represented by - grad cJT, where c denotes a constant, and £ denotes the magnetic force of the field ; such a sphere must evidently tend to move towards the places where E1 is greatest. The same analysis may be applied to explain why diamagnetic bodies tend to move, as in Faraday's experiments, from the stronger to the weaker parts of the field. * We auppaee all trmmitiom to be eontinuoua, so si 10 avoid the necessity fur writing surface -integrals separately. rCamb. and Dub. Math. Journal, ii (1847). p. 230; W. Thomson'* Paptn m EteclroMtattCM and Maguauin, p. 49S ; of. alto Phil. Mag. zzxvii (1850.1, p. 341. R 2 24:4 The Mathematical Electricians of the Two years later Thomson presented to the Koyal Society a memoir* in which the results of Poisson 'a theory of magnetism were derived from experimental data, without making use of the hypothesis of magnetic fluids; and this was followed in 1850 by a second memoir.f in which Thomson drew attention to the fact previously noticed by Poisson,J that the magnetic intensity at a point within a magnetized body depends on the shape of the small cavity in which the exploring magnet is placed. Thomson distinguished two vectors ;§ one of these, by later writers generally denoted by B, represents the magnetic intensity at a point situated in a sinall crevice in the magnetized body, when the faces of the crevice are at right angles to the direction of magnetization ; the vector B is always circuital. The other vector, generally denoted by H, represents the magnetic intensity in a narrow tubular cavity tangential to the direction of magnetization ; it is an irrotational vector. The magnetic potential tends at any point to a limit which is independent of the shape of the cavity in which the point is situated ; and the space-gradient of this limit is identical with JL Thomson called B the " magnetic force according to the electro-magnetic definition," and H the " magnetic force accord- ing to the polar definition " ; but the names magnetic induction and magnetic force, proposed by Maxwell, have been generally used by later writers. It may be remarked that the vector to which Faraday applied the term " magnetic force," and which he represented by lines of force, is not H, but B ; for the number of unit lines of force passing through any gap must depend only on the gap, and not on the particular diaphragm filling up the gap, across which the flux ia estimated ; and this can be the case only if the vector which is represented by the lines of force is a circuital vector. * Phil. Tram. , 1851, p. 213 ; Thomson's 1'uptri on Ehct. and Mag., p. 345. t Phil. Tram., 18-51, p. £69 ; Pnptrt en Mitel, and Mm/., p. 382. J Cf . p. 64, { Loo. tit., § 18 of tli e original paper, and { 617 of the lupunt^ 3,Bl,ZEdhyG00gle Middle of the Nineteenth Century. 245 Thomson introduced a number of new terms into magnetic science — as indeed he did into every science in which he was interested. The ratio of the measure of the induced magnetiza- tion I,-, in a temporary magnet, to the magnetizing force H, he named the susceptibility ; it is positive for paramagnetic and negative for diamagnetic bodies, and is connected with Poisson's constant A," by the relation 3 kp * " 4* 1 - V where « denotes the susceptibility. By an easy extension of Poisson's analysis it is Been that the magnetic induction and magnetic force are connected by the equation B = H + 4rt, where I denotes the total intensity of magnetization : so if I* denote the permanent magnetization, we have B = H + 4irli + 4rt , - „H + 4*1* where ft denotes (1 + 4jt*): ft was called by Thomson the •permeability. In 1851 Thomson extended his magnetic theory so as to include magnecrystollic phenomena. The mathematical founda- tions of the theory of magnecrystallic action had been laid by anticipation, long before the experimental discovery of the phenomenon, in a memoir read by Poisson to the Academy in February, 1824. Poisson, as will be remembered, had supposed temporary magnetism to be due to " magnetic fluids," movable within the infinitely small " magnetic elements " of which he assumed magnetizable matter to be constituted. He bad not overlooked the possibility that in crystals these magnetic elements might be non-Bpherical (e.g. ellipsoidal), and symmetri- cally arranged ; and had remarked that a portion of such a crystal, when placed in a magnetic field, would act in a manner depending on its orientation. The relations connecting * Cf . p. 66. 3,Bl,ZEdhyG00gle 246 The Mathematical Electricians a/ the the induced magnetization I with the magnetizing force H he had given in a form equivalent to I 7, - a.ff, + b'Ht + c"E„ ] It - a"Hz 4 bHt + c'H„ ( I, - a'H, + b"Hv + cH„ Thomson now* showed that the nine coefficients a, b', c" . . ., introduced by Poisaon, are not independent of each other. For a sphere composed of the magnecryBtalline substance, if placed in a uniform field of force, would be acted on by a couple : and the work done by this couple when the sphere, supposed of unit volume, performs a complete revolution round the axis of x may be easily shown to be wff (1 - HfjH') (- b" + c'). Hut this work must be zero, since the system is restored to its primitive condition ; and hence b" and c must be equal Similarly c" = a', and a" - V. By change of axes three more coefficients may be removed, so that the equations may be brought to the form Ig ■ KiHt, ly — K%Hy, I, = Kl/fl, where ku *,, k> may be called the principal magnetic suscepti- bilities. In the same year (1851) Thomson investigated the energy which, as was evident from Faraday's work on self-induction, must be stored in connexion with every electric current. He showed that, in his own words,! "the value of a current in a closed conductor, left without electromotive force, is the quantity of work that would be got by letting all the infinitely small currents into which it may be divided along the lines of motion of the electricity come together from an infinite distance, and make it up. Each of these ' infinitely small currents ' is of course in a circuit which is generally of finite length ; it is the section of each partial conductor and the strength of the current in it that must be infinitely small." •Phil. Mag. (4) i (1B51), p. 177: Paptri on E/MmliUia and Majnelum, p. 471- t Papiri on Eltctroitatiti and Magrutitm, p. 446. dhyGoogle Middle of the Nineteenth Century. 247 Discussing next the mutual energy due to the approach of a permanent magnet and a circuit carrying a current, he arrived at the remarkable conclusion that in this case there is uo electrokinetic energy which depends on the mutual action ; the energy is simply the sum of that due to the permanent magneta and that due to the currents. If a permanent magnet is caused to approach a circuit carrying a current, the electromotive force acting in the circuit is thereby temporarily increased ; the amount of energy dissipated as Joulian heat, and the speed of the chemical reactions in the cells, are temporarily increased also. But the increase in the Joulian heat is exactly equal to the increase in the energy derived from consumption of chemicals, together with the mechanical work done on the magnet by the operator who moves it ; so that the balance of energy is perfect, and none needs to be added to or taken from the electrokinetic form. It will now be evident why it was that Helniholtz escaped in this case the errors into which he was led in other cases by his neglect of electrokinetic energy ; for in this case there was no electrokinetic energy to neglect Two years later, in 1853, Thomson* gave a new form to the expression for the energy of a system of permanent and temporary magnets. We have seen that the energy of such a system is represented by where pa denotes the density of Foisson's equivalent magnetiza- tion for the permanent magnets, and 0 denotes the magnetic potential, and where the integration may be extended over the whole of apace. Substituting for p» its value - div I«,t the expression may be written in the form \ ^divlodccfytfc; .p. a sdzEdhyGoogle • Proc. Glugov Phil Soo. iii (1863}, p. 381; Kelvin'. Math. Paptrt, i, p. 621. tCl p. 14. 248 The Mathematical Electricians of the or, integrating by parts, -if|T(Io.grad$), and the other is the derivate with respect to the time (with sign reversed) of a vector- potential a ; so that if i denote the current and k the specific conductivity, Ohm's law is expressed by the equation i = k (c' grad - a). Kirchhoff calculated the value of a by aid of Weber's formula for the inductive action of one current element on another; the result is where r denotes the vector from the point (x, y, z), at which a is measured, to any other point (#', y", z") of the conductor, at which the current is i' ; and the integration is extended over the whole volume of the conductor. The remaining general equations are the ordinary equation of the electrostatic potential V'$ + 4irp = 0 (where p denotes the density of electric charge), and the equation of conservation of electricity |.divi-0. • Ann. d. Phj-«. tii (18S7), p. 6Z9: On. Abhandl., p. 16*. 3,Bl,ZEdhyG00gle Middle of the Nineteenth Century. 261 It will be seen that Kirchhoff's electrical researches were greatly influenced by those of Weber. The latter investiga- tions, however, did not enjoy unquestioned authority ; for there was still a question as to whether the expressions given by Weber for the mutual energy of two current elements, and for the mutual energy of two electrons, were to be preferred to the rival formulae of Neumann and Riemann. The matter was examined in 1870 by Helmholtz, in a series of memoirs* to which reference has already been mad&t Helmholtz remarked that, for two elements da, da', carrying currents i, %', the electro- dynamic energy is ii'jA.t.&t') r ' according to Neumann, and £(r.4i)fr.aO, according to Weber; and that these expressions differ from each other only by the quantity ii' dads' , , , , -, ,,, , > „ |- cos (as. as) + cos (r. da) ooa(r. da )\, ... ,. aV or Wdsdt •!—=-> ; dads since this vanishes when integrated round either circuit, the two formulae give the same result when applied to entire currents. A general formula including both that of Neumann and that of Weber is evidently »"( those of «'. The unknown constant * has clearly no influence ao long as closed circuits only are considered : if h be replaced by zero, the expression for the kinetic potential becomes — {axe + yy + zz - r), which, as will appear later, closely resembles the corresponding expression in the modern theory of electrons. Clausius' formula has the great advantage over "Weber's, that it does not compel us to assume equal and opposite velocities for the vitreous and resinous charges in an electric current; on the other hand, Clausius' expression involves the absolute velocities of the electrons, while Weber's depends only on then- relative motion; and therefore Clausius' theory requires the assumption of a fixed aether in space, to which the velocities v and v' may be referred. When the behaviour of finite electrical systems is predicted from the formulae of Weber, Riemann, and Clausius, the three laws do not always lead to concordant results. For instance, if a circular current be rotated with constant angular velocity round its axis, according to Weber's law there would be a development of free electricity on a stationary conductor in the neighbourhood ; whereas, according to Clausius' formula there would be no induction on a stationary body, but electrification D,Bl,ZEdhyG00gle MiddU of the Nineteenth Century. 263 would appear on a body turning with the circuit as if rigidly connected with it. Again* lot a magnet be suspended within a hollow metallic body, and let the hollow body be suddenly charged or discharged; then, according to Clausius' theory, the magnet is unaffected; but according to Weber's and Biemann's theories it experiences an impulsive couple. And again, if an electrified disk be rotated in its own plane, under certain circumstances a steady current will be induced in a neighbouring circuit according to Weber's law, but not according to the other formulae. An interesting objection to Clausius' theory was brought forward in 1879 by Frohlichf — namely, that when a charge of free electricity and a constant electric current are at rest relatively to each other, but partake together of the translatory motion of the earth in space, a force should act between them if Clausius' law were true. It was, however, shown by BuddeJ that the circuit itself acquires an electrostatic charge, partly as a result of the same action which causes the force on the external conductor, and partly as a result of electrostatic induction by the charge on the external conductor ; and that the total force between the circuit and external conductor is thus reduced to zero.§ We have seen that the discrimination between the different lawB of electrodynamic force is closely connected with the question whether in au electric current there are two kinds of electricity moving in opposite directions, or only one kind moving in one direction. On the unitary hypothesis, that the * The two following cru'ial experiments, with others, Were Buggestud by E. Bodde, Ann. d. Phis, m (1BST), p. 100. t Ann. d. Pliyu. be (18SU), p. 261. * Ann. d. Phja. z (1880), p. 663. 0 Thin cue of n charge and current moving eide by tide vu afterwards examined by Fits Gerald (Trans. Bo;. Dub. Soo. i, 1882; Scienl. Writing* *f G. F. Fitz Gerald, p. Ill) without reference to Clausius' formula, from the standpoint of Maxwell's theory. The result obtained waa the »m« — namely, that the electricity induced on the conductor carrying the current neutralize* the ponderomotive force between the current and the external charge. 3,Bl,ZEdhyG00gle 264 The Mathematical Electricians of the current consists in a transport of one kind of electricity with a definite velocity relative to the wire, it might be expected that a coil rotated rapidly about its own axis would generate a magnetic field different from that produced by the same coil at rest. Experiments to determine the matter were performed by A. Foppl* and by E. L. Nichols and W. S. Franklin,t bot with negative results. The latter investigators found that the velocity of electricity must be such that the quantity conveyed past a specified point in a unit of time, when the direction of the current was that in which the coil was travelling, did not differ from that transferred when the current and coil were moving in opposite directions by as much as one part in ten million, even when the velocity of the wire was 9096 cm./sec They considered that they would have been able to detect a change of deflexion due to the motion of the coil, even though the velocity of the current had been considerably greater than a thousand million metres per second. During the decades in the middle of the century consider- able progress was made in the science of thermo-electricity, whose beginnings we have already described.^ In Faraday's laboratory note-book, under the date July 28th, 1836, we readg : — " Surely the converse of thermo-electricity ought to be obtained experimentally. Pass current through a circuit of antimony and bismuth." Unknown to Faraday, the experiment here indicated had already been made, although its author had arrived at it by a different train of ideas. In 1834 Jean Charles Peltier|| (o. 1785, d. 1845) attempted the task, which was afterwards performed with success by Joule,1T of measuring the heat evolved by the passage of an electric current through a conductor. He found that a current produces in a homogeneous conductor an elevation * Ann. d. Fhjt. xxvii (1886), p. 410. t Amer. Jour. 3d., nirii (1889), p. 103. X Cf. pp. 92, 93. § Bailee Jones's Lift of Faraday, ii, p. 76. | Annalci de Cliimio, Wi (1834), p. 371. * Cf. p. 239. 3,Bl,ZEdhyG00gle Middle of the Nineteenth Century. 265 of temperature, which is the some in all parts of the conductor where the cross-section is the same ; but he did not succeed in connecting the thermal phenomena quantitatively with the strength of .the current — a failure which was due chiefly to the circumstance that his attention was fixed on the rise of temperature rather than on the amount of the heat evolved. But incidentally the investigation led to an important discovery — namely, that when a current was passed in succession through two conductors made of dissimilar metals, there was an evolution of heat at the junction ; and that this depended on the direction of the current ; for if the junction was heated when the current flowed in one sense, it was cooled when the current flowed in the opposite sense. This Peltier effect, as it is called, is quite distinct from the ordinary Joulian liberation of heat, in which the amount of energy set free in the thermal form is unaffected by a reversal of the current ; the Joulian effect is, hi fact, propor- tional to the square of the current-strength, while the Peltier effect is proportional to the current-strength directly. The Peltier heat which is absorbed from external sources when a current i flows for unit time through a junction from one metal B to another metal A may therefore be denoted by where T denotes the absolute temperature of the junction. The function Y1AB (T) is found to be expressible as the difference of two parts, of which one depends on the metal A only, and the other on the metal B only ; thus we can write ni(T>-n„W-n,(T). In 1851 a general theory of thermo-electric phenomena was constructed on the foundation of Seebeck's* and Peltier's dis- coveries by W. Thomson.t Consider a circuit formed of two • Of. pp. 92, S3. t Proc. R.8. Edinb. iii (1861), p. 91 ; Phil. Mag. iii (18S2), p. 629 : Kelvin'* Math, tmd Pkfe. Paper., i, p. 316. Cf. alio Tnna. E. 8. Edinb. zzi (1854), p. 123, reprinted in Paper; i, p. 232 ; and Phil. Trans., 1866, reprinted in Paper*, ii, p. 189. 3,Bl,ZEdhyG00gle 266 The Mathematical Electricians of the metals, A and B, and let one junction be maintained at a slightly higher temperature (T 4 ST) than the temperature T of the other junction. As Seebeck had shown, a thermo-electric current will be set up in the circuit. Thomson saw that such a system might be regarded as a beat-engine, which absorbs a certain quantity of heat at the hot junction, and converts part of this into electrical energy,' liberating the rest in the form of heat at the cold junction. If the Joulian evolution of heat be neglected, the process is reversible, and must obey the second law of thermodynamics ; that is, the sum of the quantities of heat absorbed, each divided by the absolute temperature at which it is absorbed, must vanish. Thus we have T + ST ' T ' so the Peltier effect Tl^(T') must be directly proportional to the absolute temperature T, This result, however, as Thomson well knew, was contradicted by the observations of Cumming, who had shown that when the temperature of the hot junction is gradually in creased, the electromotive force rises to a maximum value and then decreases. The contradiction led Thomson to predict the existence of a hitherto unrecognized thermo-electric phenomenon — namely, a reversible absorption of heat at places in the circuit other than the junctions. Suppose that a current flows along a wire which is of the same metal throughout, but varies in temperature from point to point. Thomson showed that heat must be liberated at some points and absorbed at others, so as either to accentuate or to diminish the differences of temperature at the different points of the wire. Suppose that the heat absorbed from external sources when unit electric charge passes from the absolute temperature T to the temperature (T + ST) in a metal A is denoted by S^T). ST. The thermodynamical equation now takes the corrected form UUT+BT) Ui(T) ST 3,Bl,ZEdhyG00gle Middle oflhi Nineteenth Century. 267 Since the metals A and B are quite independent, this gives ~TVtf F~ ~ *(r> T " "• 1IT\-Td 'Siffll *(r> r3?rr-r This equation connects Thomson's "specific heat of electricity" 5!i(T) with the Peltier effect In 1870 P. G. Tait* found experimentally that the specific heat of electricity in pure metals is proportional to the absolute temperature. We may therefore write SA{T) ■= a4T, where aA denotes a constant characteristic of the metal A. The thermodynamical equation then becomes rff'i T )"'* ni(2>»iJ,+ ^, where Tj denotes another constant characteristic of the metal. The chief part of the Peltier effect arises from the term vAT. By the investigations which have been described in the present chapter, the theory of electric currents was considerably advanced in several directions. In all theBa researches, how- ever, attention was fixed on the conductor carrying the current as the seat of the phenomenon. In the following period, interest was centred not so much on the conductors which carry charges and currents, as on the processes which take place in the dielectric media around them. * I'roc. B. S. Edinb. ra(l870), p. .108. Cf. also Butelli, Atti dells R. Ace. di Torino, zxii (1886), p. 4S, translated Phil. Mug. niv (1887), p. 295. 3,Bl,ZEdhyG00gle ( 268 ) CHAPTER VIII. Since the time of Descartes, natural philosophers have never ceased to speculate on the manner in which electric and magnetic influences are transmitted through apace. About the middle of the nineteenth century, speculation assumed a definite form, and issued in a rational theory. Among those who thought much on the matter was Kail Friedrich Gauss (b. 1777, d. 1855). In a letter" to Weber of date March 19, 1845, Gauss remarked that he had long ago proposed to himself to supplement the known forces which act between electric charges by other forces, such as would cause electric actions to be propagated between the charges with a finite velocity. But he expressed himself as determined not to publish his researches until he should have devised a mechanism by which the transmission could be conceived to be effected ; and this he had not succeeded in doing. More than one attempt to realize Gauss's aspiration was made by his pupil Riemann. In a fragmentary note,t which appears to have been written in 1853, but which was not published until after his death, Riemann proposed an aether whose elements should be endowed with the power of resisting compression, and also (like the elements of MaeCulIagh's aether) of resisting changes of orientation. The former pro- perty he conceived to be the cause of gravitational and electrostatic effects, and the latter to be the cause of optical and magnetic phenomena. The theory thus outlined was apparently not developed further by its author ; but in a short investigation^ which was published posthumously in 18G7J he • Qaun' Wtrkt, Y, p. 629. + Eiemmm'n Wtrkt, 2* Aufl., p. 626. X Ann. d. I'hys. ixxxi (1887}, p. 237 ; Riemann' • ftrkt, 2* Aufl., p. 283 : Phil. Hag. iniv (1867), p. 308. i It had teen presented to the Gottingen Academy in IS53, bnt afterwards withdrawn. dhyGoOglc Maxwell. 269 returned to the question of the process by which electric action is propagated through space. In this memoir he proposed to replace Poisson'B equation for the electrostatic potential, namely, by the equation according to which the changes of potential due to changing electrification would be propagated outwards from the charges with a velocity c. This, so far as it goes, is in agreement with the view which is now accepted as correct ; but Kiemann's hypothesis was too slight to serve as the basis of a complete theory. Success came only when the properties of the inter- vening medium were taken into account. In that power to which Gauss attached so much importance, of devising dynamical models and analogies for obscure physical phenomena, perhaps no one has ever excelled W. Thomson"; and to him, jointly with Faraday, is due the credit of having initiated the theory of the electric medium. In one of his earliest papers, written at the age of seventeen,! Thomson compared the distribution of electrostatic force, in a region containing electrified conductors, with the distribution of the flow of heat in an infinite solid : the equipotential surfaces in the one case correspond to the isothermal surfaces in the other, and an electric charge corresponds to a source of heat? * As will appear from the present chapter, Maxwell had the name power in a very marked degree. It has always been cultivated by the " Cambridge school " of natural philosophers. t Camb. Hath. Journal, iii (Feb. 1842). p. 71 ; reprinted in Thomson's i>B*r* on Sltttmtatia and Magiut'um, p. t. Alio Camb. and Dub. Hath. Journal, Nov., 1845 ; reprinted in tmptri, p. 16. J As regards tbis comparison, Thomson had been anticipated by Chasles, Journal de 1'Ec. Folyt. it (1837), p. 268, who had shown that attraction accord- ing to Newton's law gives rise to the same fields as the steady conduction of heat, both depending on Laplace's equation v: V - 0 . It will be remembered that Ohm bad used an analogy between thermal conduction and galvanic phen dhyGoOgk 270 Maxwell. It may, perhaps, seem as if the value of such an analogy as this consisted merely in the prospect which it offered of comparing, and thereby extending, the mathematical theories of heat and electricity. But to the physicist its chief interest lay rather in the idea that formulae which relate to the electric field, and which had been deduced from laws of action at a distance, were shown to be identical with formulae relating to the theory of heat, which had been deduced from hypotheses of action between contiguous particles. In 1846 — the year after he had taken his degree as second wrangler at Cambridge — Thomson investigated" the analogies of electric phenomena with those of elasticity. For this purpose he examined the equations of equilibrium of an incompressible elastic solid which is in a state of strain ; and showed that the distribution of the vector which represents the elastic displacement might be assimilated to the distribution of the electric force in an electrostatic system. This, however, as he went on to show, is not the only analogy which may be perceived with the equations of elasticity ; for the elastic displacement may equally well be identified with a vector a, defined in terms of the magnetic induction B by the relation curl a = B. The vector a iB equivalent to the vector-potential which had been used in the memoirs of Neumann, Weber, and Kirchhoff, on the induction of currents ; but Thomson arrived at it independently by a different process, and without being at the time aware of the identification. The results of Thomson's memoir seemed to suggest a picture of the propagation of electric or magnetic force : might it not take place in somewhat the same way as changes in the elastic displacement are transmitted through an elastic solid t These suggestions were not at the time pursued further by their author ; but they helped to inspire another young ' Climb, and Dub. Hath. Joum. ii (1 84V), p. 61 : Thomson'! JfalA. end fty>. Papert, i, p. 76. 3,Bl,ZEdhyG00gle Maxwell. 271 Cambridge man to take up the matter a few years later. James Clerk Maxwell, by whom the problem was eventually solved, was born in 1831, the son of a landed proprietor in Dumfriesshire. He was educated at Edinburgh, and at Trinity College, Cambridge, of which society he became in 1855 a Fellow ; and not long after his election to Fellowship, he communicated to the Cambridge Philosophical Society the first of bis endeavours* to form a mechanical conception of the electro-magnetic field. Maxwell had been reading Faraday's Experimental Re- searches; and, gifted as' he was with a physical imagination akin to Faraday's, he had been profoundly impressed by the theory of lines of force. At the same time, he was a trained mathematician; and the distinguishing feature of almost all his researches was the union of the imaginative and the analytical faculties to produce results partaking of both natures. This first memoir may be regarded as an attempt to connect the ideas of Faraday with the mathematical analogies which had been devised by Thomson, Maxwell considered first the illustration of Faraday's lines of force which is afforded by the lines of flow of a liquid. The lines of force represent the direction of a vector; and the magnitude of this vector is everywhere inversely proportional to the cross-section of a narrow tube formed by such lines. This relation between magnitude and direction is possessed by any circuital vector ; and in particular by the vector which represents the velocity at any point in a fluid, if the fluid be incompressible. It is therefore possible to represent the magnetic induction B, which is the vector represented by Faraday's lines of magnetic force, as the velocity of an incom- pressible fluid. Such an analogy had been indicated some years previously by Faraday himself,! who had suggested that along the lines of magnetic force there may be a " dynamic condition," analogous to that of the electric current, and * Trans. Curni). Phil. So*. X, p. 27; Maxwell'* Scientific Papers, i, p. 155. t EV. Em., i 3269 (1SG2). 3,Bl,ZEdhyG00gle 272 Maxwell. that, in fact, " the physical lines of magnetic force are currents." The comparison with the lines of flow of a liquid is applicable to electric as well aa to magnetic lines of force. In this case the vector which corresponds to the velocity of the fluid is, in free aether, the electric force £. But when different dielectrics are present in the field, the electric force is not a circuital vector, and. therefore cannot be represented by lines of force ; in fact, the equation divE-0 is now replaced by the equation div(eE) = 0, where t denotes the specific inductive capacity or dielectric constant at the place (x, y, z). It is, however, evident from this equation that the vector *E is circuital ; this vector, which will be denoted by D, bears to B a relation similar to that which the magnetic induction B bears to the magnetic force H. It is the vector D which is represented by Faraday's lines of electric force, and which in the hydrodynamics! analogy corresponds to the velocity of the incompressible fluid. In comparing fluid motion with electric fields it is necessary to introduce sources and sinks into the fluid to correspond to the electric charges ; for D is not circuital at places where there is free charge. The magnetic analogy is therefore somewhat the simpler. In the latter half of his memoir Maxwell discussed how Faraday's "electrotonic state" might be represented in mathe- matical symbols. This problem he solved by borrowing from Thomson's investigation of 1847 the vector a, which is defined in terms of the magnetic induction by the equation curl a - B ; if, with Maxwell, we call a the eketrotonic intensity, the equation is equivalent to the statement that " the entire electrotonic intensity round the boundary of any surface measures the number of lines of magnetic force which pass. 3,Bl,ZEdhyG00gle Maxwell. 273 through that surface." The electromotive force of induction at the place (x, y, z) is - 3a/3< : as Maxwell said, " the electromotive force on any element of a conductor is measured by the instantaneous rate of change of the electrotonic intensity on that element." From this it is evident that a is no other than the vector-potential which had been employed by Neumann, Weber, and Kirchhoff, in the calculation of induced currents ; and we may take* for the electrotonic intensity due to a current t' flowing in a circuit s' the value which results from Neumann's theory, namely, -'J?" It may, however, be remarked that the equation curl a - B, taken alone, is insufficient to determine a uniquely ; for we can choose a so as to satisfy this, and also to satisfy the equation div a = \p, where 1/1 denotes any arbitrary scalar. There are, therefore, an infinite number of possible functions a. With the particular value of a which has been adopted, we have .. 8 ., f dx' 3 ., f dy1 8 ., f dz' div a = r- t — + — t \ — + - f I — ']>e: - 0; so the vector-potential a which we have chosen is circuital. In this memoir the physical importance of the operators curl and div first became evidentf ; for, in addition to those applications which have been mentioned. Maxwell showed that * Cf. p. 224. t Theee openton bad, however, occurred frequently in the writing* of Stokw, especially in nil memoir of 18*9 on the Dynamical Theory of Diffractiot. dhyGoogle 274 Maxwell. he connexion between the strength 1 of a current and the magnetic field H, to which it gives rise, may be represented by the equation 4« - curl H ; this equation is equivalent to the statement that " the entire magnetic intensity round the boundary of any surface measures the quantity of electric current which passes through that surface." In the same year (1856) in which Maxwell's investigation was published, Thomson* put forward an alternative inter- pretation of magnetism. He had now come to the conclusion, from a study of the magnetic rotation of the plane of polariza- tion of light, that magnetism possesses a rotatory character; and suggested that the resultant angular momentum of the thermal motions of a body + might be taken as the measure of the magnetic moment. " Tbo explanation," he wrote, " of all phenomena of electromagnetic attraction or repulsion, or of electromagnetic induction, is to be looked for simply in the inertia or pressure of the matter of which the motions constitute heat. Whether this matter is or is not electricity. whether it is a continuous fluid interpermeating the spaces between molecular nuclei, or is itself molecularly grouped : or whether all matter is continuous, and molecular heterogeneous- ness consists in finite vortical or other relative motions of contiguous parts of a body: it is impossible to decide, and, perhaps, in vain to speculate, in the present state of science." The two interpretations of magnetism, in which the linear and rotatory characters respectively are attributed to it, occur frequently in the subsequent history of the subject. The former was amplified in 1858, when Helmholtz published his researches J on vortex motion ; for Helmholtz showed that if a *Proo. Boy. Soo. riii (1866), p. 160; si (1861), p. 327, foot-note: Phil. Hap. jriii (1857), p. IBS; Baltimart Ltetura, Appendix F. t This wu written shortly buforo the kinetic theory of gue* wu developed by Clauaius and Maxwell. •Journal fur Math. Iv (1858), p. 25; Helm bolt* 'a Witt. Abk. i, p. 101; translated Phil. Mag. ixxiii (1867), p. 486. 3,Bl,ZEdhyG00gle Maxwell. 275 magnetic field produced by electric currents is compared to the flow of an incompressible fluid, so that the magnetic vector is represented by the fluid velocity, then the electric currents correspond to the vortex- filaments in the fluid. This analogy correlates many theorems in hydrodynamics and electricity ; for instance, the theorem that a re-entrant vortex-filament is equivalent to a uniform distribution of doublets over any surface bounded by it, corresponds to Ampere's theorem of the equivalence of electric currents and magnetic shells. In his memoir of 1855, Maxwell had not attempted to construct a mechanical model of electrodynamic actions, but had expressed his intention of doing so. " By a careful study," he wrote," " of the laws of elastic solids, and of the motions of viscous fluids, I hope to discover a method of forming a mechanical conception of this electrotonic state adapted to general reasoning " ; and in a foot-note he referred to the effort which Thomson had already made in this direction. Six years elapsed, however, before anything further on the subject was published. In the meantime, Maxwell became Professor of Natural Philosophy in King's College, London — a position in which he had opportunities of personal contact with Faraday, whom he had long reverenced. Faraday had now concluded the Experimental Researches, and was living in retirement at Hampton Court ; but his thoughts frequently recurred to the great problem which he had brought so near to solution. It appears from his note-book that in 1857f he was speculating whether the velocity of propagation of magnetic action its of the same order as that of light, and whether it is affected by the susceptibility to induction of the bodies through which the action is transmitted. The answer to this question was furnished in 1861-2, when Maxwell fulfilled his promise of devising a mechanical conception of the electromagnetic field.J «M»i*tU'« SeitntiJIe Paprrt.i, p. 188. t Bene* Jones's Lifi of Faraday u, p. 379. * Phil. Hag. ni (1861), pp. 161, 281, 338; uiii (1862), pp. 12, 86; Mb* well's Scientific Papers, i, p. 461. T 2 >BizBdb¥Google 276 Maxwell. In the interval since the publication of hia previous n Maxwell had become convinced by Thomson's arguments that magnetism is in its nature rotatory. "The transference of electrolytes in fixed directions by the electric current, and the rotation of polarized light in fixed directions by magnetic force, are," he wrote, " the facts the consideration of which has induced me to regard magnetism as a phenomenon of rotation, and electric currents as phenomena of translation." This con- ception of magnetism he brought into connexion with Faraday's idea, that tubes of force tend to contract longitudinally and to expand laterally. Such a tendency may be attributed to centrifugal force, if it be assumed that each tube of force contains fluid which is in rotation about the axis of the tube. Accordingly Maxwell supposed that, in any magnetic field, the medium whose vibrations constitute light is in rotation about the lines of magnetic force; each unit tube of force may for the present be pictured as an isolated vortex. The energy of the motion per unit volume is proportional to fJP, where /i denotes the density of the medium, and H denotes the linear velocity at the circumference of each vortex. But, as we have seen* Thomson had already shown that the energy of any magnetic field, whether produced by magnets or by electric currents, is where the integration is taken over all space, and where p denotes the magnetic permeability, and H the magnetic force. It was therefore natural to identify the density of the medium at any place with the magnetic permeability, and the circum- ferential velocity of the vortices with the magnetic force. But an objection to the proposed analogy now presents itself. Since two neighbouring vortices rotate in the same direction, the particles in the circumference of one vortex most be moving in the opposite direction to the particles contiguous • Cf. pp. 248, 260. 3,Bl,ZEdhyG00gle Maxwell. 277 to them in the circumference of tbe adjacent vortex ; and it seems, therefore, as if the motion would be discontinuous. Maxwell escaped from this difficulty by imitating a well-known mechanical arrangement. When it is desired that two wheels should revolve in the same sense, an " idle " wheel is inserted between them bo as to be in gear with both. The model of the electromagnetic field to which Maxwell arrived by the intro- duction of this device greatly resembles that proposed by Bernoulli in 1736." He supposed a layer of particles, acting as idle wheels, to be interposed between each vortex and the next, and to roll without eliding on the vortices ; so that each vortex tends to make the neighbouring vortices revolve in the same direction as itself. The particles were supposed to be not other- wise constrained, so that the velocity of the centre of any particle would be the mean of the circumferential velocities of the vortices between which it is placed. This condition yields (in suitable units) the analytical equation 4*1 - curl H, where the vector l denotes the flux of the particles, so that its ^-component t, denotes the quantity of particles transferred in unit time across unit area perpendicular to the a; direction. On comparing this equation with that which represents Oersted s discovery, it is seen that the flux i of the movable particles interposed between neighbouring vortices is the analogue of the electric current. It will be noticed that in Maxwell's model the relation between electric current and magnetic force is secured by a connexion which is not of a dynamical, hut of a purely kine- matical character. The above equation simply expresses the existence of certain non-holonomic constraints within the system. If from any cause the rotatory velocity of some of tbe cellular vortices is altered, the disturbance will be propagated from that part of the model to all other parts, by the mutual • Cf. p. 100. D,Bl,ZEdhyG00gle 278 Maxwell. action of the particles and vortices. This action is determined, as Maxwell showed, by the relation juH - - curl E which connects E, the force exerted on a unit quantity of particles at any place in consequence of the tangential action of the vortices, with H, the rate of change of velocity of the neighbouring vortices. It will be observed that this equation is not kinematical but dynamical. On comparing it with the electromagnetic equations | curl a = pH, | Induced electromotive force ■= - a, it is seen that E must be interpreted electromagnetically as the induced electromotive force. Thus the motion of the particles constitutes an electric current, the tangential force with which they are pressed by the matter of. the vortex-cells constitutes electromotive force, and the pressure of the particles on each other may be taken to correspond to the tension or potential of the electricity. The mechanism must next he extended so as to take account of the phenomena of electrostatics. For this purpose Maxwell assumed that the particles, when they are displaced from their equilibrium position in any direction, exert a tangential action on the elastic substance of the cells ; and that this gives rise to a distortion of the cells, which in turn calls into play a force arising from their elasticity, equal and opposite to the force which urges the particles away from the equilibrium position. When the exciting force is removed, the cells recover their form, and the electricity returns to its former position. The state of the medium, in which the electric particles are displaced in a definite direction, is assumed to represent an electrostatic field. Such a displacement does not itself con- stitute a current, because when it has attained a certain value it remains constant; but the variations of displacement are to be regarded as currents, in the positive or negative direction according as the displacement is increasing or diminishing. 3,Bl,ZEdhyG00gle Maxwell. 279 The conception of the electrostatic state aa a displacement of something from its equilibrium position was not altogether new, although it had not been previously presented in this form. Thomson, as we have seen, had compared electric force to the displacement in an elastic solid ; and Faraday, who had likened the particles of a ponderable dielectric to small con- ductors embedded in an insulating medium * had supposed that when the dielectric is subjected to an electrostatic field, there is a displacement of electric charge on each of the small conductors. The motion of these charges, when the field is varied, is equivalent to an electric current ; and it was from Lhis precedent that Maxwell derived the principle, which became of cardinal importance in his theory, that variations of displace- ment are to be counted as currents. But in adopting the idea, be altogether transformed it ; for Faraday's conception of displacement was applicable only to ponderable dielectrics, and was in fact introduced solely in order to explain why the specific inductive capacity of snch dielectrics is different from that of free aether ; whereas according to Maxwell there is displacement wherever there is electric force, whether material bodies are present or not. The difference between the conceptions of Faraday and Maxwell in this respect may be illustrated by an analogy drawn from the theory of magnetism. When a piece of iron is placed in a magnetic field, there is induced in it a magnetic distribution, say of intensity I ; this induced magnetization exists only within the iron, being zero in the free aether outside. The vector 1 may be compared to the polarization or displacement, which according to Faraday is induced in dielectrics by an electric field; and the electric current con- stituted by the variation of this polarization is then analogous to dl/dt. But the entity which was called by Maxwell the electric displacement in the dielectric is analogous not to I, bat to the magnetic induction B : the Maxwellian displace- • C(. p. 210. 3,Bl,ZEdhyG00gle 280 Maxwell. meat-current corresponds to dBJdt, and may therefore have a value different from zero even in free aether. It may be remarked in passing that the term dxapUuxment, which was thuB introduced, and which has been retained in the later development of the theory, is perhaps not well chosen ; what in the early models of the aether was represented as an actual displacement, has in later investigations been conceived of as a change of structure rather than of position in the elements of the aether. Maxwell supposed the electromotive' force acting on the electric particles to be connected with the displacement D which accompanies it, by an equation of the form where c, denotes a constant which depends on the elastic properties of the cells. The displacement-current D must now be inserted in the relation which connects the current with the magnetic force ; and thus we obtain the equation curl H = 4vS, where the vector 8, which is called the total current, is the sum of the convection-current l and the displacement-current D. By performing the operation div on both sides of this equation, it is seen that the total current is a circuital vector. In the model, the total current is represented by the total motion of the rolling particles ; and this is conditioned by the rotations of the vortices in such a way as to impose the kinematic relation div 8 - 0. Having obtained the equations of motion of bis system of vortices and particles, Maxwell proceeded to determine the rate of propagation of disturbances through it. He considered in particular the case in which the substance represented is a dielectric, so that the conduction-current is zero. If, moreover, 3,Bl,ZEdhyG00gle Maxwell. 281 the constant fi be supposed to have the value unity, the equations may be written {div H = 0, e,' curl H - E, - curl S - H. Eliminating E, we see* that H satisfies the equations (div H - 0, ( H = e,'V,H. But these are precisely the equations which the light-vector satisfies in a medium in which the velocity of propagation is c, : it follows that disturbances are propagated through the model by waves which are similar to waves of light, the magnetic (and similarly the electric) vector being in the wave-front For a plane-polarized wave propagated parallel to the axis of z, the equations reduce to tdJZf IE, t$&M d£f d£r dKt aft agr ' ~Ci & ~ dt' c' ~dz ~ &' a* - dt' Ite = dt' whence we have CiBr « Ex> — clHI — Ej ', these equations show that the electric and magnetic vectors are at right angles to each other. The question now arises as to the magnitude of the constant Ci.\ This may be determined by comparing different expressions for the energy of an electrostatic field. The work done by an electromotive force E in producing a displacement D is E . dD or JED per unit volume, since E is proportional to D. But if it be assumed that the energy of an electrostatic field is resident in the dielectric, the amount of energy per unit volume may be * For if a denote iny vector, we We identically v Ja + gnd div * + cor) ourl * = 0. t For critieumB on lie procedure by which Maiwtll determined the Telocity of propagation of disturbance, cf. F. Duhem, Ln Thiorwt £UatyiqutM dt J. CUrk Jf«i»U, Pari*, 1602. 3,Bl,ZEdhyG00gle 282 Maxwell. calculated by considering the mechanical force required in order to increase the distance between the plates of a condenser, so as to enlarge the field comprised between them. The result is that the energy per unit volume of the dielectric is tV/fir, where i denotes the specific inductive capacity of the dielectric and E' denotes the electric force, measured in terms of the electrostatic unit : if E denotes the electric force expressed in terms of the electrodynamic units used in the present investi- gation, we have E - cE', where e denotes the constant which" occurs in transformations of this kind. The energy is therefore (E'/Httc5 per unit volume. Comparing this with the expression for the energy in terms of E and D, we have D - «l/4*t* and therefore the constant c, has the value «"'. Thus the result is obtained that the velocity of propagation of dis- turbances in Maxwell's medium is cr*, where t denotes the specific inductive capacity and c denotes the velocity for which Kohlrausch and Weber had foundf the value 31 x 10'° cmjeec Now by this time the velocity of light was known, not only from the astronomical observations of aberration and of Jupiter's satellites, but also by direct terrestrial experiments. In 1849 Hippolyte Louis FizeauJ had ' determined it by rotating a toothed wheel so rapidly that a beam of light transmitted through the gap between two teeth and reflected back from a mirror was eclipsed by one of the teeth on its return journey. The velocity of light was calculated from the dimensions and angular velocity of the wheel and the distance of the mirror; the result being 3-15 x 1010 cm./see.§ • Cf. pp. 227, 259. f Cf. p. 260. * Compto* Bendue, xxii (1849), p. 90. A determination nude by Cornu is 1874 was On tliia principle. j A different experimental method ill employed in 1362 by L£on Foucanlt (Comptae Rendus, It, pp. fiOl , 792) ; in thin a ray from an origin O wa* reflected by a revolving mirror if to a filed mirror, and so reflected back to it, and again to O. Il )■ evident that the returning ray MO mult be deviated by twice the angle through which M turns while the light paitea from If to the fixed mirror and bach. The value thus obtained by Foucanlt for the Telocity of light ni 3,Bl,ZEdhyG00gle Maxwell 283 Maxwell was impressed, as Kirchhoff had been before him, by the close agreement between the electric ratio c and the velocity of light* ; and having demonstrated that the propaga- tion of electric disturbance resembles that of light, he did not hesitate to assert the identity of the two phenomena. "We can scarcely avoid the inference," he said, " that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena." Thus was answered the question which Priestley had asked almost exactly a hundred years before :t " Is there any electric fluid mi generis at all, distinct from the aether ? " The presence of the dielectric constant t in the expression c*"', which Maxwell had obtained for the velocity of propaga- tion of electromagnetic disturbances, suggested a further test of the identity of these disturbances with light: for the velocity of light in a medium is known to be inversely proportional to the refractive index of the medium, and therefore the refractive index should be, according to the theory, proportional to the square root of the specific inductive capacity. At the time, however, Maxwell did not examine whether this relation was confirmed by experiment. In what has preceded, the magnetic permeability y. has been supposed to have the value unity. If this is not the case, the 2-98 X 10ID cm. Inc. Subsequent determinations by Micholson in 1870 (Ait. Papers of the Amer. Ephemeris, i), and bj Newconib in 1B82 {ibid., ii) depended on the mine principle. Aa was shown afterwards by Lord Bayleigh (Nature, xxiv, p. 3H2, nv, p. S2J and by Gibbs (Nature, xzziii, p. 682), the value obtained for the velocity of light bythe methoda of Pima and Fomanlt represents they roup-rehciiy, not ihe «atw- rtUxity; the eclipse* of Jupiter's satellites alao give the group -velocity, while the value deduced from the coefficient of aberration is the wave- velocity. In a non- dispenine medium, the group -velocity coincides with the wave-veloeity ; and the agreement of the values ut the velocity of light obtained by the two astronomical methods aeema to negative the possibility of any appreciable dispersion in free The velocity of light in dispersive media was directly investigated by MichaUon in 1883—4, with results in accordance with theory. * He had " worked out the formulae in the country, before seeing Weber's result." Cf. Campbell and Qarnett's Lift of MtxwM, p. 244. t Prieetley'a Biitory, p. 488. 3,Bl,ZEdhyG00gle 284 Maxwell. velocity of propagation of disturbance may be shown, by the same analysis, to be crip-i; bo that it is diminished when ft is greater than unity, i.e., in paramagnetic bodies. This inference had been anticipated by Faraday : " Nor is it likely," he wrote,* " that the paramagnetic body oxygen can exist in the air and not retard the transmission of the magnetism." It was inevitable that a theory bo novel and so capacious as that of Maxwell should involve conceptions which his contempo- raries understood with difficulty and accepted with reluctance. Of these the most difficult and unacceptable was the principle that the total current is always a circuital vector; or, as it is generally expressed, that " all currents are closed." According to the older electricians, a current which is employed in charging a condenser is not closed, but terminates at the coatings of the condenser, where charges are accumulating. Maxwell, on the other hand, taught that the dielectric between the coatings is the seat of a process — the displacement-current — which is proportional to the rate of increase of the electric force in the dielectric ; and that this process produces the same magnetic effects as a true current, and forms, so to speak, a continuation, through the dielectric, of the charging current, so that the latter may be regarded as' flowing in a closed circuit. Another characteristic feature of Maxwell's theory is the conception — for which, as we have seen, he was largely indebted to Faraday and Thomson — that magnetic energy is the kinetic energy of a medium occupying the whole of space, and that electric energy is the energy of strain of the same medium. By this conception electromagnetic theory was brought into such close parallelism with the elastic-solid theories of the aether, that it was bound to issue in an electromagnetic theory of light. Maxwell's viewB were presented in a more developed form in a memoir entitled "A Dynamical Theory of the Electro- magnetic Field," which was read to the Royal Society in 1864 rf ■ Faradny'j labu.atory note-book for 1867 : at. Benoe Jones's J.i/e of Farmini. ii, p. 380. t Phil. Tnuw. civ (1869), p. 459 : Mixwall'i Sdmt. Ftptr; i, p. 828 3,Bl,ZEdhyG00gle Maxwell. 285 in this the architecture of hia system was displayed, stripped of the scaffolding by aid of which it had been first erected. As the equations employed were for the most part the same as had been set forth in the previous investigation, they need only be briefly recapitulated. The magnetic induction pH, being a circuital vector, may be expressed in terms of a vector-potential A by the equation ,uH = curl A. The electric displacement D is connected with the volume- density p of free electric charge by the electrostatic equation div D - p. The principle of conservation of electricity yields the equation divi = - dp/dt, where i denotes the conduction-current. The law of induction of currents — namely, that the total electromotive force in any circuit is proportional to the rate of decrease of the number of lines of magnetic induction which pass through it — may be written - ourlS -pE; from which it follows that the electric force E must be expressible in the form E - - A + grad \p , where \fi denotes some scalar function. The quantities A and $ which occur in this equation are not as yet completely deter- minate ; for the equation by which A is defined in terms of the magnetic induction specifies only the circuital part of A ; and as the irrotational part of A is thus indeterminate, it is evident that }f> also must be indeterminate. Maxwell decided the matter by assuming* A to be a circuital vector; thus div A - 0, and therefore div E - - Vip, • This i« the effect of the introduction of (J", ff, B') in } 98 of the memoir; cf . al«o Maxwell'* Trntit* on BUctrttitf and Maynelum, $ 616. 3,Bl,ZEdhyG00gle 286 Maxwell. from which equation it is evident that \{. represents the electro- static potential The principle which is peculiar to Maxwell's theory must now be introduced. Currents of conduction are not tha only hind of currents ; even in the older theory of Faraday, Thomson, and Mossotti, it had been assumed that electric charges are sec in motion in the particles of a dielectric when the dielectric is subjected to an electric field ; and the prede- cessors of Maxwell would not have refused to admit that the motion of these charges is in some sense a current Suppose, then, that 8 denotes the total current which is capable of generating a magnetic field : since the integral of the magnetic force round any curve is proportional to the electric current which flows through the gap enclosed by the curve, we have in suitable units curl H = 4ttS. In order to determine 8, we may consider the case of a con- denser whose coatings are supplied with electricity by a conduction-current l per unit-area of coating. If ± a denote the surface-density of electric charge on the coatings, we have i = da/di, and o - D, where D denotes the magnitude of the electric displacement D in the dielectric between the coatings ; so i <= D. But since the total current is to be circuital, its value in the dielectric most be the same as the value l which it has in the rest of the circuit ; that is, the current in the dielectric has the value it. We shall assume that the current in dielectrics always has this value, so that in the general equations the total current must be understood to be i + D. The above equations, together with those which express the proportionality of E to D in insulators, and to i in conductors, constituted Maxwell's system for a field f&rmed by isotropic bodies which are not in motion. When the\ magnetic field is due entirely to currents (including both copduction-currente 3,Bl,ZEdfyG00gle Maxwell. 287 and displacement-currents), so that there is no magnetization, we have VA. - - curl curl A = - curl H bo that the vector-potential is connected with the total current by an equation of the same form as that which connects the scalar potential with the density of electric charge. To these potentials Maxwell inclined to attribute a physical significance; he supposed „ vfl),). The other equations are the same as in isotropic media; so that the propagation of disturbance is readily seen to depend on the equation U-ff., nM* fiJii) " - curl («,• (cnrl-ff),, c'(eurlfl")„ cJ(cutIH), I ■ Now, if ft,, fit, ft, are supposed equal to each other, this equation is the same as the equation of motion of MacCullagh's aether in crystalline media* the magnetic force H corresponding to MacCullagh's elastic displacement; and we may therefore immediately infer that Maxwell's electromagnetic equations yield a satisfactory theory of the propagation of light in crystals, provided it is assumed that the magnetic permeability ib (for optical purposes) the same in all directions, and pro- vided the plane of polarization is identified with the plane which contains the magnetic vector. It is readily shown that the direction of the ray is at right angles to the magnetic vector and the electric force, and that the wave-front is the plane of the magnetic vector and the electric displacement! After this Maxwell proceeded to investigate the propagation of light in metals. The difference between metals and dielectrics, to far as electricity is concerned, is that the former are con- ductors ; and it was therefore natural to seek the cause of the optical properties of metals in their ohmic conductivity. This idea at once suggested a physical reason for the opacity of metala — namely, that within a metal the energy of the light vibrations is converted into Joulian heat in the same way as the energy of ordinary electric currents. * Cf. pp. 154 «t sqq. tin the memoir of 1864 Maxwell left open the ehoice between the above theory and that which U obtained by assuming that in crystals the specific inductive rapacity i» (lor optical purposes) the aame in all directions, while the magnetic permeability ie aeolotropic. In the latter cane the plane of polarisation must be identified with the plane which contains the electric displacement. Nine years later, in his Trtatitt (J 791), Maxwell definitely adopted the former alternative. U 3,Bl,ZEdhyG00gle 290 Maxwell. The equations of the electromagnetic field in the metal may be written curl H - 4nS, - curl B = A, 8 =l + D-KE+ eJ5/4irc', where k denotes the ohmic conductivity ; whence it is seen that the electric force satisfies the equation *E + 4jtkc1E =e"P'E. This is of the same form as the corresponding equation in the elastic-solid theory* ; and, like it, furnishes a satisfactory general explanation of metallic reflexion. It is indeed correct in all details, so long as the period of the disturbance is not too short — Le., bo long as the light-waves considered belong to the extreme infra-red region of the spectrum ; but if we attempt to apply the theory to the case of ordinary light, we are confronted by the difficulty which Lord Rayleigh indicated in the elastic- solid theory.t and which attends all attempts to explain the peculiar properties of metals by inserting a viscous term in the equation. The difficulty is that, in order to account for the properties of ideal silver, we must suppose the coefficient of E negative — that is, the dielectric constant of the metal must be negative, which would imply instability of electrical equilibrium in the metal. The problem, as we have already remarked,? was solved only when its relation to the theory of dispersion was rightly understood. At this time important developments were in progress in the last-named subject. Since the time of Fresnel, theories of dispersion had proceeded^ from the assumption that the radii of action of the particles of luminiferous media are so large as to be comparable with the wave-length of light. It was generally supposed that the aether is loaded by the molecules • Cf. p. ISO. t Cf. p. 181. Cr. also Bayleigh, Phil. Mag. (5) xu (1881), p. 81, and H. A. Lorenti, Over de Tkterit dt Tcmg knotting, Anihcm, 1375. J Cf. p. 181. f Cf. p. 182. 3,Bl,ZEdhyG00gle Maxwell. 291 of ponderable matter, and that the amount of dispersion depends on the ratio of the wave-length to the distance between adjacent molecules. This hypothesis was, however, seen to be inadequate, when, in 1862, F. P. Leroux* found that a prism filled with the vapour of iodine refracted the red rays to a greater degree than the blue rays; for in all theories which depend on the assumption of a coarse-grained lumini- ferous medium, the refractive index increases with the frequency of the light Leroux's phenomenon, to which the name anomalous dis- persion was given, was shown by later investigate rsf to be generally associated with " surface-colour," i.e., the property of brilliantly reflecting incident light of some particular frequency. Such an association seemed to indicate that the dispersive property of a substance is intimately connected with a certain frequency of vibration which i8 peculiar to that substance, and which, when it happens to fall within the limits of the visible spectrum, is apparent in the surface-colour. This idea of a frequency of vibration peculiar to each hind of ponderable matter is found in the writings of Stokes as far back as the year 1852 ;f when, discussing fluorescence, he remarked: — " Nothing seems more natural than to suppose that the incident vibrations of the luminiferous aether produce vibratory move- ments among the ultimate molecules of sensitive substances, and that the molecules in turn, swinging on their own account, produce vibrations in the luminiferous aether, and thus cause the sensation of light. The periodic times of these vibrations depend on the periods iu which the molecules are disposed to swing, not upon the periodic time of the incident vibrations." The principle here introduced, of considering the molecules as dynamical systems which possess natural free periods, and which interact with the incident vibrations, lies at the basis of * Com p tee Bendus, It (1862), p. 123. In 1870 0. Christiansen (Ann. d. Phy*. ciii, p. 479 ; cxliii, p. 260) observed a similar effect in a solution of fuchsia. t Especially by Kuadt, in a series of papers in the Annaleo d. Phys., from vol. cxlii (1871) onwards. ♦ Phil. Trans., 1852, p. 463. Stokes's Coll. Pa/jr; iii., p. 267, U 3 3,Bl,ZEdhyG00gle 29-> Maxwell. all modern theories of dispersion. The earliest of these was devised by Maxwell, who, in the Cambridge Mathematical Tripos for 1869,* published the results of the following investigation : — A model of a dispersive medium may be constituted by embedding systems which represent the atoms of ponderable matter in a medium which represents the aether. We may picture each atomf as composed of a single massive particle supported symmetrically by springs from the interior face of a massless spherical ahull : if the shell he fixed, the particle will be capable of executing vibrations about the centre of the sphere, the effect of the springs being equivalent to a force on the particle proportional to its distance from the centre. The atoms thus constituted may be supposed to occupy small spherical cavities in the aether, the outer shell of each atom being in contact with the aether at all points and partaking of its motion. An immense number of atoms is supposed to exist in each unit volume of the dispersive medium, so that the medium as a whole is fine-grained. Suppose that the potential energy of strain of free aether per unit volume is . <)' where n denotes the displacement and E an elastic constant ; so that the equation of wave-propagation in free aether is where p denotes the aethereal density. Then if a denote the mass of the atomic particles in unit volume, (q + X,) the total displacement of an atomic particle at the place x at time t, and aprC the attractive force, it is evident that for the compound medium the kinetic energy per unit volume is ■ Cambridge Calendar, 1869 ; republished by Lord Kajleigb, Pbii. Uag. jthiii (18B9), p. 151. t Thw fflimrMion it due to W. T *w 3,Bl,ZEdhyG00gle Maxwell. 293 and the potential energy per unit volume is i* + «r)/p : for such oscilla- tions, each atomic particle and its shell move together as a rigid body, so that the effect is the same as if the aether were simply loaded by the masses of the atomic particles, its rigidity remaining unaltered. • Aon. d. Pbj». fiilv (1873), pp. 399, 620 : oxlrlt (1872), pp. S86, S26. Cf. alio Helmholti, Ann. d. Phjt. oliv (1875), p. 662. 3,Bl,ZEdhyG00gle 294 Maxwell. The dispersion of light within the limits of the visible spectrum is for most substances controlled by a natural frequency p which corresponds to a vibration beyond the violet end of the visible spectrum : so that, n being smaller than p, we may expand the fraction in the formula of dispersion, and obtain the equation , , a!. n' ft* \ fi* - 1 + - 1 + — + —+.,, , p\ p* p* J which resembles the formula of dispersion in Cauchy's theory* ; indeed, we may say that Cauchy's formula is the expansion of Maxwell's formula in a series which, as it converges only when n has values within a limited range, fails to represent the phenomena outside that range. The theory as given above is defective in that it becomes meaningless when the frequency n of the incident light is equal to the frequency p of the free vibrations of the atoms. This defect may be remedied by supposing that the motion of an atomic particle relative to the shell in which it is contained is opposed by a dissipative force varying as the relative velocity : such a force suffices to prevent the forced vibration from becoming indefinitely great as the period of the incident light approaches the period of free vibration of the atoms; its introduction is justified by the fact that vibrations in this part of the spectrum suffer absorption in passing through the medium. When the incident vibration is not in the same region of the spectrum as the free vibration, the absorption is not of much importance, and may be neglected. It is shown by the spectroscope that the atomic systems which emit and absorb radiation in actual bodies possess more than one distinct free period. The theory already given may, however, readily be extended-)* to the case in which the atoms have several natural frequencies of vibration ; we have only to suppose that the external massless rigid shell is connected by springs to an interior massive rigid shell, and that this again • Cf. p. 183. t Thit lubji'ct wa* deielopod by Lord Kelvin in Ihs' Baltimore Zeettirrt. 3,Bl,ZEdhyG00gle Maxwell. 295 is connected by springs to another massive shell inside it, and so on. The corresponding extension of the equation for the refractive index is i -. c' fr jji* — n pf — n* where pi.pt,... denote the frequencies of the natural periods of vibration of the atom. The validity of the Maxwell- Sellmeier formula of disper- sion was strikingly confirmed by experimental researches in the closing years of the nineteenth century. In 1897 Rubens* showed that the formula represents closely the refractive indices of sylvin (potassium chloride) and rock-su.lt, with respect to light and radiant heat of wave-lengths between 4,240 A.U. and 223,000 A.U. The constants in the formula being known from this comparison, it was possible to predict the dispersion for radiations of still lower frequency ; and it was found that the square of the refractive index should have a negative value (indicating complete reflexion) for wave- lengths 370,000 A.U. to 550,000 A.U. in the case of rock-salt, and for wave-lengths 450,000 .to 670,000 A.U. in the case of sylvin. This inference was verified experimentally in the following year.t It may seem strange that Maxwell, having successfully employed his electromagnetic theory to explain the propagation of light in isotropic media, in crystals, and in metals, should have omitted to apply it to the problem of reflexion and refrac- tion. This is all the more surprising, as the study of the optics of crystals had already revealed a close analogy between the electromagnetic theory and MacGullagh's elastic-solid theory; and in order to explain reflexion and refraction electro- magnetically, nothing more was necessary than to transcribe MacCullagh's investigation of the same problem, interpreting o (the time-flux of the displacement of MacCullagh's aether) as the magnetic force, and curl e as the electric displacement. As ■ Ann. d. Phyi. Ix (1897), p. 154. t Rubens and Aichkinau, Ann. d. Fliyn. hi* (1898). 3,Bl,ZEdhyG00gle 296 Maxwell. in MaoCullagh's theory the difference between the contiguous media ia represented by a difference of their elastic constants, so in the electromagnetic theory it may he represented by a difference in their specific inductive capacities. From a letter which Maxwell wrote to Stokes in 1864, and which, has been preserved * it appears that the problem of reflexion and refrac- tion was engaging Maxwell's attention at the time when he was preparing his Eoyal Society memoir on the electromagnetic field; but he was not able to satisfy himself regarding the conditions which should be satisfied at the interface between the media. He seems to have been in doubt which of the rival elastic-solid theories to take as a pattern ; and it is not unlikely that he was led astray by relying too much on the analogy between the electric displacement and an elastic displacement, t For in the elastic-solid theory all three components of the dis- placement must be continuous across the interface between two contiguous media ; but Maxwell found that it was impossible to explain reflexion and refraction if all three components of the electric displacement were supposed to be continuous across the interface ; and, unwilling to give up the analogy which had hitherto guided him aright, yet unable to disprove? the Greenian conditions at bounding surfaces, he seems to have laid aside the problem until some new light should dawn upon it. This was not the only difficulty which beset the electro- magnetic theory. The theoretical conclusion, that the specific inductive capacity of a medium should be equal to the square of its refractive index with respect to waves of long period, was not as yet substantiated by experiment; and the theory of ' displacement-currents, on which everything else depended, was * Stokes's Snmtijtc Corrupmidtnci, ii, pp. 2fi, 26. t It must, be remembered that Maxwell pictured the electric displacement aa a real displacement of a medium. " My theory of electrical force*," be wrote, " ia that they are oalled into play in insulating media by alight electric displacements, which put certain email portion* of the medium into a state of distortion, which, being resisted by the elasticity of the medium, produces an electromotive force." Campbell and Oamett'a Lift of MaxusiU, p. 244. J The letter to Stokes already mentioned appears to indicate that Maxwell for a lime doubted the correctness of Green's conditions. 3,Bl,ZEdhyG00gle Maxwell. 297 unfavourably received by the most distinguished of Maxwell's contemporaries. Helmholtz indeed ultimately accepted it, bnt only after many years ; and W. Thomson (Kelvin) seems never to have thoroughly believed it to the end of his long life. In 1888 be referred to it as a "curious and ingenious, but not wholly tenable hypothesis,"* and proposed! to replace it by an extension of the older potential theories. In 1896 he had some inclination* to speculate that alterations of electrostatic force due to rapidly-changing electrification are propagated by con- densational waves in the luminiferouB aether. In 1904 he admittedjj that a bar-magnet rotating about an axis at right angles to its length is equivalent to a lamp emitting light of period equal to the period of the rotation, bnt gave his final judgment in the sentence!! : — " The so-called electromagnetic theory of light has not helped us hitherto." Thomson appears to have based his ideas of the propagation of electric disturbance on the case which had first become familiar to him — that of the transmission of signals along a wire. He clung to the older view that in such a disturbance ^ the wire is the actual medium of transmission ; whereas in Maxwell's theory the function of the wire is merely to guide the disturbance, which is resident in the surrounding dielectric. This opinion that conductors are the media of propagation of electric disturbance was entertained also by Ludwig Lorenz (i. 1829, d. 1891), of Copenhagen, who independently developed an electromagnetic theory of lightll a few years after the publication of Maxwell's memoirs. The procedure which Lorenz followed was that which Riemann bad suggested" in 1858 — namely, to modify the accepted formulae of electro- dynamics by introducing terms which, though too small to be > Nature, xziviii (1888) p. 6T1. t Brit. Amoc. Report, 1888, p. 667. J Of. Bottomley, in Nature, liii [1896J, p. 268 ; Kelvin, ib., p. 316 ; J. Willard Gibba, ib., p. 609. j BaUimort Listuru [ad. IBM), p. 3T6. || Ibid., preface, p. 7. 1 Ortnigt ovtr dit K. dantki Vid. StUkapi Fvrhandlingrr, 1867, p. 26 ; Aimi,!. derPbja. ciixi|18G7), p. 241; Phil. Mag., uiit (1867), p. 287. *' Cf. p. 268. Riemann't memoir wan, however, published only in the israe year (1867) a* Lomnc'e. dhyGoOglc 298 Maxwell. appreciable in ordinary laboratory experiments, would be capable of accounting for the propagation of electrical effects through space with a finite velocity. We have seen that in Neumann's theory the electric force E was determined by the equation E^grad0-a, (1) where 0 denotes the electrostatic potential defined by the equation *-J]J - 0. (4) From equations (1), (2), (4), we may readily derive the equation divE = 4ir; (1) and from (1), (3), (4), we have curl H = E/e* + 4iri, (II) where H or curl a denotes the magnetic force : while from (1} we have curl S - - H. (Ill) The equations (I), (II), (III) are, however, the fundamental equations of Maxwell's theory; and therefore the theory of L. Lorenz is practically equivalent to that of Maxwell, so far as concerns the propagation of electromagnetic disturbances through free aether. Lorenz himself, however, does not appear to have clearly perceived this ; for in his memoir he postulated the presence of conducting matter throughout space, and was consequently led to equations resembling those which Maxwell had given for the propagation of light in metals. Observing that his equations represented periodic electric currents at right angles to the direction of propagation of the disturbance, he suggested that all luminous vibrations might be constituted by electric currents, and hence that there was " no longer any reason for maintaining the hypothesis of an aether, since we can admit that space contains sufficient ponderable matter to enable the disturbance to be propagated." Lorenz was unable to derive from his equations any explana- tion of the existence of refractive indices, and his theory lacks 3,Bl,ZEdhyG00gle 300 Maxwell. the rich physical suggestiveness of Maxwell's ; the value of his memoir lies chiefly in the introduction of the retarded potentials. It may be remarked in passing that Lorenz's retarded potentials are not identical with Maxwell's scalar and vector potentials ; for Lorenz's a is not a circuital vector, and Lorenz's is not, like Maxwell's, the electrostatic potential, but depends on the positions occupied by the charges at certain previous instants. For some years no progress was made either with Maxwell's theory or with Lorenz's. Meanwhile, Maxwell had in 1865 resigned his chair at King's College, and had retired to his estate in Dumfriesshire, where he occupied himself in writing a connected account of electrical theory. In 1871 he returned to Cambridge as Professor of Experimental Physics; and two years later published his Treatise on Electricity and Magnetism. In this celebrated work is comprehended almost every branch of electric and magnetic theory; but the intention of the writer was to discuss the whole as far as possible from a single point of view, namely, that of Faraday; so that little or no account was given of the hypotheses which had been pro- pounded in the two preceding decades by the great German electricians. So far as Maxwell's purpose was to disseminate the ideas of Faraday, it was undoubtedly fulfilled ; but the Treatise was less successful when considered as the exposition of its author's own views. The doctrines peculiar to Maxwell — the existence of displacement-currents, and of electromagnetic vibrations identical with light — were not introduced in the first Volume, or in the first half of the second volume; and the account which was given of them was scarcely more complete, and was perhaps less attractive, than that which had been furnished in the original memoirs. Some matters were, however, discussed more fully in the Treatise than in Maxwell's previous writings ; and among these was the question of stress in the electromagnetic Geld. It will be remembered* that Faraday, when studying the • Cf. p. 209. - 3,Bl,ZEdhyG00gle Maxwell. 301 curvature of lines of force in electrostatic fields, had noticed an apparent tendency of adjacent lines to repel each other, as if each tube of force were inherently disposed to distend laterally ; and that in addition to this repellent or diverging force in the transverse direction, he supposed an attractive or contractile force to be exerted at right angles to it, that is to say, in the direction of the lines of force. Of the existence of these pressures and tensions Maxwell was fully persuaded ; and he determined analytical expressions suitable to represent them. The tension along the lines of force must be supposed to maintain the ponderomotive force which acts on the conductor on which the lines of force terminate ; and it may therefore be measured by the force which is exerted on unit area of the conductor, ie., tE^/Swc' or 4DE. The pressure at right angles to the lines of force must then be determined so as to satisfy the condition that the aether is to be in equilibrium. For this purpose, consider a thin shell of aether included between two equipotential surfaces. The equilibrium of the. portion of this shell which is intercepted by a tube of force requires (as in the theory of the equilibrium of liquid films) that the resultant force per unit area due to the above- mentioned normal tensions on its two faces shall have the value T(l/p, + l(pi), where p, and p* denote the principal radii of curvature of the shell at the place, and where T denotes the lateral stress across unit length of the surface of the shell,. T being analogous to the surface-tension of a liquid film. Now, if t denote the thickness of the shell, the area inter- cepted on the second face by the tube of force bears to the area intercepted on the first face the ratio (pi + t)(pt + t)/ptp, ; and by the fundamental property of tubes of force, D and B vary inversely as the cross-section of the tube, so the total force on the second face will bear to that on the first face the ratio />ipi/(/>i + '] (pi + t), or approximately 3,Bl,ZEdhyG00gle 302 Maxwell. the resultant force per unit axes along the outward normal is therefore - JDE . t . [1/Pl + 1/p,), and ao we have * r = -£DE.£; or the pressure at right angles to the lines of force is |DE per unit area — that is, it is numerically equal to the tension along the lines of force. The principal stresses in the medium heing thus determined, it readily follows that the stresB across any plane, to which the unit vector H is normal, is (D. H)E- J (D.E)H. Maxwell obtained* a similar formula for the case of magnetic fields ; the ponderomotive forces on magnetized matter and on conductors carrying currents may be accounted for by assuming a stress in the medium, the stress across the plane V being represented by the vector i(B.H).H-l(B.H).I. This, like the corresponding electrostatic formula, represents a tension across planes perpendicular to the lines of force, and a pressure across planes parallel to them. It may be remarked that Maxwell made no distinction between stress in the material dielectric and stress in the aether : indeed, so long as it was supposed that material bodies when displaced carry the contained aether along with them, no distinction was possible. In the modifications of Maxwell's theory which were developed many years afterwards by his followers, stresses corresponding to those introduced by Maxwell were assigned to the aether, na distinct from ponderable matter ; and it was assumed that the only stresses set up in material bodies by the electromagnetic Held are produced indirectly: they may be calculated by the methods of the theory of elasticity, from a knowledge of the ponderomotive forces exerted on the electric charges connected with the bodies. * Maxwell's Treatise an Electricity and Mayutitm, } 613. 3,Bl,ZEdhyG00gle Maxwell. 303 Another remark suggested by Maxwell's theory of stress in the medium la that he considered the question from the purely statical point of view. He determined the stress so that it might produce the required forces on ponderable bodies, and be self-equilibrating in free aether. But* if the electric and magnetic phenomena are not really statical, but are kinetic in their nature, the stress or pressure need not be self-equilibrating. This may be illustrated by reference to the hydrodynamical models of the aether shortly to be described, in which perforated solids are immersed in a moving liquid : the ponderomotive forces exerted on the eolids by the liquid correspond to those which act on conductors carrying currents in a magnetic field, and yet there is no stress in the medium beyond the pressure of the liquid. Among the problems to which Maxwell applied his theory of stress in the medium was one which had engaged the attention of many generations of his predecessors. The ad- herents of the corpuscular theory of light in the eighteenth century believed that their hypothesis would be decisively con- finned if it could be shown that rays of light possess momentum : to determine the matter, several investigators directed powerful beams of light on delicately-suspended bodies, and looked for evidences of a pressure due to the impulse of the corpuscles. Such an experiment was performed in 1708 by Homberg.t who imagined that he actually obtained the effect in question ; but Mairan and Du Fay in the middle of the century, having repeated his operations, failed to confirm hie conclusion.* The subject was afterwards taken up by Michell, who "some years ago," wrote Priestley)} in 1772, " endeavoured to ascertain the momentum of light in a much more accurate manner than those in which M. Romberg and M. Mairan had attempted it." He exposed a very thin and delicately-suspended copper plate * CI. V. Bjerknm, Phil. Mag. iz (1906), p. 191. t HUtoire de l'Acad., 1708, p. 21. J J. J. de Mairan, Traiti de i'Aurare boriale, p. 370. { HUtorn #/ Vition, i, p. 387. 3,Bl,ZEdhyG00gle 304 Maxwell. to the rays of the aun concentrated by a mirror, and observed a deflexion. He was not satisfied that the effect of the heating of the air had been altogether excluded, bat " there seems to be no doubt," in Priestley's opinion, " but that the motion above mentioned is to be ascribed to the impulse of the rays of light" A similar experiment was made by A. Bennet,* who directed the light from the focus of a large lens on writing-paper delicately suspended ' in an exhausted receiver, but " could not perceive any motion distinguishable from the effeete of heat." " Perhaps," he concluded, " sensible heat and light may not be caused by the influx or rectilineal projections of fine particles, but by the vibrations made in the universally diffused calorie or matter of heat, or fluid of light." Thus Bennet, and after him Young, t regarded the non-appearance of light-repulsion in this experiment as an argument in favour of the nndulatory system of light. " For," wrote Young, " granting the utmost imaginable subtility of the corpuscles of light, their effects might naturally be expected to bear some proportion to the effects of the much less rapid motions of the electrical fluid, which are so very easily perceptible, even in their weakest Btates." This attitude is all the more remarkable, because Euler many years before had expressed the opinion that light-pressure might be expected just as reasonably on the undulatory as on the corpuscular hypothesis. "Just as," he wrote,} "a vehement sound excites not only a vibratory motion in the particles of the air, but there is also observed a real movement of the small particles of dust which are suspended therein, it is not to be doubted but that the vibratory motion set up by the light causes a similar effect." Euler not only inferred the existence of light-pressure, but even (adopting a suggestion of Kepler's) accounted for the tails of comets by supposing that the solar rays, impinging on the atmosphere of a comet, drive off from it the more subtle of its particles. 3,Bl,ZEdhyG00gle The question was examined by Maxwell' from the point of view of the electromagnetic theory of light; which readily furnishes reasons for the existence of light-pressure. For suppose that light falls on a metallic reflecting surface at perpendicular incidence. The light may be regarded as con- stituted of a rapidly-alternating magnetic field ; and this musit induce electric currents in the surface layers of the metal. But a metal carrying currents in a magnetic field is acted on by a ponderomotive force, which is at right anglea to both the magnetic force and the direction of the current, and is there- fore, in the present case, normal to the reflecting surface : this ponderomotive force is the light-pressure. Thus, according to Maxwell's theory, light-pressure is only an extended case of effects which may readily be produced in the laboratory. The magnitude of the light-pressure was deduced by Maxwell from his theory of stresses in the medium. We have seen that the stress across a plane whose unit-normal is H is represented by the vector (D.*).I-i(D.E).*+l(B.jr).H-i(B.H).H. Now, suppose that a plane wave is incident perpendicularly on a perfectly reflecting metallic sheet: this sheet must support the mechanical stress which exists at its boundary in the aether. Owing to the presence of the reflected wave, D is zero at the surface ; and B is perpendicular to B", so (B . K) vanishes. Thus the stress is a pressure of magnitude (l/8ir) (B . H) uormal to the surface: that is, the light-pressure is equal to the density of the aethereal energy in the region immediately outside the metal. This was Maxwell's result. This conclusion has been reached on the assumption that the light is incident normally to the reflecting surface. If, on the other hand, the surface is placed in an enclosure completely surrounded by a radiating shell, so that radiation falls on it from all directions, it may be shown that the light-pressure is measured by one-third of the density of aethereal energy. * Maxwell'* TrtatUt , the electrokinetic energy of the system may be represented by where ti, -i, denote the strengths of the currents; and the condition' that the number of lines of force linked with each circuit is to be invariable gives the equations Z,»i + Ll%i, - constant, -£ii*i + -£**i = constant. * Thomson's Reprint of Paptrtm Elect, and Mag.. f$ 573, 733, 761 (187ft- 1872). t Maxwell's Trtatiw on Sleet, and Mag., S 664. J For tl»i» reason W. Thomson cnllscl it perfect MMiriurtnr »n Uml nlnw dinnwgittic. D,Bl,ZEdhyG00gle 314 Models of the Aether. It is evident that, when the system is considered from the point of view of general dynamics, the electric currents must be regarded as generalized velocities, and the quantities (£,ii + £„*,) and (Zuij + At,) as momenta. The electromagnetic ponderomotive force on the rings tending to increase any coordinate x is dT/dx. In the analogous hydrodynamical system, the fluid velocity corresponds to the magnetic force: and therefore the circulation through each ring (which is defined to be the integral jvdg, taken round a path linked once with the ring) corresponds kinematically to the electric current ; and the flux of fluid through each ring corresponds to the number of lines of magnetic force which pass through the aperture of the ring. But in the hydro- dynamical problem the circulations play the part of generalized momenta ; while the fluxes of fluid through the rings play the part of generalized velocities. The kinetic energy may indeed be expressed in the form K= J (JW + 2JV1)KlKl + N,*S), where K], ki, denote the circulations (so that «i and e, are proportional respectively to i, and t,), and Nu N~„, N„ depend on the positions of the rings ; but this is the Hamiltonian (as opposed to the Lagrangian) form of the energy-function,* and the ponderomotive force on the rings tending to increase any coordinate x is - dKldx. Since dK/dx is equal to dT/dx, we see that the ponderomotive forces on the rings in any position in the hydrodynamical system are equal, but oppositf, to the ponderomotive forces on the rings in the electric system. The reason for the difference between the two cases may readily be understood The rings cannot cut through the lines of magnetic force in the one system, but they can cut through the stream-lines in the other : consequently the flux of fluid through the rings is not invariable when the rings are moved, the invariants in the hydrodynamical system being the circulations. •Cf. Whittaker, Analytical Dynamic, \ 109. 3,Bl,ZEdhyG00gle Models of the Aether. 3 1 5 If a thin ring, for which the circulation is zero, is introduced into the fluid, it will experience no ponderomotive forces ; but if a ring initially carrying no current is introduced into a magnetic field, it will experience ponderomotive forces, owing to the electric currents induced in it by its motion. Imperfect though the analogy is, it is not without interest. A bar-magnet, being equivalent to a cm-rent circulating in a wire wound round it, may be compared (as W. Thomson remarked) to a straight tube immersed in a perfect fluid, the fluid entering at one end and flowing out by the other, so that the particles of fluid follow the lines of magnetic force. If two such tubes are presented with like ends to each other, they attract ; with unlike ends, they repel. The forces are thus diametrically opposite in direction to those of magnets ; but in other respects the laws of mutual action between these tubes and between magnets are precisely the sama" * The mathematical analysis in this case is very simple. A narrow tub* through which water ii flowing may be regarded as equivalent to a sourie at one end of the tube and a link at the other ; and the problem may therefore be reduced to the consideration of linka in an unlimited fluid. If there are two sinks in such a fluid, ib tit and m', the velocity -potential it [ dxdydz, where r and r denote distance from the sinks. The kinetic energy per unit volume of the fluid it where p denote* the density of the fluid; whence it is easily seen that the total energy of the fluid, when the two sinks are at a distance i apart, exceeds the total energy when the; are at an infinite distance apart by an amount the integration being taken throughout the whole volume of the fluid, except two small sphere* i, •', surrounding the sinks. By Green's theorem, this expression where the integration is taken over » and »', and i» denotes the interior normal to j ore*. The integral taken over «' vanishes ; evaluating the remaining integral, we The energy of the fluid is therefore greater when sinks of strengths m, m' are at a 3,Bl,ZEdhyG00gle 316 Models of the Aether. Thomson, moreover, investigated* the ponderomotive forces which act between two solid bodies immersed in a fluid, when one of the bodies is constrained to perform small oscillations. If, for example, a small sphere immersed in an incompressible fluid is compelled to oscillate along the line which joins its centre to that of a much larger sphere, which is free, the free sphere will be attracted if it is denser than the fluid ; while if it is less dense than the fluid, it will be repelled or attracted according as the ratio of its distance from the vibrator to its radius is greater or less than a certain quantity depending on the ratio of its density to the density of the fluid. Systems of this kind were afterwards extensively investigated by C. A. Bjerknes-t Bjerknes showed that two spheres which are immersed in an incompressible fluid, and which pulsate (i.e., change in volume) regularly, exert on each other (by the mediation of the fluid) an attraction, determined by the inverse square law, if the pulsations are concordant; and exert on each other a repulsion, determined likewise by the inverse square law, if the phases of the pulsations differ by half a period. It is necessary to suppose that the medium is incom- pressible, so that all pulsations are propagated instantaneously : otherwise attractions would change to repulsions and vice versa at distances greater than a quarter wave-length.} If the spheres, instead of pulsating, oscillate to and fro in straight lines about their mean positions, the forces between them are proportional in magnitude and the same in direction, but mutual distance I than when sinks of the Mine strengths are at infinite distance apart by an amount i-rp •iim'jl. Since, in the case of the tubal, the quantities ■ correspond to the fluzea of fluid, this eipreasiun corresponds to the Lagrangian form of the kinetic energy ; and therefore the force tending to increase the co-irdi- natex of one of the sinks is (S/3se) (*«■(> mm'/fl- Whence it is seen that the Utteadt of twu tubes attract, and the unlike end* ripel, according to the inverse square lair. * Phil. Hag. ili (1870), p. 427. t Itepertorium d. Hathematik too Koniaberger iind Zeuner (1876), p. 268. GGttingar Naohrichten, 1876, p. 245. Comptea Kendua, lxxxtr (1877), f. 1177- Cf. Nature, »tiv (18811, p. 360. X On the mathematical theory of the force between two pulsating tpnem in ■ fluid, cf. W. M. Hicks, Proo. Camb. PhiL Soc. iii (18TB), p. 276 ; it (1880). p. S>. dhyGoogle Models of the Aether. 3 1 7 opposite in sign, to those which act between two magnets oriented along the directions of oscillation* The results obtained by Bjerknes were extended by A. H. Leahyf to the case of two spheres pulsating in an elastic medium ; the wave-length of the disturbance being supposed large in comparison with the distance between the spheres. For this ajstem Bjerknes' results are reversed, the law being now that of attraction in the case of unlike phases, and of repulsion in the case of like phases : the intensity is as before proportional to the inverse square of the distance. The same author afterwards discussedj the oscillations which may be produced in an elastic medium by the displacement, in the direction of the tangent to the cross- section, of the surfaces of tubes of small sectional area : the tubes either forming closed curves, or extending inde- finitely in both directiona The direction and circumstances of the motion are in general analogous to ordinary vortex- motions in an incompressible fluid ; and it was shown by Leahy that, if the period of the oscillation be such that the waves produced are long compared with ordinary finite distances, the displacement due to the tangential disturbances is proportional to the velocity due to vortex-rings of the same form as the tubular surfaces. One of these " oscillatory twists," as the tubular surfaces may be called, produces a displacement which is analogous to the magnetic force due to a current flowing in a curve coincident with the tube ; the strength of the current being proportional to Vw sinpt, where b denotes the radius of the twist, and a sin pt its angular displacement. If the field of vibration is explored by a rectilineal twist of the same period as that of the vibration, the twist will experience a force * A theory of graritation haa been based by Kara on the assumption iLnl gravitating particle! resemble slightly compressible spheres immersed in aa incom- pressible perfect Quid : the sphere* extoute pulsations, whose intensity corresponds t« the mass of the gravitating puiticles, and thus force* of the Newtonian kind are produced between them. Cf. Korn, Bin* Tkeuri* tier Gravitation and der eltet. S'teheimmgm, Berlin, 189S. t Trans. Camb. Phil. Sod. xiv (1B84), p. 45. ; Tnms. Climb. Phil. Son. xiv (1885), p. 188. DinlizBdbyGoOgk 3 1 8 Models of the Aether. at right angles to the plane containing the twist and the direction of the displacement which would exist if the twist were removed ; if the displacement of the medium be repre- sented by F sin pt, and the angular displacement of the twist by \* id dm &u> 3u 3m5p -"y- W +W +{*) + 255T S^ %£; and on subtracting the forms which this equation takes in the two cases, we have dy dx which, when the turbulent motion is fine-grained, so thai f{y, i) in sensibly constant over ranges within which «', v, »r pass through all their values, may be written Moreover, we have n a 1 - 8(*'»') 3{tt'») 3(m'd) 8p' , 8p'i for positive and negative values of u', v, ware equally probable ; and therefore the value of the second member of this equation is doubled by adding to itself what it becomes when for u', «, w we substitute - it', -v, -w; which (as may be seen by inspection of the above equation in Vp) does not change the value ofp'. 3,Bl,ZEdhyG00gle Models of the Aether. 33 1 Comparing this equation with that which determines the value of Q, we have or substituting for to, % \ a* »al a> The isotropy with respect to * and z gives the equation + (-'s+"4s)||v^ But by integration by parts we obtain the equation Ai*a* "•*)%• ' •• — '(is + si*™- and by the condition of incompressibility the second member may be written A.Qnflti.Q/afl.V**, or -A.v..ffl3tf).V*v,i so we have 9..-%mA.u' + **)\.v~v. 3y I Va? *' dy'JI On account of the isotropy, we may write J for and, therefore, The deviation from isotropy shown by this equation is very small, because of the amallneBS of df(y, t)idy. The equation is therefore not restricted to the initial values of the two members, 3,Bl,ZEdhyG00gle 332 Models of the Aether. for we may neglect an infinitesimal deviation from (2/9).ff in the first factor of the second member, in consideration of the smallness of the second factor. Hence for all values of t we have the equation which, in combination with (1), yields the result g/(iK>-4*£u«r.O; the form of this equation shows that laminar disturbances are propagated through the vortex-sponge in the same manner as waves of distortion ill a homogeneous elastic solid. The question of the stability of the turbulent motion remained undecided ; and at the time Thomson seems to have thought it likely that the motion would suffer diffusion. But two years later* he showed that stability was ensured at any rate when space is filled with a set of approximately straight hollow vortex filaments. Fitz Geraldf subsequently determined the energy per unit- volume in a turbulent liquid which is transmitting laminar waveB. Writing for brevity t2/9)i?=r*, /(y,0--P. a^ A(u'v)-y, the equations are dP dy , dy „,3P *-*• and aT-^e* If the quantity P + 7VP-22 is integrated throughout space, and the variations of the integral with respect to time are determined, it is found that J"&**-J[J(,,¥+ *&)*** -I('S-*S)*** * Pro*. Boy. Irish Acad. (3) i (1SSS), p. 340 ; Kelvin*! Mali, undlliy,. frnptn, ', p. 202. t Brit. Auoc. Sep., 18B9. Filz Gerald' i Scientific iFrUiHft, p. 464. 3,Bl,ZEdhyG00gle Models of the Aether. 333 Integrating the second term under the integral by parts, and omitting the superficial terms (which may be at infinity, or wherever energy enters the space under consideration), we have Hence it appears that the quantity 2, which is of the dimensions of energy, must be proportional to the energy per unit-volume of the medium — a result which shows that there is a pronounced similarity between the dynamics of a vortex-sponge and of Maxwell's elastic aether. A definite vortex-sponge model of the aether was described by Hicks in his Presidential Address to the mathematical section of the British Association in 1895.* In this the small motions whose function is to confer the qnasi-rigidity were not completely chaotic, but were disposed systematically. The medium was supposed to be constituted of cubical elements of fluid, each containing a rotational circulation complete in itself : in any element, the motion close to the central vertical diameter of the element is vertically upwards : the fluid which is thus carried to the upper part of the element flows outwards over the top, down the sides, and up the centre again. In each of the six adjoining elements the motion is similar to this, but in the reverse direction. The rotational motion in the elements confers on them the power of resisting distortion, so that waves may be propagated through the medium as through an elastic solid ; but the rotations are without effect on irrotational motions of the fluid, provided the velocities in the irrotational motion are slow compared with the velocity of propagation of distortional vibrations. A different model was described four years later by Fitz GeralcLf Since the distribution of velocity of a fluid in the • Brit. Assoc. Rep., 18H5, p. 695. t Pro*. Boy. Dublin Sot., December 12, 1899; FitsQcnld'i Sritnti/le Writing; p. *72. D,Bl,ZEdhyG00gle 334 Models of the Aether. neighbourhood of a vortex filament is the same as the distribu- tion of magnetic force around a wire of identical form carrying an electric current, it ia evident that the fluid has more energy when the filament has the form of a helix than when it is straight; so if space were filled with vortices, whose axes were all parallel to a given direction, there would be an increase in the energy per unit volume when the vortices were bent into a spiral form ; and this could be measured by the square of a vector— Bay, E — which may be supposed parallel to this direction. If now a single spiral vortex is surrounded by parallel straight ones, the latter will not remain straight, but will be bent by the action of their spiral neighbour. The transference of spirality may be specified by a vector H, which will be dis- tributed in circles round the spiral vortex ; its magnitude will depend on the rate at which spirality is being lost by the original spiral, and can be taken such that its square is equal to the mean energy of this new motion. The vectors E and H will then represent the electric and magnetic vectors; the vortex spirals representing tubes of electric force. Fitz Gerald's spirality is essentially similar to the laminar motion investigated by Lord Kelvin, since it involves a flow in the direction of the axis of the spiral, and such a flow cannot take place along the direction of a vortex filament without a spiral deformation of a filament. Other vortex analogues have been devised for electro- statical systems. One such, which was described in 1888 by W. M. Hicks,* depends on the circumstance that if two bodies in contact in an infinite fluid are separated from each other, and if there be a vortex filament which terminates on the bodies, there will be formed at the point where they separate a hollow vortex filamentf stretching from one to the other, with rotation * Brit. Aasoc. Bep., 1888, p. 677. t A hollow TOttex is a cyclic motion exutingin a fluid without tbe presence of nny actual rotational filaments. On the general theory cf. Hicks, Phil. Traiu . nlixv (1883), p. 161 ; clxxvi (188S), p. 726 ; cicii [1898), p. 33. 3,Bl,ZEdhyG00gle Models of the Aether. 335 equal and opposite to that of the original filament. As the bodies are moved apart, the hollow vortex may, through failure of stability, dissociate into a number of smaller ones ; and if the resulting number be very large, they will ultimately take up a position of stable equilibrium. The two sets of filaments — the original filaments and their hollow companions — will be intermingled, and each will distribute itself according to the same law as the lines of force between the two bodies which are equally and oppositely electrified. Since the pressure inside a hollow vortex is zero, the portion of the surface on which it abuts experiences a diminution of pressure ; the two bodies are therefore attracted. Moreover, as the two bodies separate further, the distribution of the filaments being the same as that of lines of electric force, the diminution of pressure for each line is the same at all distances, and there- fore the force between the two bodies follows the same law as the force between two bodies equally and oppositely electrified. It may be shown that the effect of the original filaments is similar, the diminution of pressure being half as large again as for the hollow vortices. If another surface were brought into the presence of the others, those of the filaments which encounter it would break off and rearrange themselves so that each part of a broken filament terminates on the new body. This analogy thus gives a complete account of electrostatic actions both quantitatively and qualitatively : the electric charge on a body corresponds to the number of ends of filaments abutting on it, the sign being determined by the direction of rotation of the filament as viewed from the body. A magnetic field may be supposed to be produced by the motion of the vortex filaments through the stationary aether, the magnetic force being at right angles to the filament and to its direction of motion. Electrostatic and magnetic fields thus correspond to states of motion in the medium, in which, how- ever, there is no bodily flow; for the two kinds of filament produce circulation in opposite directions. 3,Bl,ZEdhyG00gle 386 Models of the Aether. It is possible that hollow vortices are better adapted than ordinary vortex-filaments for the construction of models of the aether. Such, at any rate, was the opinion of Thomson (Kelvin) in his later years." The analytical difficulties of the subject are formidable, and progress is consequently alow ; but among the many mechanical schemes which have been devised to represent electrical and optical phenomena, none possesses greater interest than that which pictures the aether as a vortex -sponge. * Proc. Roy. Irish Acad., November 30, 1889 ; Kelvin's Math, end Pk„i. Paper; iv, p. 202. " Rotational vortex-ooree," be wrote, "murt be »b«olutttr tiiscardcd : and we mint bare nothing but irrolational revolution end vncuoiu 3,Bl,ZEdhyG00gle ( 337 ) THE FOLLOWERS OF MAXWELL. The most notable imperfection in the electromagnetic theory of light, as presented in Maxwell's original memoirs, was the absence of any explanation of reflexion and refraction. Before the publication of Maxwell's Treatise, however, a method of supplying the omission was indicated by Helmholtz.* The principles on which the explanation depends are that the normal component of the electric displacement D, the tangential components of the electric force E, and the magnetic vector B or H, are to be continuous across the interface at which the reflexion takes place; the optical difference between the con- tiguous bodies being represented by a difference in their dielectric constants, and the electric vector being assumed to be at right angles to the plane of polarization.f The analysis required is a mere transcription of MacCulIagh's theory of reflexion,J if the derivate of MacCulIagh's displacement e with respect to the time be interpreted as the magnetic force, fi curl e as the electric force, and curl e as the electric displace- ment. The mathematical details of the solution were not given by Helmholtz himself, but were supplied a few years later in the inaugural dissertation of H. A. Lorentz.Si In the years immediately following the publication of Maxwell's Treatise, a certain' amount of evidence in favour of * Journal fiir Hath. Uxii (1870), p. 68, note. t Helmholtx (loc. cit.) pointed out that if the optical difference between the media were assumed to be due to a difference in their magnetic permeabilities, it would be necessary to suppoae the magnetic vector at right angles to the plane of polarisation in order to obtain Fresnel'a due and tangent formulae of reflexion. X Cf. pp. 148, 149, 104-166. $ Zoitichrift fiir Hath. u. Phya. nii (1877), pp. I, 20a : Over ie thnrii dtr ttrngkaatting en breiinf van hit litht, Amhem, 1876. Lorentx'a work was baaed on Helmholti'a equations, but remains aubatantially unchanged when Maxwell's formulae ate aubatituted. 3,Bl,ZEdhyG00gle 338 The Followers of Maxwell. his theory was furnished by experiment. That an electric field is closely concerned with the propagation of light was demon- strated in 1875, when John Kerr* showed that dielectrics subjected to powerful electroatatic force acquire the property of double refraction, their optical behaviour being similar to that of uniaxal crystals whose axes are directed along the lines of force. Other researches undertaken at this time had a more direct bearing on the questions at issue between the hypothesis of Maxwell and the older potential theories. In 1875-6 Helmholtzt and his pupil Schiller* attempted to discriminate between the various doctrines and formulae relative to unclosed circuits by performing a crucial experiment. It was agreed in all theories that a ring-shaped magnet, which returns into itself so as to have no poles, can exert no ponderomotive force on other magnets or on closed electric currents. Helmholtzg had, however, shown in 1873 that accord- ing to the potential-theories such a magnet would exert a ponderomotive force on an unclosed current The matter was tested by suspending a magnetized steel ring by a long fibre in a closed metallic case, near which was placed a terminal of a Holtz machine. No ponderomotive force could be observed when the machine was put in action so as to produce a brush discharge from the terminal : from which it was inferred that the potential-theories do not correctly represent the phenomena, at least when displacement-currents and convection-currents (such as that of the electricity carried by the electrically repelled air from the terminal) are not taken into account. The researches of Helmholtz and Schiller brought into prominence the question as to the effects produced by the • Phil. Mag. (i) 1 (1876), pp. 337, 446 ; (6) viu (1879), pp. 85, 229 ; nii (188!), pp. 163, 248. t Monat»berioht« d. Acid, d. Berlin,' 1875, p. 400. Ann. d. Phys., cWiii (1 8T6S p. 87. I Ann- d. Phys. clix (1876), pp. 4S6, 637 ; cli (1877), p. 3St- i The valuable memoirs by Helmholtz in Journal fur Hath, Urii (IS7«). p. 57 ; Izit (1873), p. 3a ; luviii (1874), p. 273, to which reference hu ilrMdr been muds, contain a full discussion of the various possibilities or the poucusl- 3,Bl,ZEdhyG00gle The Followers of Maxwell. 339 translator^ motion of electric charges. That the convection of electricity is equivalent to a current had been suggested long before by Faraday* "If," he wrote in 1838, "a ball be electrified positively in the middle of a room and be then moved in any direction, effects will be produced as if a current in the same direction had existed." To decide the matter a new experiment inspired by Helmholtz was performed by H. A. Rowlandf m 1876. The electrified body in Rowland's disposition was a disk of ebonite, coated with gold leaf and capable of turning rapidly round a vertical axis between two fixed plates of glass, each gilt on one side. The gilt faces of the plates could be earthed, while the ebonite disk received electricity from a point placed near its edge ; each coating of the disk thus formed a condenser with the plate nearest to it An astatic needle waa placed above the upper condenser-plate, nearly over the edge of the disk; and when the disk was rotated a magnetic field was found to be produced. This experiment, which has since been repeated under improved conditions by Rowland and Hutchinson ,J H. Pender§, and £ichenwald,|| shows that the " convection-current " produced by the rotation of a charged disk, when the other ends of the lines of force are on an earthed stationary plate parallel to it, produces the same mag- netic field as an ordinary conduction-current flowing in a circuit which coincides with the path of the convection-current. When two disks forming a condenser are rotated together, the magnetic action is the sum of the magnetic actions of each of the disks separately. It appears, therefore, that electric charges cling to the matter of a conductor and move with it, so far as Rowland's phenomenon is concerned. The first examination of the matter from the point of view of Maxwell's theory was undertaken by J. J. Thomson,! in 1881. If an electrostatically charged body is in motion, the change in • Bxptr. St,., j 164*. tHonBtiberiehUd.Akmd.d- Berlin, 1878, p. 211: Ann. d. Phyi. olriii (1878), p. 487 : AnnalM do Chin.. eC de Phyi. xii (1877) p. 119. J Phil. Hag. KTii (1889), p. 446. f «tf.ii(I90I),p. 178: y (1903), p. 34. | Ann. d. Phji. zi (1901), p. 1. 1 Phil. Mag. si (1881), p. 229. Z 2 3,Bl,ZEdhyG00gle 340 The Followers of Maxwell, the location of the charge must produce a continuous alteration of the electric field at any point in the surrounding medium ; or, in the language of Maxwell's theory, there must be displacement- currents in the medium. It was to these displacement-currents that Thomson, in his original investigation, attributed the magnetic effects of moving charges. The particular aystem which he considered was that formed by a charged spherical conductor, moving uniformly in a straight line. It was assumed that the distribution of electricity remains uniform over the surface during the motion, and that the electric field in any position of the sphere in the same as if the sphere were at rest ; these assumptions are true so long as quantities of order (r/c)* are neglected, where v denotes the velocity of the sphere and c the velocity of light. Thomson's method was to determine the displacement- currents in the space outside the sphere from the known values of the electric field, and then to calculate the vector- potential due to theae displacement-currents by means of the formula where 8' denotes the displacement-current at (ai'yV). The magnetic field was then determined by the equation H - curl A. A defect in this investigation was pointed out by Fitz Gerald, who, in a short but most valuable note,* published a few months afterwards, observed that the displacement-currents of Thomson do not satisfy the circuital condition. This is most simply seen by considering the case in which the system consists of two parallel plates forming a condenser; if one of the plates ia fixed, and the other plate is moved towards it, the electric field is annihilated in the space over which the moving plate travels: this destruction of electric displacement constitutes a displace- ment-current, which, considered alone, is evidently not a closed 3,Bl,ZEdhyG00gle The Followers of Maxwell. 341 current. The defect, as Fitz Gerald showed, may be immediately removed by assuming that a moving charge itself is to be counted as a current-element : the total current, thus composed of the displacement- currents and the convection-current, is circuital. Making this correction, Fitz Gerald found that the magnetic force due to a sphere of charge e moving with velocity v along the axis of z is curl (0, 0, ev/r) — a formula which shows that the displacement-currents have no resultant magnetic effect, since the term ev(r would be obtained from the convection-current alone. The expressions obtained by Thomson and Fitz Gerald were correct only to the first order of the small quantity vje. The effect of including terms of higher order was considered in 1889 by Oliver Heaviside* whose solution may be derived in the following manner: — Suppose that a charged system is in motion with uniform velocity v parallel to the axis of z ; the total current consists of the displacement-current E/4wv. Eliminating B between this and the equation curl E = - H, and remembering that H is here circuital, we have H/e' - V'H - 4jr curl p*. If, therefore, a vector-potential a be defined by the equation i/c1 - V1* = 4irpv, the magnetic force will be the curl of a ; and from the equation for a it is evident that the components ax and a, are zero, and that a, is to be determined from the equation Hifc* - V'a, = 4wpt>. • Phil. Mag. ixrii (1889), p. 334. 3,Bi,zEdhyG00gre 342 The Followers of Maxwell. Now, let (x , y, £) denote coordinates relative to axea which are parallel to the axes (x,y,z), and which move with the charged bodies ; then a, is a function of {x, y, £) only ; so we have s-* and u=~vrv and the preceding equation is readily seen to be equivalent to B'a, d*aM 8*a, where £, denotes (1 - t**/**)-*?. But this is simply Poisson's equation, with £, substituted for z; so the solution may be transcribed from the known solution of Poisson's equation : it is p'v' dx' dy' At,' -In J l(«. -«.'>■ + (* -*')' + (y -sO'l1' the integrations being taken over all the space in which there are moving charges ; or fff pVMdi/d? "■ "JJJ t(t-r>' + (i - "m^—rr * (i - «'/'*■) (y-»W If the moving system consists of a single charge e at the point X, = 0, this gives "* r(l -•■ sin* tf/cO*' where Bin* & = (x* + y*)/r*. It is readily seen that the lines of magnetic force due to the moving point-charge are circles whose centres are on the line of motion, the magnitude of the magnetic force being ev (1 - nfo1) sin 9 ^(l-^sin'ft/c*)!" The electric force is radial, its magnitude being «fts(l-yy. xli (1890), |>. 400. t Tnum. Roy. Dublin Soo. iii (1883) ; Fit* Gerald'* Scitnt. Writing; p. 121. 3,Bl,ZEdhyG00gle 346 The Followers of Maxwell. the strength of the current is varied according to the simple periodic law. The circuit will be supposed to be ■ circle of small area S, whose centre is the origin and whose plane is the plane of xy ; and the surrounding medium will be supposed to be free aether. The current may be taken to be of strength A cos (2wt/T), so that the moment of the equivalent magnet is SA cos (2wt/T). Now in the older electrodynamics, the vector-potential due to a magnetic molecule of (vector) moment ■ at the origin is (1/4tt) curl (H/r)> where r denotes distance from the origin. The vector-potential due to Fitz Gerald's magnetic oscillator would therefore be fl/lir) curl E, where X denotes a vector parallel to the axis of 2, and of magnitude (1/r) SA cos (2jrf/ T). The change which is involved in replacing the assumptions of the older electrodynamics by those of Maxwell's theory is in the present case equivalent" to retarding the potential ; so that the vector-potential a due to the oscillator is (1/4*-) curl K where E is still directed parallel to the axis of 2, and is of magnitude _ SA W r\ The electric force E at any point of space is - a, and the magnetic force H is curl a: so that these quantities may be calculated without difficulty. The electric energy per unit volume is E'/Swc* : performing the calculations, it is found that the value of this quantity averaged over a period of the oscillation and also averaged over the surface of a sphere of radiuB r is The part of this which is radiated is evidently that which is proportional to the inverse square of the distance,! M ^e • Cf. pp. 208, 399. + The other term, whiuh is neglected, U very small compared to the to* retained, at great distances from the origin ; it is what would be obtained if the effect's of induction of the displacement- currents wen neglected : i.e. it is the energy of the forced displacement -currents which are produced directly by the Tariation of the primary current, and which originate the radiating displacemeal- 3,Bl,ZEdhyG00gle The Followers of Maxwell. 347 average value of the radiant energy of electric type at distance r from the oscillator is 2x*A*S'l3clr'Tt per unit volume. The radiant energy of magnetic type may be calculated in a similar way, and is found to have the same value ; so the total radiant energy at distance r is ■iit*Ax3'/l&rtTl per unit volume; and therefore the energy radiated in unit time is 16irM'i?/3e*I\ This is small, unless the frequency is very high ; so that ordinary alternating currents would give no appreciable radia- tion. Fitz Gerald, however, in the same year* indicated a method by which the difficulty of obtaining currents of sufficiently high frequency might be overcome: this was, to employ the alternating currents which are produced when a condenser is discharged. The Fits Gerald radiator constructed on this principle is closely akin to the radiator afterwards developed with such success by Hertz : the only difference is that in Fitz Gerald's arrangement the condenser is used merely as the store of energy (its plates being so close together that the electrostatic field due to the charges is practically confined to the space between them), and the actual source of radiation is the alternating magnetic field due to the circular loop of wire: while in Hertz's arrangement the loop of wire is abolished, the condenser plates are at some distance apart, and the source of radiation is the alternating electrostatic field due to their charges. In the study of electrical radiation, valuable help is afforded by a general theorem on the transfer of energy in the electro- magnetic field, which was discovered in 1884 by John Henry Poynting.f We have seen that the older writers on electric currents recognized that an electric current is associated with the transport of energy from one place (e,g. the voltaic cell which maintains the current) to another (e.g. an electromotor which is worked by the current) ; but they supposed the energy to be conveyed by the current itself within the wire, in much * Brit. AMM. Hep., 1883 ; FitiQenild'i Scientific Writing; p. \%9. t Phil. Timni. olzxv (18M), p. 343. 3,Bl,ZEdhyG00gle 348 . The Follcmers of Maxwell. the same way as dynamical energy is carried by water flowing in a pipe ; whereas in Maxwell's theory, the storehouse and vehicle of energy is the dielectric medium surrounding the wire. What Foynting achieved -(Was to show that the flux of energy at any place might be expressed by a simple formula in terms of the electric and magnetic forces at the place. Denoting as usual by E the electric force, by D the electric displacement, by H the magnetic force, and by B the magnetic induction, the energy stored in unit volume of the medium is* iKD+(l/8w)BH; so the increase of this in unit time is (since in isotropic media D is proportional to E, and B is proportional to H) ED + (1/4tt)HB or B (8 - l) + (l/4,r) HE, where S denotes the total current, and l the current of conduction ; or (in virtue of the fundamental electromagnetic equations) - (E . l) + (1/4*-) [E . curl H) - (l/4w) (H . curl E;, or - (E . i) - (1/4t) div [E . H]. Now (E . l) is the amount of electric energy transformed into heat per unit volume per second; and therefore the quantity - (1/4jt) div [E . H] must represent the deposit of energy in unit volume per second due to the streaming of energy; which shows that the flux of energy is represented by the vector (1/4*-) [E.H].f This is Poynting's theorem: thai the fivx of energy at any place is represented by the vector-product of the electric and magnetic forces, divided by 4w.* • Cf- pp. 248, 250, 282. t Of course any circuital vector may be added. II. M. Macdonald, Eltctric Wmtt, p. T2, propounded u form whidi differs from Poynling'a by a non-circuital Teetor. J The analogue of Poynting'e theorem in the theory of the Titration* of an isotropic elastic solid may be easily obtained ; for from tbe equation of motion of an elastic solid, »*"=-(* + 4n/3) grad dir • — » ourl ourl a, it folio**, thnt |(i>* + H* + J") (dW •)* + J* (curl •)»( - - div W, B*V 3,Bl,ZEdhyG00gle The Followers of Maxwell. 349 In the special case of tbe field which surrounds a straight wire carrying a continuous current, the lines of magnetic force are circles round the axis of the wire, while the lines of electric force are directed along the wire ; hence energy must be flowing in the medium in a direction at right angles to the axis of the wire. A current in any conductor may therefore be regarded as consisting essentially of a convergence of electric and magnetic energy from the medium upon the conductor, and its trans- formation there into other forms. This association of a current with motions at right angles to the wire in which it flows doubtless suggested to Poynting the conceptions of a memoir which he published" in the following- year. When an electric current flowing in a straight wire is gradually increased in strength from zero, the surrounding space becomes filled with lines of magnetic force, which have the form of circles round the axis of the wire. Poynting, adopting Faraday's idea of the physical reality of lines of force, assumed that these lines of force arrive at their places by moving out- wards from the wire ; so that the magnetic field grows by a con- tinual emission from the wire of lines of force, which enlarge and spread out like the circular ripples from the place where a stone is dropped into a pond. The electromotive force which is. associated with a changing magnetic field was now attributed directly to the motion of the lines of force, so that wherever electromotive force is produced by change in the magnetic field, or by motion of matter through the field, the electric intensity is equal to the number of tubes of magnetic force intersected by unit length in unit time. A similar conception was introduced in regard to lines of electric force. It was assumed that any change in the total where W denotes the vector - (* 4 **jZ) div • . • + n [our! « . *] ; ■nd tinea the eipremion which i> differentiated with respect to t represents the turn of the kinetic end potential energise per unit volume of the solid (wva for terms which give only surface-integrals), it is seen that W Is the analogue of the Pointing vector. Cf . L. Donati, Bologna Hem. (5) vii (1899), p. 033. • Phil. Trans, clurri (1886), p. 277. 3,Bl,ZEdhyG00gle 350 The Followers of Maxwell. electric induction through a curve is caused by the passage of tubes of force in or out across the boundary ; bo that whenever magnetomotive force is produced by change in the electric field, or by motion of matter through the field, the magnetomotive force is proportional to the number of tubes of electric force intersected by unit length in unit time. Poynting, moreover, assumed that when a steady current C flows in a straight wire, C tubes of electric force close in upon the wire in unit time, and are there dissolved, their energy appearing as heat. If E denote the magnitude of the electric force, the energy of each tube per unit length is \E, so the amount of energy brought to the wire is \CE per unit length per unit time. This is, however, only half the energy actually transformed into heat in the wire : bo Poynting further assumed that E tubes of magnetic force also move in per unit length per unit time, and finally disappear by contraction to in finitely small rings. This motion accounts for the existence of- the electric field; and since each tube (which is a closed ring) contains energy of amount \C, the disappearance of the tubes accounts for the remaining \CE units of energy dissipated in the wire. The theory of moving tubes of force has been extensively developed by Sir Joseph Thomson." Of the two kinds of tubes — magnetic and electric — which had been introduced by Faraday and used by Poynting, Thomson resolved to discard the former and employ only the latter. This was a distinct departure from Faraday's conceptions, in which, as we have seen, great significance was attached to the physical reality of the magnetic lines ; but Thomson justified his choice by inferences drawn from the phenomena of electric conduction in liquids and gases. As will appear subsequently, these phenomena indicate that molecular structure is closely connected with tubes of electro- static force — perhaps much more closely than with tubes of magnetic force ; and Thomson therefore decided to regard uXkat.amd 3,Bl,ZEdhyG00gle The Followers of Maxwell. 351 magnetism as the secondary effect, sod to ascribe magnetic fields, not to the presence of magnetic tubes, but to the motion of electric tubes. In order to account for the fact that magnetic fields may occur without any manifestation of electric force, he assumed that tubes exist in great numbers everywhere in space, either in the form of closed circuits or else terminating on atoms, and that electric force is only perceived when the tubes have a greater tendency to lie in one direction than in another. In a steady magnetic field the positive and negative tubes might be conceived to be moving in opposite directions with equal velocities. A beam of light might, from this point of view, be regarded simply as a group of tubes of force which are moving with the velocity of light at right angles to their own length. Such a conception almost amounts to a return to the corpuscular theory; but since the tubes have definite directions per- pendicular to the direction of propagation, there would now be no difficulty in explaining polarization. The energy accompanying all electric and magnetic pheno- mena was supposed by Thomson to be ultimately kinetic energy of the aether ; the electric part of it being represented by rota- tion of the aether inside and about the tubes, and the magnetic part being the energy of the additional disturbance set up in the aether by the movement of the tubes. The inertia of this latter motion he regarded as the cause of induced electromotive There was, however, one phenomenon of the electromagnetic field as yet unexplained in terms of these conceptions — namely, the ponderomotive force which is exerted by the field on a conductor carrying an electric current. Now any pondero- motive force consists in a transfer of mechanical momentum from the agent which exerts the force to the body which experiences it ; and it occurred to Thomson that the pondero- motive forces of the electromagnetic field might be explained if the moving tubes of force, which enter a conductor carrying a current and are there dissolved, were supposed to 3,Bl,ZEdhyG00gle 352 The Followers of Maxwell. mechanical momentum, which could be yielded up to the conductor. It is readily seen that such momentum must be directed at right angles to the tube and to the magnetic induction — a result which suggests that the momentum stored in unit volume of the aether may be proportional to the vector- product of the electric and magnetic vectors. For this conjecture reasons of a more definite kind may be given* We have already seen-f that the ponderomotive forces on material bodies in the electromagnetic field may be accounted for by Maxwell's supposition that across any plane in the aether whose unit normal is V, there in a stress represented by P» - (B.K) E - J (D.E)H + (1/4jt) (B.H)H - (1/8*-) (B.HjH. So long as the field is steady (i.e. electrostatic or magnetostatic) the resultant of the stresses acting on any element of volume of the aether is zero, bo that the element is in equilibrium. But when the field is variable, this is no longer the case. The resultant stress on the aether contained within a surface S is // Pw . OS integrated over the surface : transforming this into a volume- integral, the term (D . N ) E gives a term div D . E + (D . V)E, where V denotes the vector operator (3/3*, 3/9y, 3/3z) ; and the first of these terms vanishes, since D is a circuital vector; the term - £ (D . E) V gives in the volume-integral a term \ grad (D . E) ; and the magnetic terms give similar results. So the resultant force on unit-volume of the aether is (D . V) E + \ grad (D . E) + (l/4r) (B . V) E + (l/8») grad (B . H), which may be written [curl E . D] + (1/4tt) [curl H . B] ; * The hypothesis that the tether is h atorehotue of mechanical momentum, which wu lint advanced by J. J. Thomson {Kiciut Renartku in Eltet. and Mq. (1893), p. 13), vae aiterwarda developed by H. Poincare, Archive* Need. (3)v (1900), p. 262, and by M. Abraham, G8U, XncD., 1902, p. 20. t Cf. p. 302. .yGoogle The Followers 0/ Maxwell. 353 or, by virtue of the fundamental equations for dielectrics, [-h.D] + [b.B], or 0/W)[D.B]. This result compels ub to adopt one of three alternatives : either to modify the theory so as to reduce to zero the resultant force on an element of free aether ; this expedient has not met with general favour ;• or to assume that the force in question sets the aether in motion : this alternative was chosen by Helmholtz,f but is inconsistent with the theory of the aether which was generally received in the closing years of the century; or lastly, with Thomson,* to accept the principle that the aether is itself the vehicle of mechanical momentum, of amount [D . B] per unit volume. Maxwell's theory was now being developed in ways which could scarcely have been anticipated by its author. But although every year added something to the superstructure, the founda- tions remained much as Maxwell had laid them ; the doubtful argument by which he had sought to justify the introduction of displacement- currents was still all that was offered in their defence. In 1884, however, the theory was established^ on a different basis by a pupil of Helmholtz', Heinrich Hertz (b. 1857, d. 1894). The train of Hertz' ideas resembles that by which Ampere, on hearing of Oersted's discovery of the magnetic field produced by electric currents, inferred that electric currents should exert ponderomotive forces on each other. Ampere argued that a current, being competent to originate a magnetic field, must be equivalent to a magnet in other respects ; and therefore that currents, like magnets, should exhibit forces of mutual attraction and repulsion. * It woe, however, adopted by Q. T. WiJker, Aberration and tin EltclromagMtU Fuld, Cunb., 1900. t Berlin Sitiungsberichte, 1803, p. 619; Ami. d. Phys. liii (1384), p. 135. Helmholtx supposed the aether to behave as a frictionlesi incompressible fluid. J Lot. cit. £■ Ann. d. Phye. ixiii (1884), p. 84: English version in Hertz's Mitettituuom Papen, translated by D. E. Jones and G. A. Schott, p. 273. 3,Bl,ZEdhyG00gle 354 The Followers of Maxwell. Ampere's reasoning rests on the assumption that the mag- netic field produced by a current is in all respects of the same nature as that produced by a magnet ; in other words, that only one kind of magnetic force exists. This principle of the " unity of magnetic force" Hertz now proposed to supplement by assert- ing that the electric force generated by a changing magnetic field is identical in nature with the electric force due to electro- static charges; this second principle he called the "unity of electric force." Suppose, then, that a system of electric currents 1 exists in otherwise empty apace. According to the older theory, these currents give rise to a vector-potential a, , equal to Pot 1 ;* and the magnetic force HL is the curl of ai : while the electric force E, at any point in the field, produced by the variation of the currents, is - a\. It is now assumed that the electric force so produced is indistinguishable from the electric force which would be set up by electrostatic charges, and therefore that the Bystem of varying currents exerts ponderomotive forces on electrostatic charges; the principle of action and reaction then requires that electrostatic charges should exert ponderomotive forces on a system of varying currents, and consequently (again appealing to the principle of the unity of electric force) that two systems of varying currents should exert on each other ponderomotive forces due to the variations. But just as Helmholtz,f by aid of the principle of conser- vation of energy, deduced the existence of an electromotive force of induction from the existence of the ponderomotive forces between electric currents (i.e. variable electric systems), so from the existence of ponderomotive forces between variable systems of currents (i.e. variable magnetic systems) we may infer that variations in the rate of change of a variable magnetic system give rise to induced magnetic forces in the surrounding space. The analytical formulae wliich determine these forces * a = Pot B >B ; and the electric force B, will then be a,. This system is not, however, final ; for we must now perform the process again with these improved values of the electric and magnetic forces and the vector-potential ; and so we obtain for the magnetic force the value curl a,, and for the electric force the value - a,, where ■»-*i- 7—3 w Pot »> = *' " T~* Si Pot •• + 77^ Si Pot Pot a' ■ 4nr& dt* (4JTC1)' dt* This process must again be repeated indefinitely ; so finally we obtain for the magnetic force H the value curl a, and for the electric force E the value - a, where -(5sy.|Potp°'Pot'' + - •■ 3,Bl,ZEdhyG00gle 356 The Followers of Maxwell. It is evident that the quantity a thus defined satisfies the equation V,-l|.--4«. This equation may be written ouriH = (l/^)E + 4«i, while the equations H - curl a, E - - a give curl E - - H. These are, however, the fundamental equations of Maxwell's theory in the form given in his memoir of 1868* That Hertz's deduction is ingenious and interesting will readily be admitted. That it is conclusive may scarcely be claimed : for the argument of Helmholtz regarding the induc- tion of currents is not altogether satisfactory ; and Hertz, in following his master, is on no surer ground. In the course of a discussionf on the validity of Hertz's assumptions, which followed the publication of his paper, E. Aulinger{ brought to light a contradiction between the principles of the unity of electric and of magnetic force and the electrodynamics of Weber. Consider an electrostatically charged hollow sphere, in the interior of which is a wire carrying a variable current According to Weber's theory, the sphere would exert a turning couple on the wire; but according to Hertz's principles, no action would be exerted, since charging the sphere makes no difference to either the electric or the magnetic force in its interior. The experiment thus suggested would be a crucial test of the correctness of Weber's theory ; it has the advantage of requiring nothing but closed currents and electrostatic charges at rest ; but the quantities to be observed would be on the limits of observational accuracy. • Cf. p. 287. t Lorberg, An . d. Phy«. XXVii (1886), p. 666 ; xui (1887), p. 1*1 tiniunn, ibid. XI n (1886), p. 598. % Ann. d. Phj.. uvii (1S86), p. 119 3,Bl,ZEdhyG00gle The Followers of Maxwell. 357 After hia attempt to justify the Maxwellian equations on theoretical grounds, Hertz turned his attention to the possibility of verifying them by direct experiment. His interest in the matter had first been aroused some years previously, when the Berlin Academy proposed as a prize subject " To establish experimentally a relation between electromagnetic actions and the polarization of dielectrics." Helmholtz suggested to Hertz that he should attempt the solution ; but at the time he saw no way of bringing phenomena of this kind within the limits of observation. From this time forward,however, the idea of electric oscillations was continually present to his mind ; and in the spring of 1886 he noticed an effect* which formed the starting- point of his later researches. When an open circuit was formed of a piece of copper wire, bent into the form of a rectangle, bo that the ends of the wire were separated only by a short air- gap, and when this open circuit was connected by a wire with any point of a circuit through which the spark- discharge of an induction-coil was taking place, it was found that a spark passed in the air-gap of the open circuit. This was explained by supposing that the change of potential, which is propagated along the connecting wire from the induction-coil, reaches one end of the open circuit before it reaches the other, so that a spark passes between them; and the phenomenon therefore was regarded as indicating a finite velocity of propagation of electric potential along wires.t "Ann. d. Phya. xxxi (1887), p. 431. Hertz's Electric Want, translated by D. E. Jonee, p. 28. t Unknown to Haiti, the transmission of electrio waves along wire* had been observed in 1ST*) by Wilholm Ton Beiald, Munchen Sitzungsberichie, i (1870), p. 113; Phil. Mag. xl (1870), p. 42. "If," ha wrote at the conclusion of a aerie* of experiment*, " electrical wave* be sent into a wire insulated at the end, the; ■ill be reflected at that end. The phenomena which aocompsny this process in alternating discharge* appear to owe their origin to the interference of the advancing and reflected waves,'* and, " an electric discharge travel* with the ■aire rapidity in wires of equal length, without reference to the material* of winch tlieee wires are made." The subject was investigated by 0. 1. Ledge and A. P. Chattock at almost the same time aa Hertz's experiments were being curried out : mention was made of their researches at the meeting of the British Association in 1888. 3,Bl,ZEdhyG00gle 358 The Followers o/ Maxwell. Continuing his experiments, Hertz* found that a epark could be induced in the open or secondary circuit even when it was not in metallic connexion with the primary circuit in which the electric oscillations were generated; and he rightly inter- preted the phenomenon by showing that the secondary circnit was of such dimensions as to make the free period of electric oscillations in it nearly equal to the period of the oscillations in the primary circuit ; the disturbance which passed from one circuit to the other by induction would consequently be greatly intensified in the secondary circuit by resonance. The discovery that sparks may be produced in the air-gap of a secondary circuit, provided it has the dimensions proper for resonance, was of great importance: for it supplied a method of detecting electrical effects in air at a distance from the primary disturbance; a suitable detector was in fact all that was needed in order to observe the propagation of electric waves in free space, and thereby decisively test the Maxwellian theory. To this work Hertz now addressed himself.t The radiator or primary source of the disturbances studied by Hertz may be constructed of two sheets of metal in the same plane, each sheet carrying a stiff wire which projects towards the other sheet and terminates in a knob ; the sheets are to be excited by connecting them to the terminals of an induction coil. The sheets may be regarded as the two coatings of a modified Leyden jar, with air as the dielectric between them ; the electric field is extended throughout the air, instead of being confined to the narrow space between the coatings, as in the ordinary Leyden jar. Such a disposition ensures that the system shall lose a large part of its energy by radiation at each oscillation. t Sir Oliver Lodge was about this time independently studying electric oacilla- tiocs in air in connexion with the theory of lightning -conductor" ! of. Lodge, Phil. Mag. xxvi (1888), p. 217. So long before a* 1812, Joseph Hsnry, of Washington, bad noticed that the inductive effect* of the Leyden jar diaebarje could be observed at considerable distances, and had even euggeeted a comparison with " a epark from flint and steel in the case of light." D,Bl,ZEdhyG00gle The Followers of Maxwell. 859 Aa in the jar discharge," the electricity surges from one sheet to the other, with a period proportional to (CX)*, where G denotes the electrostatic capacity of the system formed by the two sheets, and L denotes the self-induction of the connexion. The capacity and induction should be made as small as possible in order to make the period small. The detector used by Hertz was that already described, namely, a wire bent into an incompletely closed curve, and of such dimensions that its free period of oscillation was the same as that of the primary oscillation, so that resonance might take place. Towards the end of the year 1887, when studying the sparks induced in the resonating circuit by the primary disturbance, Hertz noticedf that the phenomena were distinctly modified when a large mass of an insulating substance was brought into the neighbourhood of the apparatus; thus confirming the principle that the changing electric polarization which is pro- duced when an alternating electric force acts on a dielectric is capable of displaying electromagnetic effects. Early in the following year (1888) Hertz determined to verify Maxwell's theory directly by showing that electro- magnetic actions are propagated in air with a finite velocity.} For this purpose he transmitted the disturbance from the primary oscillator by two different paths, viz., through the air and along a wire ; and having exposed the detector to the joint influence of the two partial disturbances, he observed inter- ference between them. In this way he found the ratio of the velocity of electric waves in air to their velocity when conducted by wires ; and the latter velocity he determined by observing the distance between the nodes of stationary waves in the wire, and calculating the period of the primary oscillation. The velocity of propagation of electric disturbances in air was in • Cf. p. MS. t Ann. A. Phjt. mil, p. 373. Sltttric Waft (Euglieh edition), p. 96. * Ana. d. Phyt mix (1888), p. 661 . Bltetrio Watu (English edition) p. 107. dhyGoogle 360 The Followers of Maxwell. this way shown to be finite and of the same order as the velocity of light" Later in 1888 Hertzf showed that electric waves in air are reflected at the surface of a wall ; stationary waves may thus be produced, and interference may be obtained between direct and reflected beams travelling in the same direction. The theoretical analysis of the disturbance emitted by a Hertzian radiator according to Maxwell's theory was given by Hertz in the following year.* The effects of the radiator are chiefly determined by the free electric charges which, alternately appearing at the two sides, generate an electric field by their presence and a magnetic field by their motion. In each oscillation, as the charges on the poles of the radiator increase from zero, lines of electric force, having their ends on these poles, move outwards into the surrounding Bpace. When the charges on the poles attain their greatest values, the lines cease to issue outwards, and the existing lines begin to retreat inwards towards the poles ; but the outer lines of force contract in such a way that their upper and lower parts touch each other at some distance from the radiator, and the remoter portion of each of these lines thus takes the form of a loop ; and when the rest of the line of force retreats inwards towards the radiator, this loop becomes detaohed and is propagated outwards as radiation. In this way the radiator emits a series of whirl-rings, which as they move grow thinner and wider; at a distance, the disturbance * Hertz 'e experiments gave the Value 46/28 for the ratio of the velocity of electric wuvos in air to the velocity of electric wave* conducted fay the wine, and 2 x 10'° cms. per see. far the latter velocity. These numbers were afterwards found to be open to objection : Poincarfi (Comptes Rendus, cxi (18S0), p. 323) showed that the period calculated by Hertz, waa V2 x the true period, which would malte the velocity of propagation in air equal to that of light x V2. Ernst Lecher (Wiener Berichte, Hay 8, 18BD ; Phil. Mag. in (1890), p. 138), experimenting on the velocity of propagation of eleotrio vibrations in wires, found instead of Hertz's 2 x 10'° emu. per sec, a value within two per cent, of the velocity of light. E. S»ra«mandL. De I* Rive at Geneva (Archives dee 8c. Phys.xxix{lB»3)) finally proved that the velocities of propagation in air and along wires are equal. tAnn. d. Ptays.xxxiv(I888),p.610. £/«i™ (Paw (Engliah edition), p. 1S4. I Ibid., xxxvi (1889), p. 1. SUetrU Want (English edition), p. 137. 3,Bl,ZEdhyG00gle The Followers of Maxwell. 361 is approximately a plane wave, the opposite sides of the ring representing the two phases of the wave. When one of theBe rings has become detached from the radiator, the energy con- tained may subsequently be regarded as travelling outwards with it. To discuss the problem analytically* we take the axis of the radiator as axis of z, and the centre of the spark-gap as origin. The field may be regarded as due to an electric doublet formed of a positive and an equal negative charge, displaced from each other along the axis of the vibrator, and of moment Ae-'t* Bin (2-*ctj\), the factor e~*"' being inserted to represent the damping. The simplest method of proceeding, which was suggested by Fitz Gerald.f is to form the retarded potentials $ and a of L. Lorenz.J These are determined in terms of the charges and their velocities by the equations +., at* „.,«£* ■r r ~* r whence it is readily shown that in the present case - - dFfdz, a = (0, 0, dF/dt), where F - sin — {ci - r). The electric and magnetic forces are then determined by the equations E = c* grad 0 - a, H = curl a. It is found that the electric force may be regarded as com- pounded of a force i^i, parallel to the axis of the vibrator and depending at any instant only on the distance from the vibrator, together with a force $>, Bin H acting in the meridian plane • Cf. Earl Pennon and A. Lee, Phil. Tram, oxoiii (1899), p. 165. + Brit. Auoo. Bep., Lerd* (1800), p. 785. J Cf. p. 298. The rue of retarded potential! was also recommended in the following year by Poincari, Comptes Rendu s, oiiit (1861), p. 616. 3,Bl,ZEdhyG00gle 362 The Followers of Maxwell. perpendicular to the radius from the centre, where $, depends at any instant only on the distance from the vibrator, and 0 denotes the angle which the radius makes with the axis of the oscillator. At points on the axis, and in the equatorial plane, the electric force is parallel to the axis. At a great distance from the oscillator, ^, is small compared with £,, so the wave is purely transverse. The magnetic force is directed along circles whose centres are on the axis of the radiator; and its magnitude may be represented in the form , depends only on r and t ; at great distances from the radiator, cf, is approximately equal to f„ If the activity of the oscillator he supposed to be continually maintained, bo that there is no damping, we may replace p, by zero, and may proceed as in the case of the magnetic oscillator* to determine the amount of energy radiated. The mean out- ward flow of energy per unit time is found to be ^A' {2*f\)*; from which it is seen that the rate of loss of energy by radiation increases greatly as the wave-length decreases. The action of an electrical vibrator may be studied by the aid of mechanical models. In one of these, devised by Larmor.f the aether is represented by an incompressible elastic solid, in which are two cavities, corresponding to the conductors of the vibrator, filled with incompressible fluid of negligible inertia. The electric force is represented by the displacement of the solid. For such rapid alternations as are here considered, the metallic poles behave as perfect conductors; and the tangential components of electric force at their surfaces' are zero. This condition may be satisfied in the model by suppos- ing the lining of each cavity to be of flexible sheet-metal, so aa to be incapable of tangential displacement ; the normal displace- ment of the lining then corresponds to the surface-density of electric charge on the conductor. In order to obtain oscillations in the solid resembling those of an electric vibrator, we may suppose that the two cavities " Cf . p. 8*8. t Proc. Ciimb. Phil. Soc. fii (1891), p. 166. 3,Bl,ZEdhyG00gle The Followers of Maxwell. 36.1 have the form of semicircular tubes forming the two halves of a complete circle. Each tube is enlarged at each of its ends, so as to present a front of considerable area to the corresponding front at the end of the other tube. Thus at each end of one diameter of the circle there is a pair of opposing fronts, which are separated from each other by a thin sheet of the elastic solid. The disturbance may be originated by forcing an excess of liquid into one of the enlarged ends of one of the cavities. This involves displacing the thin sheet of elastic solid, which separates it from the opposing front of the other cavity, and thus causing a corresponding deficiency of liquid in the enlarged end behind this front. The liquid will then surge backwards and forwards in each cavity between its enlarged ends ; and, the motion being communicated to the elastic solid, vibrations will be generated resembling those which are produced in the aether by a Hertzian oscillator. 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 ; of hard pitah.f •Ann. d. Phva. xxxti (1889), p. 769; EUttrie Want (English td.), p. 172. | 0. J. Lodge and J. L. Howard in the nng year (towed that elactrio radiation might bo refracted and coiicsu tinted by means of large lenaea. Cf. Phil. Mag. xxrii (1889), p. 48. 3,Bl,ZEdhyG00gle 364 The Followers of Maxwell. The old question aa 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,t 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. WienerJ 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 — La, 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 Btjq. t Nature, rail (1889), p. 391. J Ann. d. PhjB. zl (1890), p. 203. Cf. ■ controversy regarding the results: Comptes Sendus, cxii{1891), pp.188, 325, 329, 36G, 383,466; and Awi.d.Pbrt. xli(189u), p. 16* ; xliii(1891), p. 177; ilviii (1893), p. 119. $ Gnu. Nach. 1890, p. 106; Ann. d. Phya. xl (1890), p. S77; BUttrU Warn (English ed.), p. 196. In this memoir Hertx advocated the form of the equation which Maxwell had used in his paper of 1888 (cf. mprm, p. 287) in p the earlier form, which involved the scalar and vector potentials. 3,Bl,ZEdhyG00gle 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 systems! 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 B 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. Phya. ill (1890), p. 369 ; Shetric Warn (English ad.), p. 341. The propagation of light through a moving dielectric bad been diacuued prericmily, on the basis of Maxwell's equations for moving bodies, by J. J. Thoinsuu, Phil. Hag. ix (1880), p. 28* ; Proc. Camb. Phil. Hoc. v (1886), p. 250. tCI. p. 288. 3,Bl,ZEdhyG00gle 366 The Followers of Maxwell. Maxwell made no change in the other electromagnetic equations, which therefore retained the customary forms D = tE'/W, div D = 0, 4w(i 4 i>) = 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 4w [D . w] is generated in a dielectric which moves with velocity win 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," sucli 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 he 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 ivt- * Ht-nvimde'a general theory wan published in s series of papers in Ihe El e< triii an, from 1886 onwards. His earlier work was republished in hit EL-ctricol PapeTi (2 vols., 1883), and his Electromagnetic Tktary (2 Tols., 1891). Mention may be specially made of a memoir in Phil. Trans, clxxziii (1893), p. 423. 3,Bl,ZEdhyG00gle 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 8 = D + i + pw + curl [D . w] , where pw denotes the conduction-current ; and the equation connecting current with magnetic force is curl (H'-h„)- -In*, where h„ 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 X by adding to the kinetic energy of the aether, which is represented by Jpe\ a term £\ We may therefore write B - H -i- a curl [K. D], where a denotes a constant. But if this be combined with the customary electromagnetic equations curl H - 4*D^ curl E = - B, D - eB/4nc', and all the vectors except B be eliminated (X being treated as a constant), we obtain the equation B - («r» V'B + 0/4*) curl (PBfffiff), where 3/90 stands for (K£jfc + K$dy + K£fat) ; 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 Rowland, 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. D,Bl,ZEdhyG00gle 370 The Followers of Maxwell. metals, were extensively studied* in the years following the publication of Rowland's memoir: but it was not until the modem 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, clinii (1891}, p. 371 ) was, lika 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 forte was not continuous acroM the interface between a magnetised and an unmagnetiied medium ; but Basaot subsequently overcame tlii* difficulty (Nature, lii (1896}, p. SIS; liii (1866), p. 130; Amei. Jour. Hath, xix (1897), p. 60)— the effect analogous to Hall's being introduced into the equation connecting electric din placement with electric force, so that the equation took the form Basset, in 1893 (Froc. Camb. Phil. Sou. viii, p. 68), derived analytical expressions which represent Kerr's magneto-optic phenomenon by substituting • 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 B. T. Gluabrook, Phil. Hag. xi (1881), p. 397 ; by J. 1. Thomson, Recent Rmarchts, p. 182 : by D. A. Goldhammer, Ann. d. Phys. xlvi (1892}, p. 71; xlvii (1892), p. 346; xlviii (1893), p. 710; 1 (1863), p. 772; by P. Drnde, Ann. d. Phys. xlvi (1892), p. 363; ilviii (1893), p. 122; ilii (1893), p. 690 ; lii (1894)) p. 466: by C. E. Wind, VeraUtgen Eon. Akad. Amsterdam. 29th Sept., 1864: by Heiff, Ann. d. Phys. Ivii (1896), p. 281: by J. O. Leathern, Phil. Trans, cxe (1897), p. 89; Trans. Camb. Phil. Soc. xvii (1898), p. 16: and by W. Voigt in many memoirs, and in his treatise, Magneto- und Eltitn-tptii. 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 equation* curl H ■•-- 4*8, - curl S = H ; the total current 8 being assumed to consist of a part (the displacement -current) proportional to E, a part (the conduction -current) proportional to Z, and a put 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. Loud. Hath. See. vi (1875)) investigated the propagation of waves of displacement along a stretched chain whose links contain rotating fly-wheels : cf. also Laraior, Proc. Lond. Hath. Soc. xxi. (1880), p. 423; niii (1891), p. 127; F. Basenohrl, Wien Siuungsberkhto orii, 2a (1868), p. 1016 ; W. Thomson (Kelvin), Phil. Hag. ilviii (1899), p. 236, and BaUinwn Ledum; and Fit* Gerald, Electrician, Aug. 4, 1899, Fit/Geralds Stitnlijlc Writing; p. 481. t Journal de Physique (3) i, pp. 20a, 345. 3,Bl,ZEdhyG00gle 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 - (47re70 D + k curl D, id place of the customary equation E-(4«V<)1>: the constant le being a measure of the natural rotatory power of the substance concerned. The remaining equations are as usual, curl H = 4wD, - curl E - H, Eliminating H and E, we have B - (e7«) VT> + tA/4») V* curl D. For a plane wave which is propagated parallel to the axis of x, this equation reduces to Ip' " t cte' ~ 4w fttP 3* J. e' 31 J, k_&P*. df " i cO" + 4ir ete" ' and, as MacCullagh had shown in lS36,t 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 iu 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. tW. p. 175. 2B2 3,Bl,ZEdhyG00gk ( 372 ) CHAPTER 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 ti rst indicated, iii 1850, by Alex. Williamson,t 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 I* regarded as entirely free. In 1857 this idea was made by * Cf. p. 78. t I'liil. Mag. xzivii (I860), p. 360; Liebig'i Annul™ d. Chan. u. Phuiu. liirii (18fil) p. S7. dhyGoOgk Conduction in Solutions and Gases, etc. 373 R. Clauaiufl * 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 "WiUiftmaon-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 Faradayt in 1835 ; and in 1844 it was further investigated by Frederic Daniell and W. A. Miller,} 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. • Ana. i. Phy«. ci (1887), p. 338 ; Phil. Mag. »i (1858), p. 94. t Ezpcr. Bn. \\ 625-53C. % Phil. Trans., 18(4, p. 1. Cf. alio Pouillet, Comptw Rendna xi (1646), p. 1644. 3,Bl,ZEdhyG00gle 374 Conduction in Solutions and Gases, This idea was adopted by W. Hittorf, of Mitnster, 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 : w. 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 vju in a large number of cases ot eleetrolysis.t The theory of ionic movements was advanced a further stage by F. W. KohlrauschJ (J. 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 kfm, 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 two> ions move independently under the influence of the electric force. Let « 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 + r„ where E denotes the electric charge carried by one gramme- * Aon. a. Phye. lxxxiz (1863), p. 177; xcviii (1856), p. 1 ; oiii (1858), p. 1 : CTi(185B), pp. 337, 613. t The ratio e/(u 4 i>) was termed by Hittorf tlie trmtpert nttnhtr of the anion. I Amid. Phye. vi (1878), pp. 1, 1+5. The chief resulta had been comm unrated to the Academy of Giittingen in 1876 and 1877. i A gramme-equivalent means a man of the salt whose weigh! in gramme* U the molecular weight divided by tbe valency of the ione. 3,Bl,ZEdhyG00gle from Faraday to % % Thomson. 375 equivalent of ion* Thus mE (v, + v) - total current - k • m\, or X ■ JS(n + v). The determination of vju by the method of Hittorf, and of {v, + 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 currents as it is called— is observed ; but after a short time it ceases; and after its cessation the state of the Bystem 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 muBt 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 96680 coulombs. t The phenomenon of Toltaic polarization was discovered by Bitter in 1S03. Bitter explained it by comparing the action of the polnriting current to that of a current which is used to ohnrge a condenser. Volte in 1806 put forward the alternative explanation, that the products of decomposition set up a reverse electromotiTe force. 3,Bl,ZEdhyG00gle 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 JIQ has a large surface, whilejthe cathode S has a variable surface S small in comparison. When the external electromotive force is applied, it is easily seen thst 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 H* to be zero, so that the potential of H is - V. The state of the system may be varied by altering either for S; we assume that these • Phil. Tnm». olii (1871), p. 129. t Complei Rendui Ixxri (1873), p. 1407. Phil. Mig. alvii (1871), p. 281. Ann. de Chim. el de Pbj«. v (1S76), p. «*, wi (1877), p. 268. 3,Bl,ZEdhyG00gle from Faraday to J. J. Thomson. 377 alterations may be performed independently, reversibly, and isothermally, and that the state of the large electrode Sn is not altered thereby. Let de denote the quantity of electricity which passes through the cell from ft,, 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 de-d [Sf(V)\; BO 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 3,Bl,ZEdhyG00gle 378 Conduction in So/utions 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 V was sensibly independent of V, bo that the curve which represent* the relation between 7 and V 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 dyjdV 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 * LippmaD, Compte* Bendui, xcv j 1382), p. 688. t Train. Conn. Acad, iii (1876-18T8), pp. 108, 343; Gibb»" Scientific &p*n, i, p. 66. X This ii umbodied in equation (690] of Oibba' memoir. $ Berlin Mon&taber., 1881, p. 946 ; Wit. Abh. i, p. 926 ; Ann. d. Phj*. xn. (1882), p. 31. Cf. alia Planck, Ann. d. Phvs. sift (1891), p. 386. 3,Bl,ZEdhyG00gle 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 =- dy/dV. This equation shows that when drytdV 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 H„ ; 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 ; bo 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. A further consequence deduced by Helmholtz from this theory leads to a second method of determining the difference of potential between mercury and an electrolyte. If a mercury surface is rapidly extending, and electricity is not rapidly transferred through the electrolyte, the electric surface-density in the double layer must rapidly decrease, since the same quantity of electricity is being distributed over an increasing area. Thais it may be inferred that a rapidly extending mercury-surface in an electrolyte is at the same potential as the electrolyte. This conception is realized in the droppiiig-electrodc, in * The conception of double layers of electricity at the surface of separation of two bodie* had been already applied by Helmhotti to explain various other phenomena — e.g., theVolta cont act- difference of potential of two metals, fractional electricity, and "electric endoamose," or the transport of fluid which occur* when an electric current is paued through two conducting liquids separated by a porous Wrier. Cf. Helmhulti, Berlin Monataherichte, February 37, 1879 ; Ann. d. Phy*. Tii {1879}, p. 337 ; Helmholtx, Win. AM. i, p. 855. DindzBdbyGoOgk 380 Conduction in Solutions and Gases, which a jet of mercury, falling from a reservoir into an electro- lytic solution, is bo adjusted that it breaks into drops when the jet touches the solution. According to Helmholtz'B conclusion there is no difference of potential between the drops and the electrolyte ; and therefore the difference of potential between the electrolyte and a layer of mercury underlying it in the same vessel is equal to the difference of potential between this layer of mercury and the mercury in the upper reservoir, which difference is a measurable quantity. It will be seen that according to the theories both of Gibbs and of Helmholtz, and indeed according to all other theories on the subject," dy/dV is zero for an electrode whose surface is • E.g., thntof Warburg, Ann. d. Phys. xli (1890), p. 1. In this it is assumed that the electrolytic solution near the electrodes originally contains a salt of mercury in solution. When the external electromotive force ii applied, a conduc- tion-current panei through the electrolyte, which in the body of the electrolyte in carried by the acid and hydrogen ioun. Warburg supposed that at the cathode the hydrogen ions react with the salt of mercury, reducing it to metallic mercury, which is deposited on the electrode. Thus a considerable change in concentration of the salt of mercury is caused at the cathode. At the anode, the acid iona carrying the current attack the mercury of the electrode, and thus increase the local concentration of (he mercuric salt ; but On account of the sine of the anode this increase is trivial and may be neglected. Warburg thus supposed that the electromotive force of the polarised cell is really that of a concentre ti< in cell, depending on the different concentration! of mercuric salt at the electrodes. He found dyfdV to he equal to the amount of mercuric salt at the cathode per unit area of cathode, divided by the electro- chemical equivalent of mercury. The equation previously obtained is thus presented in a new physical interpretation. Warburg connected the increase of the surface-tension with the fact that the surface-tension between mercury and a solution always increases when the con- centration of the solution is diminished. Hie theory, of course, leads to no conclusion regarding the absolute potential difference between the mercury and (he solution, as Helmholtz' does. At an electrode whose surface is rapidly increasing — e.g., a dropping electrode — Warburg supposed that the surface-density of mercuric salt tends to zero, so •T/sTiisMv, The explanation of dropping electrodes favoured by Nemst, Btiift « elm Ann. d. Pkg: Iviii (1808), is that the difference of potential corresponding to the equilibrium between the mercury and the electrolyte is instantaneously established ; but that ions are withdrawn from the solution in order to form the double layer necessary for this, and that these ions are carried down with the drops 3,Bl,ZEdhyG00gle from Faraday to J. J. Thomson. 381 rapidly increasing— e.g., a dropping electrode; that is to say, the difference of potential between an ordinary mercury electrode and the electrolyte, when the surface-tension haa ite maximum value, is equal to the difference of potential between a dropping-electrode and the same electrolyte. This result has been experimentally verified by various investigators, who have shown that the applied electromotive force when the surface- tension has ite maximum value in the capillary electrometer, is equal to the electromotive force of a oell having as electrodes a large mercury electrode and a dropping electrode. Another memoir which belongs to the same period of Helmholtz' career, and which has led to important develop- ments, was concerned with a special class of voltaic cells. The most usual type of cell is that in which the positive electrode is composed of a different metal from the negative electrode, and the evolution of energy depends on the difference in the chemical affinities of these metals for the liquids in the cell. But in the class of cells now considered* by Helmholtz, the two electrodes are composed of the same metal (say, copper) ; and the liquid (say, solution of copper sulphate) is more con- centrated in the neighbourhood of one electrode than in the neighbourhood of the other. When the cell is in operation, the salt passes from the places of high concentration to the places of low concentration, bo as to equalize its distribution ; and this process is accompanied by the flow of a current in the outer circuit between the electrodes. Such cells had been studied experimentally by James Moser a short time previouslyt to Helmholtz' investigation. The activity of the cell is due to the fact that the available energy of a solution depends on its concentration ; the molecules of mercury, until the upper layer of the aolution U so much impoverished that the double layer can no longer be formed. The impoverishment of the upper layer of the solution haa actually been observed by Palmier, Zeitach. Phya. Chem. XXT (1898). p- S66 ; xiTiii (1899), p. M7 ; inn (1901), p. 664. * Berlin Honataber. , 1877, p. 713 ; Phil. Hag. (o) v (1878), p. 348 ; reprinted kith addition* in Ann. d. Phys. iii (1878), p. 301. t Ann. d. Phya. iii (1878), p. 216. 3,Bl,ZEdhyG00gle 382 Conduction in Solutions and Gases, of salt, in passing from a high to a low concentration, are therefore capable of supplying energy, just as a compressed gas is capable of supplying energy when its degree of compression is reduced. To examine the matter quantitatively, let nf(njV) denote the term in the available energy of a solution, which is due to the diseolntion of n gramme-molecules of salt in a volume V of pure solvent ; the function / will of course depend also on the temperature. Then when dn gramme-molecules of solvent are evaporated from the solution, the decrease in the available energy of the system is evidently equal to the available energy of dn gramme-molecules of liquid solvent, less the available energy of dn gramme-molecules of the vapour of the solvent, together with nf(n/V) less nf\n/(V-vdn)], where v denotes the volume of one gramme-molecule of the liquid. But this decrease in available energy must be equal to the mechanical work supplied to the external world, which is dn .p, (v' - v), if p, denote the vapour -pressure of the solution at the temperature in question, and v' denote the volume of one gramme-molecule of vapour. We have therefore dn . p, (v' -v) = - available energy of dn gramme-molecules of solvent vapour + available energy of dn gramme-molecules of liquid solvent + n/(«/F) - nf{n/(V- vdn)\. Subtracting from this the equation obtained by making n zero, we have dn . (j>, - p.) (v' - o) = nf(n/V) - n/|n/( V - vdn)\, where p0 denotes the vapour-pressure of the pure solvent at the temperature in question ; so that (P, - P.) (•-•)-- (»V V')f («/ V)v. Now, it is known that when a salt is dissolved in water, the vapour-pressure is lowered in proportion to the concentration of the salt — at any rate when the concentration is small : in 3,Bl,ZEdhyG00gle front Faraday to y. f. Thomson. 3t)3 fact, by the law of Raoult, (p0-pi)/p-> is approximately equal to mf V ; so that the previous equation becomes P*V{if-v)-nf{nlV). Neglecting v in comparison with v, and making use of the equation of state of perfect gases (namely, p^ = BT. where T denotes the absolute temperature, and B denotes the constant of the equation of state), we have f(n/V) - RTVjn, and therefore f(n/V)-RTlog(n/V). Thus in the available energy of one gramme-molecule of a dissolved salt, the term which depends on the concentration is proportional to the logarithm of the concentration ; and hence, if in a concentration-cell one gramme-molecule of the salt passes from a high concentration c, at one electrode to a low concentration c, at the other electrode, its available energy is thereby diminished by an amount proportional to log (ca/cj. The energy which thus disappears is given up by the system in the form of electrical work; and therefore the electromotive force of the concentration-cell must be proportional to log (ci/c,). The theory of solutions and their vapour-pressure was not at the time sufficiently developed to enable Helmholtz to determine precisely the coefficient of log (c,/c,) in the expression.* An important advance in the theory of solutions was effected in 1887, by a young Swedish physicist, Svante Arrheniua.t * The foimulii given by HelmholtE wu that the electromotive force of the celt U equal to i(l -»)olog(«/ei), where rj mid ci denote the concentrations of the solu- tion at the electrode!, e denote* the volume of one gramme of vapour in equilibrium with the water at the temperature in question, n denotes the transport number for the cation (Hittorf's 1/n), and & denotes 9 x the lowering of vaponr-pressure when □ne gramme- equivalent of salt in dissolved in q grammes of water, where q denotes a large number. t Zeitsohrift fur phys. Chem. i (1887), p. 631. Previous investigations, in which the theory was to some extent foreshadowed, were published in Bibang till Sreuska Vet. Ak. Forh. viii (1884), Not. 13 and 14. 3,Bl,ZEdhyG00gle 384 Conduction in Solutions and Gases, Interpreting the properties discovered by Kohl^a^8ch• in the light of the ideas of Williamson and Clausius regarding the spontaneous dissociation of electrolytes, Arrhenius inferred that in very dilute solutions the electrolyte is completely dissociated into ions, but that in more concentrated solutions the salt is less completely dissociated ; and that as in all solutions the transport of electricity in the solution is effected solely by the movement of ions, the equivalent conductivity! must be pro- portional to the fraction which expresses the degree of ionization. By aid of these conceptions it became possible to estimate the dissociation quantitatively, and to construct a general theory of electrolytes. Contemporary physicists and chemists found it difficult at first to believe that a Bait exists in dilute solution only in the form of ions, ag. that the sodium and chlorine exist separately and independently in a solution of common Bait. But there is a certain amount of chemical evidence in favour of Arrhenius' conception. For instance, the testa in chemical analysis are really tests for the ions ; iron in the form of a fer- rocyanide, and chlorine in the form of a chlorate, do not respond to the characteristic tests for iron and chlorine respectively, which are really the tests for the iron and chlorine ions. The general acceptance of Arrhenius' views was hastened by the advocacy of Ostwald, who brought to light further evidence in their favour. For instance, all permanganates in dilute solution show the same purple colour ; and Ostwald considered their absorption-spectra to be identical £ tbiB identity is easily accounted for on Arrhenius' theory, by supposing that the spectrum in question is that of the union which corresponds to the acid radicle. The blue colour which is observed in dilute solutions of copper salts, even when the strong solution is not blue, may in the same way be • Cf. p. 874. t I.e. tht ohmio specific, conductivity or the solution divided by the number of gramme-equivalent* of rait per unit volume. ; Examination of the spectra with higher dispersion doe* not altogether confirm this conclusion. 3,Bl,ZEdhyG00gle from Faraday to J .J. Thomson. 385 ascribed to a blue copper cation, A striking instance of the aame kind is afforded by ferric sulphocyanide ; here the strong solution shows a deep red colour, due to the salt itself ; but on dilution the colour disappears, the ions being colourless. If it be granted that ions can have any kind of permanent existence in a salt solution, it may be shown from tbenno- djnamical considerations that the degree of dissociation must increase as the dilution increases, and that at infinite dilution there must be complete dissociation. For the available energy of a dilute solution of volume V, containing n, gramme-molecules of one substance, w, gramme-molecules of another, and so on, is (as may be shown by an obvious extension of the reasoning already employed in connexion with concentration-cells)* 2»r$r (T) + RfS-rir log (V V) + the available energy possessed by the solvent before the introduction of the solutes, where (T) + RT log (*, r/nrih) - RT, or 7ti Vlfwat - a function of T only. * Cf. pp. 382-383. D,Bl,ZEdhyG00gle 386 Conduction in Solutions and Gases, Since in a neutral solution the number of anions is equal to the number of cations, this equation may be written m* = Vn, x a function of T only ; it shows that when V is very large (so that the solution is very dilute), re, is very large compared with re, ; that ie to say, the salt tends towards a state of complete dissociation. The ideas of Arrhenius contributed to the success of Walther Nernst* in perfecting Helmholtz' theory of concentration cells, and representing their mechanism in a much more definite fashion than had been done heretofore. In an electrolytic solution let the drift-velocity of the cations under unit electric force be u, and that of the anions be v, bo that the fraction «/(* + v) of the current is transported by the cations, and the fraction v/(u + v) by the anions. If the concentration of the solution be c, at one electrode, and c, at the other, it follows from the formula previously found for the available energy that one gramme - ion of cations, in moving from one electrode to the other, is capable of yielding up an amouutf ST log (c,/c,) of energy; while one gramme-ion of anions going in the opposite direction must absorb the same amount of energy. The total quantity of work furnished when one gramme-molecule of salt is transferred from concentration e, to concentration e, is therefore u + v 6 es The quantity of electric charge which passes in the circuit when one gramme-molecule of the salt ia transferred is pro- portional to the valency v of the ions, and the work furnished is proportional to the product of this charge and the electro *Zeitachr. fiir pliys. Cbem. ii (1888), p. 613; iv (1889), p. 129; Berlin Sitiungaberiohte, 1889, p. 83 ; Ann. d. Phy*. ilv (1891), p. 360. Cf. ■!" Max PUnck, Ann. d. Phji. mix (1890), p. 161 ; zl (1890), p. 661. t Tli e correct law of dependence at the available energy on tbe tempentuie *" by ttui time known. 3,Bl,ZEdhyG00gle from Faraday to J. J. Thomson. 387 motive force E of the cell; so that in suitable units we have V U + V c, A typical concentration-cell to which this formula may be applied may be constituted in the following way : — Let a quantity of zinc amalgam, in which the concentration of zinc ia c, be in contact with a dilute solution of zinc sulphate, and let this in turn be in contact with a quantity of zinc amalgam of concentration c,. When the two masses of amalgam are con- nected by a conducting wire outside the cell, an electric current fluws in the wire from the weak to the strong amalgam,* while zinc cations pass through the Bolution from the strong amalgam to the weak. The electromotive force of such a cell, in which the current may be supposed to be carried solely by cations, is Not content with the derivation of the electromotive force from considerations of energy, Nernst proceeded to supply a definite mechanical conception of the procesB of conduction in electrolytes. The ions are impelled by the electric force asso- ciated with the gradient of potential in the electrolyte. But this is not the only force which acts on them ; for, since their available energy decreases as the concentration decreases, there must be a force assisting every process by which the concentra- tion is decreased. The matter may be illustrated by the analogy of a gas compressed in a cylinder fitted with a piston ; the available energy of the gas decreases as its degree of compression decreases; and therefore that movement of the piston which tends to decrease the compression is assisted by a force— the " pressure " of the gas on the pUton. Similarly, if a solution were contained within a cylinder fitted with a piston which is permeable to the pure solvent but not to the solute, and if the whole were immersed in pure solvent, the available energy of * It will hardly ba nuceuary to remark that this mppoaed direction of the current it purely convention*! - D,Bl,ZEdhyG00gle 388 Conduction in Solutions and Gases, the system would be decreased if the piston were to move outwards so as to admit more solvent into the solution; and therefore this movement of the piston would be assisted by a force— the "osmotic pressure of the solution," as it is called.* Consider, then, the case of a single electrolyte supposed to be perfectly dissociated ; its state will he supposed to be the same at all points of any plane at right angles to the axis of x. Let v denote the valency of the ions, and V the electric potential at any point. Sincef the available energy of a given quantity of a substance in very dilute solution depends on the concentration iu exactly the same way as the available energy of a given quantity of a perfect gas depends on its density, it follows that the osmotic pressure p for each ion is determined in terms of the concentration and temperature by the equation of state of perfect gases Up - RTc, where M denotes the molecular weight of the salt, and e the mass of salt per unit volume. Consider the cations contained in a parallelepiped at the place x, whose cross-section is of unit area and whose length is dx. The mechanical force acting on them due to the electric field is - (vcjM) d Vjdx . dx, and the mechanical force on them due to the osmotic pressure iB - dp/dx .dx. If « denote the velocity of drift of the cations in a field of unit electric force, the total amount of charge which would be transferred by cations across unit area in unit time under the influence of the electric forces alone would be - (uvcjM) d Vfdx ; so, under the influence of both forces, it is _ uvc tdV_ £T dc\ M \dx cv dx) Similarly, if v denote the velocity of drift of the anions in a * Cf . Tan't Hnff, Sveaaku Vet.-Ai. Handlingar ui (188G), No. IT; Zmbchrift fur Phye. Chetn. i (1887), p. 4S1. t A» follows from the axprauion obtained, mpra, p. 383. 3,Bl,ZEdhyG00gle from Faraday to f. J, Thomson. 389 unit electric field, the charge transferred across unit area in unit time by the anions is M \ dx cv dx) We have therefore, if the total current be denoted by i, JtTde ' M dx v" J) M dx' or dV, Mdx . u-v ST dc . - -j- dx - — t + — dx. dx {v, + v)vc u +v vc dx The first term on the right evidently represents the product of the current into the ohmic resistance of the parallelepiped dx, while the second term represents the internal electromotive force of the parallelepiped. It follows that if r denote the specific resistance, we must have u + v - M/rvc, in agreement with Kohlrausch s equation ;* while by integrating the expression for the internal electromotive force of the parallelepiped dx, we obtain for the electromotive force of a cell whose activity depends on the transference of electrolyte between the concentrations c, and c,, the value v ST f 1 dc . u-v ST j" U + V V J ■ in agreement with the result already obtained. It may be remarked that although the current arising from a concentration cell which is kept at a constant temperature is capable of performing work, yet this work is provided, not by any diminution in the total internal energy of the cell, but by the abstraction of thermal energy from neighbouring bodies. This indeed (as may be seen by reference to W. Thomson's general • Cf. p. in. 3,Bl,ZEdhyG00gle 390 Conduction in Solutions and Gases, equation of available energy)" roust be the case with any system whose available energy is exactly proportional to the absolute temperature. The advances whieh were effected in the last quarter of the nineteenth century in regard to the conduction of electricity through liquids, considerable though these advances were, may be regarded as the natural development of a theory which had long been before the world. It was otherwise with the kindred problem of the conduction of electricity through gases : for although many generations of philosophers had studied the remarkable effects which are presented by the passage of a current through a rarefied gas, it was not until recent times that a satisfactory theory of the phenomena was discovered. Some of the electricians of the earlier part of the eighteenth century performed experiments in vacuous spaces ; in particular, Hauksbeef in 1705 observed a luminosity when glass is rubbed in rarefied air. But the first investigator of the continuous discharge through a rarefied gas seems to have been Watson^ who, by means of an electrical machine, sent a current through an exhausted glass tube three feet long and three inches in diameter. "It was," he wrote, "a most delightful spectacle, when the room was darkened, to see the electricity in tta passage ; to be able to observe not, as in the open air, its brushes or pencils of rays an inch or two in length, but here the coruscations were of the whole length of the tube between the plates, that is to say, thirty-two inches." Ite appearance he described as being on different occasions " of a bright silver hue," " resembling very much the most lively coruscations of the awrora borealis," and " forming a continued arch of lambent flame." His theoretical explanation was that the electricity "is seen, without any preternatural force, pushing itself on through the vacuum by its own elasticity, in order to maintain the tPhil. Tnn* xiiv (1706), p. 216fi. ' lfc». Haukatwe, liync-JbdmM Eiptrimmit, London, 1709. * Cf. p. 841. tPbil. Tnni oirimmli, Lon t Phil. Tr»m. x\, (1748), p. 93, ilvK (1752), p 3,Bl,ZEdhyG00gle from Faraday to J, J. Thomson. 391 equilibrium in the machine " — a conception which follows naturally from the combination of Watson's one-fluid theory with the prevalent doctrine of electrical atmospheres.* A different explanation was put forward by Noilet, who performed electrical experiments in rarefied air at about the same time as Wataon.f and saw in them a striking confirmation of bis own hypothesis of efflux and afflux of electric matter.} According to Noilet, the particles of the effluent stream collide with those of the affluent stream which is moving in the opposite direction ; and being thus violently shaken, are excited to the point of emitting light. Almost a century elapsed before anything more was dis- covered regarding the discharge in vacuous spaces. But in 1838 Faraday,§ while passing a current from the electrical machine between two brass rods in rarefied air, noticed that the purple haze or stream of light which proceeded from the positive pole stopped short before it arrived at the negative rod. The negative rod, which was itself covered with a con- tinuous glow, was thus separated from the purple column by a narrow dark space: to this, in honour of its discoverer, the name Faraday's dark space has generally been given by subsequent writers. That vitreous and resinous electricity give rise to different types of discharge had long been known ; and indeed, as we have seen,] I it was the study of these differences that led Franklin to identify the electricity of glass with the superfluity of Quid, and the electricity of amber with the deficiency of it. But phenomena of this class are in general much more complex than might be supposed from the appearance which they present at a first examination ; and the value of Faraday's discovery of the negative glow and dark space lay chiefly in the simple and definite character of these features of the discharge, which indicated them as promising subjects for further research. Faraday himself felt the importance of •Cf. oh. ii. t Noilet, lUehtrtlitt nar V Elettrieite, 17*9, troiiieme diaeour*. t Cf. p. 40. } Phil. Trans. , 1838 ; Zxptr. Stt, i, ) 1526. | Cf. p. 44. 3,Bl,ZEdhyG00gle 392 Conduction in Solutions and Gases, investigations in this direction. " The results connected with the different conditions of positive and negative discharge," he wrote,* " will have a far greater influence on' the philosophy of electrical science than we at present imagine." Twenty more years, however, passed before another notable advance was made. That a subject so full of promise should progress so slowly may appear strange ; hut one reason at any ■ rate iB to be found in the incapacity of the air-pumps then in use to rarefy gases to the degree required for effective study of the negative glow. The invention of GeiBsler's mercurial air-pump in 1855 did much to remove this difficulty ; and it was in Oeissler's exhausted tubes that Julius Plucker,t of Bonn, studied the discharge three years later. It had been shown by Sir Humphrey Davy in 1821J that one form of electric discharge — namely, the arc between carbon poles — is deflected when a magnet is brought near to it Plucker now performed a similar experiment with the vacuum discharge, and observed a similar deflexion. But the most interesting of his results were obtained by examining the behaviour of the negative glow in the magnetic field ; when the negative electrode was reduced to a single point, the whole of the negative light became concentrated along the line of magnetic force passing through this point. In other words, the negative glow disposed itself as if it were constituted of flexible chains of iron filings attached at one end to the cathode. Plucker noticed that when the cathode was of platinum, small particles were torn off it and deposited on the walls of the glass bulb. " It iB most natural," he wrote, " to imagine that the magnetic light is formed by the incandescence of these platinum particles as they are torn from the negative electrode." He likewise observed that during the discharge the walls of * Exptr. Ret., $ 1S23. t Ann. d. Phys. ciii (1868), pp. SB, 151; civ (1858), pp. 113, 622 ; er (18581, p. 67; orii (1869), p. 77. Phil. Hag. x. 601. 3,Bl,ZEdhy'G00gle from Faraday to J. f. Thomson. 409 indefinite time after the substance had been removed from the sunlight, and after the luminosity which properly constitutes phosphorescence had died away ; and he was thus led to con- clude that the activity was spontaneous and permanent. It was soon found that those salts of uranium which do not phosphoresce- -e.g., the uranous salts, — and the metal itself, all emit the rays ; and it became evident that what Becquerel had discovered was a radically new physical property, possessed by the element uranium in all its chemical compounds. Attempts were now made to trace this activity in other substances. In 1.898 it was recognized in thorium and its compounds ;* and in the same year P. Curie and Madame Sklodowska Curie announced to the French Academy the separation from the mineral pitchblende of two new highly active elements, to which they gave the names of poloniumf and radium. J A host of workers was soon engaged in studying the properties of the Becquerel rays. The discoverer himself had shown § in 1896 that these rays, like the X- and cathode rays, impart conductivity to gases. It was found in 1899 by Eutherfordll that the rays from uranium are not all of the same kind, but that at least two distinct types are present ; one of these, to which he gave the name o-rays, is readily absorbed ; while another, which he named 0-radiatiou, has a greater penetrating power. It was then shown by Giesel, Becquerel, and others, that part of the radiation is deflected by a magnetic field,1f and part is not.** After this Monsieur and Madame Curieft found that the deviable rays carry negative electric charges, * By Schmidt, Ann. A. Phys., liv (1898), p. Ml ; and by MaJiims Curie, Comptee Rendu), cxxvi (1998), p. 1101. t ComptM Bendue, cxnil (189H), p. 176. J Ibid. , c»*vii (1888), p. 12i5. f Ibid., cixii (189B), p. 6S9. | Phil. Mag. (5). ilvii (1899), p. 109. 1 Giesel, Ann. d. Pby». Ixix (1899), p. S3! (working with polonium); Becquerel, Compter Bendue, cxxii (1899), p. 990 (working wiih radium); Meyer nndv. Schweidler, Phys. Zehsohr. i (1899), p. 113 (working with polonium and radium). •• Becquerel, CompttJ Bendui, cixix (1889), p. 1205) ; mi* (1900), pp. 206, 372. Curie, ibid, c-xxx (1900), p. 73. ft Comptes lteiidua, cm (1900), p. 617. 3,Bl,ZEdhyG00gle 410 Conduction in Solutions and Gases. and Becqnerel* succeeded in deviating them by an electrostatic field. The deviable or /3- rays were thus clearly of the same nature as cathode rays ; and when measurements of the electric and magnetic deviations gave for the ratio m/e a value of the order 10"', the identity of the 0-particlee with the cathode-ray corpuscles was fully established. The subsequent history of the new branch of physica thus created falls outside the limits of the present work. We must now consider the progress which was achieved in the general theory of aether and electricity in the last decade of the nineteenth century. • Comptes Eendus, c»u (1900), p. 809. 3,Bl,ZEdhyG00gle ( +11 ) CHAPTER XII. THE THEORY OP AETHER AND ELECTRONS IN THE CLOSING YEARS OF THE NINETEENTH CENTURY. The attempts of Maxwell* and of Hertzf to extend the theory of the electromagnetic field to the case in which ponderable bodies are in motion had not been altogether successful. Neither writer had taken account of any motion of the material particles relative to the aether entangled with them, so that in both investigations the moving bodies were regarded simply as homogeneous portions of the medium which fills all space, distinguished only by special values of the electric and magnetic constants. Such an assumption is evidently incon- sistent with the admirable theory by which FresnelJ had explained the optical behaviour of moving transparent bodies ; it was therefore not surprising that writers subsequent to Hertz should have proposed to replace his equations by others designed to agree with Freeuel's formulae. Before discussing these, however, it may be well to review briefly tbe evidence for and against the motion of the aether in and adjacent to moving ponderable bodies, as it appeared in the last decade of the nineteenth century. The phenomena of aberration had been explained by Youngg on the assumption that the aether around bodies is unaffected by their motion. But it was shown by Stokes;; in 1H45 that this is not the only possible explanation. For suppose that the motion of the earth communicates motion to the neighbour- ing portions of the aether ; thiB may be regarded as Buperposed on the vibratory motion which the aethereal particles have • C( p. 288. t CC p. 36S. X Cf. p. 118. { Cf. p. 115. || Phil. Mug. uvii (1815), p. 8 : xx*iii (18*6), p. 76 ; xxix (1848), p. 6. 3,Bl,ZEdhyG00gle ■412 The Theory of Aether and Electrons in the when transmitting light : the orientation of the wave-fronts of the light will consequently in general be altered ; and the direc- tion in which a heavenly body ia seen, being normal to the wave- fronts will thereby be affected. But if the aethereal motion is irrotational, so that the elements of the aether do not rotate, it is easily seen that the direction of propagation of the light in space is unaffected ; the luminous disturbance is still propagated in straight lines from the star, while the normal to the wave-front at any point deviates from this line of propagation by the small angle ujc, where u denotes the component of the aethereal velocity at the point, resolved at right angles to the line of propagation, and e denotes the velocity of light. If it be supposed that the aether near the earth is at rest relatively to the earth's surface, the star will appear to be displaced towards the direction in which the earth in moving, through an angle measured by the ratio of the velocity of the earth to the velocity of light, multiplied by the sine of the angle between the direction of the earth's motion and the line joining the earth and star. This is precisely the law of aberration. An objection to Stokes's theory has been pointed out by several writers, amongst others by H. A. Lorentz* This is, that the irrotational motion of an incompressible fluid is completely determinate when the normal component of the velocity at its boundary is given : so that if the aether were supposed to have the same normal component of velocity as the earth, it would not have the same tangential component of velocity. It follows that no motion will in general exist which satisfies Stokes's conditions ; and the difficulty is not solved in any very satisfactory fashion by either of the suggestions whicb. have been proposed to meet it. One of these is to suppose tbat the moving earth does generate a rotational disturbance, which, however, being radiated away with the velocity of Ught, does not affect the steadier irrotational motion ; the other, which was • ArrhiTB. Need, xri (1890), p. 103. 3,Bl,ZEdhyG00gle Closing Years of the Nineteenth Century. 413' advanced by Planck,- is that the two conditions of Stokes's theory — namely, that the motion of the aether is to be irrotational and that at the earth's surface its velocity is to be the same as that of the earth — may both be satisfied if the aether is supposed to be compressible in accordance with Boyle's law, and subject to gravity, so that round the earth it is compressed like the atmosphere ; the velocity of light being supposed independent of the condensation of the aether. Lorentz,f in calling attention to the defects of Stokes's theory, proposed to combine the ideas of Stokes and Fresnel, by assuming that the aether near the earth is moving irrotationally (as in Stokes's theory), but that at the surface of the earth the aethereal velocity is not necessarily the same as that of ponder- able matter, and that (as in Fresnel's theory) a material body imparts the fraction Iji1 - l;/u* of its own motion to the aether within it. Fresnel's theory is a particular case of this new theory, being derived from it by supposing the velocity-potential to be zero. Aberration is by no means the only astronomical phenomenon which depends on the velocity of propagation of light ; we have indeed seenj that this velocity was originally determined by observing the retardation of the eclipses of Jupiter's satellites. It was remarked by Maxwell§ in 1H79 that these eclipses furnish, theoretically at least, a means of determining the velocity of the solar system relative to the aether. For if the distance from the eclipsed satellite to the earth be divided by the observed retardation in time of the eclipse, the quotient represents the velocity of propagation of light in this direction, relative to the solar system; and this will differ from the velocity of propagation of light relative to the aether by the component, in this direction, of the sun's velocity relative to the aether. By taking observations when Jupiter is in different signs of the ■ Cf. Lorentz, Proa. Amsterdam Aoad. (English ed,), i (1699), p. 413. t Archives Neerl. xxi (1886), p. 103 1 of. olio Zittinsgeveisl. Son. Ale. Amder- dam, 18B7-88, p. 266. J Cf. p. 22. i Proa. E. 8. ui (1880), p. 108. 3,Bl,ZEdhyG00gle 414 The Theory of Aether and Electrons in the zodiac, it should therefore be possible to determine the sun's velocity relative to the aether, or at least that component of it which lies in the ecliptic. The same principles may be applied to the discussion of other astronomical phenomena. Thus the minimum of a variable star of the Algol type will be retarded or accelerated by an interval of time which is found by dividing the projection of the radius from the sun to the earth on the direction from the sun to the Algol variable by the velocity, relative to the solar system, of propagation of light from the variable ; and thus the latter quantity may be deduced from observations of the retardation* Another instance in which the time taken by light to cross an orbit influences an observable quantity is afforded by the astronomy of double stars. Savaryt long ago remarked that when the plane of the orbit of a double star is not at right angles to the line of sight, an inequality in the apparent motion must be caused by the circumstance that the light from the remoter star has the longer journey to make. Yvon Villarceau* showed that the effect might be represented by a constant alteration of the elliptic elements of the orbit (which alteration is of course beyond detection), together with a periodic inequality, which may be completely specified by the following statement : the apparent coordinates of one star relative to the other have the values which in the absence of this effect they would have at an earlier or later instant, differing from the actual time by the amount m, - ;«, z 7»] + 771, C where m, and m, denote the masses of the stars, c the velocity of light, and s the actual distance of the two stars from each • The velocity of light waa found from obaerratione of Algol, by C. V. L. Chnrlier, Ofvenigt af K. Vet.-Ak. Forhaniil. XM (1889), p. S23. t Conn, dee Tempt, 1830. X Addition! i In Connausance dec Tempe, 1878 : an improved deduction «u given by H. Seeliger, Sitxungaberichte d. K. Ak. zu Miumhen, xix (1SSBI, p. 19. dhyGoOgk Closing Years of the Nineteenth Century. 415 other at the time when the light was emitted, resolved along the line of sight. In the existing state of double-star astronomy, this effect would be masked by errors of observation- Yillarceau also examined the consequences of supposing that the velocity of light depends on the velocity of the source by which it is emitted. If, for instance, the velocity of light from a star occulted by the moon were less than the velocity of light reflected by the moon, then the apparent position of the lunar disk would be more advanced in it« movement than that of the star, so that at emersion the star would first appear at some distance outside the lunar disk, and at immersion the star would be projected on the interior of the disk at the instant of its disappearance. The amount by which the image of the star could encroach on that of the disk on this account could not be so much as 0"*71 ; encroachment to the extent of more than 1" has been observed, but is evidently to be attributed for the most part to other causes, Among the consequences of the finite velocity of propagation of light which are of importance in astronomy, a leading place must be assigned to the principle enunciated in 1842 by Christian Doppler," that the motion of a source of light relative to an observer modifies the period of the disturbance which is received by him. The phenomenon resembles the depression of the pitch of a note when the source of sound is receding from the observer. In either case, the period of the vibrations perceived by the observer is (c + v)/c x the natural period, where v denotes the velocity of separation of the source and observer, and c denotes the velocity of propagation of the disturbance. If, e.g., the velocity of separation is equal to the orbital velocity of the earth, the D lines of sodium in the spectrum of the source will be displaced towards the red, as compared with lines derived from a terrestrial sodium flame, bs about one-tenth of the distance between them. The application of this principle to the determination of the relative velocity of * Abhandl. dm- £. Hshm. Gee. der Wuuiiioh. (S) ii (18*S), p. 466. 3,Bl,ZEdhyG00gle 416 The Theory of Aether and Electrons in the stars in the line of sight, which has proved of great service in astrophyaical research, was suggested by Fizeau in 1848.* Passing now from the astronomical observatory, we must examine the information which has been gained in the physical laboratory regarding the effect of the earth's motion on optical phenomena. We have alreadyt referred to the investigations by which the truth of. Fiesnel's formula was tested. An experiment of a different type was suggested in 1852 by Fizeau,*. who remarked that, unless the aether is carried along by the earth, the radiation emitted by a terrestrial source should have different intensities in different directions. It was, how- ever, shown long afterwards by Lorentzg that such an experiment would not he expected on theoretical grounds to yield a positive result; the amount of radiant energy imparted to an absorbing body is independent of the earth's motion. A few years later Fizeau investigated!! another possible effect. If a beam of polarized light is sent obliquely through a glass plate, the azimuth of polarization is altered to an extent which depends, amongst other things, on the refractive index of the glass. Fizeau performed this experiment with sunlight, the light being sent through the glass in the direction of the terrestrial motion, and in the opposite direction ; the readings seemed to differ in the two cases, but on account of experimental difficulties the result was indecisive. Some years later, the effect of the earth's motion on the rotation of the plane of polarization of light propagated along the axis of a quartz crystal was investigated by Mascart.1t The result was negative, Mascart stating that the rotation could not have been altered by more than the (l/40,000)th part when the orientation of the apparatus was reversed from that of * An apparatus fur demons (rating the Doppler- Fizeau effect in the laboratory was constructed by Bulopoliky, Aitropbyi. Journal liii (1901), p. 15. t Cf. pp. 117-120. * Ann. d. Phys. xcii (18M), p. OS*. 4 Proc. Amsterdam Acad. (English edition), iv (1902) p. 678. || Annates de Chim. (3) Ixviii (I860), p. 129; Anil. d. Phyis. «i? (1861;. p. 564. T Annalei de l'Ec. Norm. (2) i (1872), p. 157. 3,Bl,ZEdhyG00gle Closing Years of the -Nineteenth Century. 417 the terrestrial motion to the opposite direction. This was afterwards confirmed by Lord Bayleigh * who found that the alteration, if it existed, could not amount to (1/I00,000)th part. In terrestrial methods of determining the velocity of light the ray is made to retrace its path, so that any velocity which the earth might possess with respect to the luminiferous medium would affect the time of the double passage only by an amount proportional to the square of the constant of aberration.! In 1881, however, A. A. MicbelsonJ remarked that the effect, though of the second order, should be manifested by a measur- able difference between the times for rays describing equal paths parallel and perpendicular respectively to the direction of the earth'B motion. He produced interference-fringes between two pencils of light which had traversed paths perpendicular to each other ; but when the apparatus was rotated through a light angle, bo that the difference would be reversed, the expected displacement of the fringes could not be perceived. This result , was regarded by Michelson himself as a vindication of Stokes's theory,§ in which the aether in the neighbourhood of the -•earth is supposed to be set in motion. Lorentz||, however, showed that the quantity to be measured had only half the ' value supposed by Michelson, and suggested that the negative result of the experiment might be explained by that combina- tion of Fresnel's and Stokes's theories which was developed in his own memoirl ; since, if the velocity of the aether near the earth were (Bay) half the earth's velocity, the displacement of Michelson's fringes would be insensible. - Phil. Mag. it. (1902), p. 21fi. t The constant of abarmtion is the ratio of the earth's orbital velocity til the velocity of light j of. "up™, p. 100. J Anier. Journ. Sti. xxii (1881), p. 20. Hi* method waa afterwards improved : if. Michelson and Morley, Ainer. Journ. 8ci. Iliiv (1887), p. 333; Phil. Hag. niv (1887), p. 448. I Of. p. ill. || Arch. Neerl. iii (1886), p. 103. On the Hichelion- Morley experiment el. ■W Hicfci, Phil. Mag. iii (1902), p. 9. * Cf. p. 413. 2 E DBlpahyG00gle 418 The Theory of AetheK and Electrons in the A sequel to the experiment of Michelson and Morley was performed in 1897, when Miohelson* attempted to determine by experiment whether the relative motion of earth and aether varies with the vertical height above the terrestrial surface. Ho result, however, could be obtained to indicate that the velocity of light depends on the distance from the centre of the earth ; and Michelson concluded that if there were no choice - a, h - curl a ; and the functions a and m&7 in turn he expressed in terms of the electric charges by the equations » - flf l(pr*y/r) dxTdy'dz', $ - jtf \(p)'lr\ dxdy'M, where the bars indicate that the values of (pv»)' and (p)' refer to the instant (t - r/e). Comparing these formulae with those given above for Clausius' potentials, we see that the only change which it is necessary to make in Clausius' theory is that of retarding the potentials in the way indicated by L. Lorenz.? The electric and magnetic forces, thus defined in terms of the • Cf. pp. 298, 299. t We shall um the small letters d anil h in place of I and H, when we are concerned with Lorents' fundamental cue, in which the system consists solely of free aether and isolated electrons. J Cf. p. 298. 3,Bl,ZEdhyG00gle 422 The Theory of Aether and Blecirotis in the position and motion of the charges, satisfy the Maxwellian equations div d = 4jrc*p, div h - 0, curl d - - H, curl h = a/e* + 4wpT. The theory of Lorentz is baaed on these four aethereal equations of Maxwell, together with the equation which deter- mines the ponderomotive force on a charged particle ; this. which we shall now derive, is the contribution furnished by Glausius' theory. The Lagrangian equations of motion of the electron e are ±(3L\ U dt \UJ " 3a; " ' and two similar equations, where L denotes the total kinetic otential due to all causes, electric and mechanical. The ponderomotive force exerted on the electron by the electro- magnetic field has for its ^-component ac ~ dt \d&)' /8«, . da. 3o.. da, -'sr which, since da m reduces to 8o. So.. t)-<- or id, + e(yhl -ii,>. so that the force in questi on is ?d + «[t H This was Lorentz' expression for the ponderomotive force on an 3,Bl,ZEdhyG00gle dosing Years of ihe Nineteenth Century. 42$ electrified corpuscle of charge e moving with velocity t in a field defined by the electric force d and magnetic force h. In Lorentz* fundamental case, which has thus been examined, account has been taken only of the ultimate constituents of which the universe is supposed to be composed, namely, cor- puscles and the aether. We must now see how to build up from these the more complex systems which are directly presented to our experience. The electromagnetic field in ponderable bodies, which to our senses appears in general to vary continuously, would present a different aspect if we were able to discern molecular structure ; we should then perceive the individual electrons by which the field is produced, and the rapid fluctuations of electric and magnetic 'force between them. As it is, the values furnished by our instruments represent averages taken over volumes which, though they appear small to us, are large compared with molecular dimensions.* We shall denote an average value of this kind by a bar placed over the corresponding symbol. Lorentz supposed that the phenomena of electrostatic charge and of conduction-currents are due to the presence or motion of simple electrons such as have been considered above. The part of p arising from these is the measurable density of electrostatic charge ; this we shall denote by p,. If w denote the velocity of the ponderable matter, and if the velocity v of the electrons be written w + u, then the quantity j/v, ao far as it arises from electrons of this type, may be written pi w + pa. The former of these terms represents the convection-current, and the latter the conduction-current. Consider next the phenomena of dielectrics. Following Faraday, Thomson, and Mossotti.t Lorentz supposed that each dielectric molecule contains corpuscles charged vitreously and also corpuscles charged resinously. These in the absence of an * These principles had been enunciated, and to aome extent developed, by J. WiUard Qibba in 1882-:! : Amer. Joura. Sci. xiiii, pp. 261, 460, xiv, p. 107 ; Gibl*' Scunlifie Faptri, ii, pp. 182, 195,211. t Cf. pp. 210, 211. 3,Bl,ZEdhyG00gle 424 The Theory of Aether and Electrons in the external field are so arranged as to neutralize each otbei'a electric fields outside the molecule. For simplicity we may suppose that in each molecule only one corpuscle, of charge e, is capable of being displaced from its position ; it follows from what has been assumed that the other corpuscles in the molecule exert the same electrostatic action as a charge - e situated at the original position of this corpuscle. Thus if t is displaced to an adjacent position, the entire molecule becomes equivalent to an electric doublet, whose moment is measured by the- product of t and the displacement of e. The molecules in unit volume, taken together, will in this way give rise to a (vector) electric moment per unit volume, P, which may be compared to the (vector) intensity of magnetization in Poisson's theory of magnetism* As in that theory, we may replace the doublet-distribution T of the scalar quantity p by a volume-distribution of p, determined by the equation! p - - div P. This represents the part of p due to the dielectric molecules Moreover, the scalar quantity pwx has also a doublet-distri- bution, to which the same theorem may be applied ; the average value of the part of pw„ due to dielectric molecules, is therefore determined by the equation pwt- - div (w„F) - - wm div P -(P. V)w„ or pv - - divp.w- (P.V)w. We have now to find that part of pu which is due to dielectric molecules. For a single doublet of moment p we have, by differentiation, /// pu dxdydz - dp/dl, where the integration is taken throughout the molecule; so that HI pudxdydz- (d/dl) ( FP), where the integration is taken throughout a volume V, which •Cf. p. 64. t We iMiimfl all transition) gradual, so as to avoid s u rf hob- dirt ributions. 3,Bl,ZEdhyG00gle Closing Years of the Nineteenth Century. 426 encloses a large number of molecules, but which is small com- pared with measurable quantities; and this equation may be written Now, if P refers to differentiation at a fixed point of space (as opposed to a differentiation which accompanies the moving body j, we have (d/w = P + curl [p. w]. This equation determines the part of pv which arises from the dielectric molecules. The general equations of the aether thus become, when the averaging process is performed, div d - 4fn-c'pl - 4n-c* div P, div h - 0, curl d - - h, — i- I conTection -current -t- conduction- current , curl h - (1/c1) d + 4ir • . r_ . ■ v ' ' | + P + curl [P . w] I In order to assimilate these to the ordinary electromagnetic equations, we must evidently write d = E, the electric force; (1/4ttc*)E + P - D, the electric induction ; h - H, the magnetic vector. The equations then become (writing p for />„ as there is no longer any need to use the subscript), div D - p, - curl E = H, , div H - 0, curl H = 4*-B, where S — conduction -current + convection-current + D + curl [P . w]. 3,Bl,ZEdhyG00gle 426 The Theory of Aether and Electrons in the The term D in 8 evidently represents the displacement- current of Maxwell; and the term curl [P . w] will be recognized as a modified form of the term curl [D . w], which was first introduced into the equations by Hertz* It will be remembered that Hertz supposed this term to repre- sent the generation of a magnetic force within a dielectric which is in motion in an electric field ; and that Heaviside,f bj adducing considerations relative to the energy, showed that the term ought to be regarded as part of the total current, and inferred from its existence that a dielectric which moves in an electric field is the seat of an electric current, which produces a magnetic field in the surrounding space. The modification introduced by Lorentz consisted in replacing D by P to the vector-product ; this implied that the moving dielectric does not carry along the aethereal displacement, which is represented by the term E/4jrc* in D, but only carries along the charges which exist at opposite ends of the molecules of the ponderable dielectric, and which are represented by the term P. The part of the total current represented by the term curl [P - w] is generally called the current of dielectric convection. That a magnetic field is produced when an uncharged dielectric is in motion at right angles to the lines of force of a constant electrostatic field had been shown experimentally in 1888 by Kontgen.J His experiment consisted in rotating a dielectric disk between the plates of a condenser ; a magnetic field was produced, equivalent to that which would be produced by the rotation of the " fictitious charges " on the two faces of the dielectric, i.e., charges which bear the same relation to the dielectric polarization that Foisson's equivalent surface- density of magnetism§ bears to magnetic polarization. If V denote the difference of potential between the opposite coatings of the condenser, and t the specific inductive capacity of the dielectric, the surface -density of electric charge on the coatings • Cf. p. 366. t Cf. p. 36T I Ann. d. Plivu. xx xv {1888}, p. 381 xl (1890), p. 93. j Cf. P. 6*. 3,Bl,ZEdhyG00gle Closing Years of the Nineteenth Century. 427 is proportional to ± ill, and the fictitious charge on the sur- faces of the dielectric is proportional to T (i — 1) U. It is evident from this that if a plane condenser is charged to a given difference of potential, and is rotated in its own plane, the magnetic Geld produced is proportional to t if (as in Rowland's experiment*) the coatings are rotated while the dielectric remains at rest, but ib in the opposite direction, and iB propor- tional to (i - 1) if (as in Kbntgen's experiment) the dielectric is rotated while the coatings remain at rest. If the coatings and dielectric are rotated together, the magnetic action (being the sum of these) should be independent of e — a conclusion which was verified later by Eiehenwald.f Hitherto we have taken no account of the possible mag- netization of the ponderable body. This would modify the equations in the usual manner,? so that they finally take the form div L - p, (I) div B - 0, (II) curl H = 4n-S, (III) - curLB = B, (IV), where 8 denotes the total current formed of the displacement- current, the convection-current, the conduction-current, and the current of dielectric convection. Moreover, since 8 - pv + d/4irc", we have div 8 - div pY + (1/4JTC1) div (3d/3r) - div pv + 3p/2/, ■Cf. p. 339. t Ann. d. Phja. xi (1903), p. 421 ; liii (1904), p. 919. Eiehenwald performed other experiment) of a similar character, e.g. ha observed the nmgnelio field due to tlie changes of polarization in a dielectric which vaa mored in a non- homogeneoua electric field. | It is possible to construct a purely electronic theory of magnetization, a. magnetic molecule being supposed to contain electrons in orbital revolution. It then appears that the vector which represents the average value of > is not B, but B. DinlizBdbyGoOgk 428 The Theory of Aether and Electrons in the which vanishes by virtue of the principle of conservation of electricity. Thus div S - 0, (V) or the total current is a circuital vector. Equations (I) to (V) are the fundamental equations of Lorentz' theory of electrons. We have now to consider the relation by which the polari- zation P of dielectrics is determined. If the dielectric is moving with velocity w, the ponderomotive force on unit electric charge moving with it is (as in all theories)" E' = B+[w.B]. (1) In order to connect P with B', it is necessary to consider the motion of the corpuscles. Let e denote the charge and m the mass of a corpuscle, (£, n, £) its displacement from its position of equilibrium, k* (£, ij, Z) the institutive force which retains it in the vicinity of this point ; then the equations of motion of the corpuscle are m£ + A'S ■ bEt', and similar equations in >j and X,. When the corpuscle is set in motion by light of frequency n passing through the medium, the displacements and forces will be periodic functions of nt— say, _ Substituting these values in the equations of motion, we obtain AQ*-l)I'; (2) from this equation, Freanels formula for the velocity of light in a moving dielectric may be deduced. For, let the axis of z be taken parallel to the direction of motion of the dielectric, which is supposed to be also the direction of propagation of the light; and, considering a plane-polarized wave, take the axis of x parallel to the electric vector, so that the magnetic vector must be parallel to the axis of y. Then equation (III) above - affy/fe = 4iri), + 4m« 3.Pa/3z ; equation (IV) becomes (assuming B equal to H, as is always the case in optics), - SSJdt = Er The equation which defines the electric induction gives Dm-(ll4*t?)E* + Pm; and equations (1) and (2) give Wi*. = E, / , a /a 3\« or, neglecting «**/«*, VE, ffffX. 2w(u*-\)??E, J Substituting EI-*e"{t '/F)V l, so that V denotes the velocity of light in the moving dielectric with respect to the fixed aether we have • CI. p. in. t Thi» equation in fint given u a result of the theory of election! by Lamti in the hut chapter of bis memoir of 1892, Arch. Neerl. XXV, p. 526. It wualw gi»en by Larmor, Phil. Trana., cliixv (189*), p. 821. 3,Bl,ZEdhyG00gle Closing Years of the Nineteenth Century. 431 or (neglecting vfji?) which ia the formula of Fresnel * The hypothesis of Fresnel, \ that a ponderable body in motion carries with it the excess of aether which it contains as compared with space free from matter, is thus seen to be transformed in Lorentz' theory into the supposition that the polarized molecules of the ' dielectric, like so many small condensers, increase the dielectric constant, and that it is (bo to apeak) this augmentation of the dielectric constant which travels with the moving matter. One evident objection to Fresnel's theory, namely, that it required the relative velocity of aether and matter to be different for light of different colours, is thus removed ; for the theory of Lorentz only requires that the dielectric constant should have different values for light of different colours, and of this a satisfactory explanation is provided by the theory of dispersion. The correctness of Lorentz1 hypothesis, as opposed to that of ' Hertz (in which the whole of the contained aether was supposed to be transported with the moving body), was afterwards confirmed by various experiments. In 1901 R. Blondlotf drove a current of air through a magnetic field, at right angles to the lines of magnetic force. The air-current was made to pass between the faces of a condenser, which were connected by a wire, so as to be at the same potential. An electromotive force E' would be produced in the air by its motion in the magnetic field ; and, according to the theory of Hertz, this should produce an electric induction D of amount (e/47rc') E' (where t denotes the specific inductive capacity of the air, which is practically unity) ; so that, according to Hertz, the faces of the condenser should become charged. According to Lorentz' theory, on the other hand, the electric induction D is determined by the equation 4ffCD = E + (e-l)E' • Cf. p. 117. t Comptea Rendu* cndii (1901), p. 778. 3,Bl,ZEdhyG00gle 432 The Theory of Aether and Electrons in the where E denotes the electric force on a charge at rest, which is zero in the present case. Thus, according to Lorentz' theory, the charges on the faceB would have only (e - l)/e of the values which they would have in Hertz' theory ; that is, they would be practically zero. The result of Blondlot's experiment was in favour of the theory of Lorentz. An experiment of a similar character was performed in 1905 by H. A. Wilson.' In this, the space between the inner and outer coatings of a cylindrical condenser was filled with the dielectric ebonite. When the coatings of such a con- denser are maintained at a definite difference of potential, charges are induced on them ; and if the condenser be rotated on its axis in a magnetic field whose lines of force are parallel to the axis, these charges will be altered, owiog to the additional polarization which is produced in the dielectric molecules by their motion in the magnetic field. As before, the value of the additional charge according to the theory of Lorentz is (e - l)/e times its value as calculated by the theory of Hertz. The result of Wilson's experiments was, like that of Blondlot's, in favour of Lorentz. The reconciliation of the electromagnetic theory with Fresnel's law of the propagation of light in moving bodies was a distinct advance. But the theory of the motionless aether was hampered by one difficulty : it was, in its original form, incompetent to explain the negative result of the experiment of Michelson and Morley.f The adjustment of theory to observation in this particular was achieved by means of a remarkable hypothesis which must now be introduced. In the issue of " Nature " for June 16th, 1892^ Lodge mentioned that Fitz Gerald had communicated to him a new suggestion for overcoming the difficulty. This was, to suppose that the dimensions of material bodies are slightly altered when they are in motion relative to the aether. Five months afterwards, this hypothesis of Fitz Gerald's was adopted by 3,Bl,ZEdhyG00gle Closing Years of the Nituteenth Century. 433 Lorentz, in a communication to the Amsterdam Academy;* after which it won favour in a gradually widening circle, until eventually it came to be generally taken as the basis of all theoretical investigations on the motion of ponderable bodies through the aether. Let us first see how it explains Michelson's result. On the supposition that the aether is motionless, one of the two portions into which the original beam of light is divided should accomplish its journey in a time less than the other by w>V/e*, where w denotes the velocity of the earth, c the velocity of light, and I the length of each arm. This would be exactly compensated if the arm which is pointed in the direction of the terrestrial motion were shorter than the other by an amount uflj'le1; as would be the case if the linear dimensions of moving bodies were always contracted in the direction of their motion in the ratio of (1 - w*/2c*) to unity. This is Fitz Gerald's hypothesis of contraction. Since for the earth the ratio w/c is only 30 km. /sec 300,000 km./sec.' the fraction to'/c' is only one hundred-millionth. Several further contributions to the theory of electrons in a, motionless aether were made in a short treatisef which was published by Lorentz in 1895. One of these related to the explanation of an experimental result obtained some years previously by Th. des Coudres^ of Leipzig. Des Coudres had observed the mutual inductance of coils in different circum- stances of inclination of their common axis to the direction of the earth's motion, but had been unable to detect any effect depending on the orientation. Lorentz now showed that this could be explained by considerations similar to those which • Veralagen d. Eon. Alt. Tan Wcttiisuliuppen, 1892-3, p. 71 (November 2Slh 1892). t Vtriueh finer Thiarii Jtr eleetrUehm und opHteA'n Krich$\nun$m in bwegttn Korpern, von H. A. Lorenti ; Leiden, E. J. Brill. It was reprinted by Teubner, of l-eipzig, in 1906. % Ann. d. Phja. xz»BI (1889), p. 73. 2 F 3,Bl,ZEdhyG00gle 434 The Theory 0/ Aether and Electrons in the Budde and Fitz Gerald* had advanced in a similar case; a conductor carrying a constant electric current and moving with the earth would exert a force on electric charges at relative rest in its vicinity, were it not that this force induces on the surface of the conductor itself a compensating electrostatic charge, whoae action annuls the expected effect. The most satisfactory method of discussing the influence of the terrestrial motion on electrical phenomena is to transform the fundamental equations of the aether and electrons to axes moving with the earth. Taking the axis of x parallel to the direction of the earth's motion, and denoting the velocity of the earth by w, we write x = x, + wt, y = ylf % = z,, so that (x,, y„ *,) denote coordinates referred to axes moving with the earth. Lorentz completed the change of coordinates by introducing in place of the variable to." local time " t„ defined by the equation t = fi + «ar,/c*. It is also necessary to introduce, in place of d and h, the electric and magnetic forces relative to the moving axes : these aret di-d+[w.h] h, « h. + (l/