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Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/collectedscientiOOpoynuoft COLLECTED SCIENTIFIC PAPERS CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, Manager LONDON : FETTER LANE, E.C.4 LONDON : H. K. LEWIS AND CO., Ltd., 136, Gower Street, W.C. i LONDON : WILLIAM WESLEY AND SON, 28, Essex Street, Strand, W.C. 2 NEW YORK : G. P. PUTNAM'S SONS BOMBAY ^ CALCUTTA I MACMILLAN AND CO., Ltd. MADRAS j TORONTO : J. M. DENT AND SONS, Ltd. TOKYO : MARUZEN-KABUSHIKI-KAISHA ALL RIGHTS KESEKVED yfrl^cnyiA^'SuZ^ COLLECTED SCIENTIFIC PAPERS BY JOHN HENRY POYNTING, Sc.D., F.R.S. Mason Professor of Physics in the University of Birmingham Formerly Fellow of Trinity College, Cambridge CAMBRIDGE AT THE UNIVERSITY PRESS 1920 PREFACE IN the summer of 1914 Sir Oliver Lodge summoned a meeting of the colleagues and friends of the late Professor John Henry Poynting to consider how best a suitable memorial could be established to perpetuate his memory. The following committee was appointed to carry out the purpose of the meeting : Sir Oliver Lodge (chairman), Guy Barlow, Neville Chamber- lain, P. F. Frankland, Sir R. T. Glazebrook, R. S. Heath, George Hookham, Sir Joseph Larmor, Sir Napier Shaw, Sir J. J. Thomson, Sir Richard Threlfall, and T. Sydney Walker, with G. H. Morley as Secretary and G. A. Shakespear as Treasurer. A fund was opened and subscriptions were invited, and the Committee decided that there could be no better memorial than the publication in a collected form of the Scientific Papers of John Henry Poynting, and the distribution of copies to certain scientific institutions throughout the world. The task of editing and proof correction was gladly undertaken by his junior colleagues G. A. Shakespear and Guy Barlow who have throughout had the benefit of the advice of Sir Oliver Lodge. Biographical and critical notices by Sir Oliver Lodge, Sir Joseph Larmor, Sir J. J. Thomson and G. A. Shakespear have been inserted as an introduction to the volume. The papers have been arranged in groups, with the object of bringing together those dealing with kindred subjects. In each group the papers are in chronological order. In such collections there is inevitably a certain amount of repetition or overlapping, since a subject is often dealt with from more than one point of view — the strictly scientific and the popular — but since this variety of treat- ment is helpful, in cases of doubt the decision has generally been given in favour of inclusion. The popular discourses and general articles have for the most part been relegated to the last section of the volume. His books, and other publications which are easily accessible, are not included; but a complete list of his works in chronological order is given at the end of the volume. The papers have been reproduced as originally published except for small verbal corrections here and there. In certain cases mistakes have been y| PREFACE corrected after reference to the original manuscripts which were kindly put at our disposal by Mrs. Poynting ; but, in all cases, corrections of any import- ance have been indicated in footnotes, and editorial comments have been included in square brackets and marked Ed. We wish to acknowledge most warmly the unstinted generosity with which permission to republish the several articles has been given both by joint authors and by the original publishers, including the representatives of the following : The Birmingham Natural History and Philosophical Society. The British Association. Le Bureau International des Poids et Mesures. The Electrician. The Encyclopaedia of Religion and Ethics (Messrs T. and T. Clark). The Hibbert Journal. The Indiarubber Journal. The Inquirer. The Mason College Magazine. The Manchester Literary and Philosophical Society. Nature. The Philosophical Magazine. The Royal Astronomical Society. The Royal Society. The Royal Statistical Society. The Royal Institution. La Societe de Physique. : , The Committee was fortunate in securing the services of the Cambridge University Press as publishers, and the editors take this opportunity of thanking them for their unfailing courtesy and for invaluable help in proof- correcting. G. A. SHAKESPEAR. GUY BARLOW. The University. Birmingh.\m. 1920. A PERSONAL NOTE John Henry Poynting was a man admired of all who knew him, beloved of all who knew him well. Of somewhat less than middle height and sturdy thickset build, his general appearance was suggestive rather of rural than academic interests, »but even a casual observer would probably have been struck with the sense of power indicated by his fine head. His face, which was of a meditative cast when in repose, lighted with a genial and friendly warmth when he was conversing with friends, and his grey-blue eyes were expressive of the kindly gentleness of his nature. In habit he was methodical, and indeed the condition of his health was for many years such that he could not have accomplished so much had he not economised effort with method. (He told me once that he had never been able to do more than six hours' useful work a day.) He had, moreover, great power of concentration. He was a remarkably clear thinker and had that characteristic insight into fundamental ideas which intuitively distinguishes between hypothesis and fact ; and it was probably for this reason that he viewed with suspicion some of the more recent developments of mathematical physics. He withheld his judgment when the experimental foundations were either wanting or else inadequate to bear the superstructure erected upon them. Himself a man of wide interests and sympathies, and with a finely balanced sense of proportion, he was keenly alive to the danger of that too-exclusive specialisation which so frequently makes a man incapable of conversation except in his awn particular line of work. He felt the need of guarding against atrophy of the spiritual side of his character in his outlook on life, and sought in the reading, of fiction, and even more of poetry, the complement to the intellectual stimulus of scientific work. Indeed of poetry he read much, though he spoke of it but rarely even with his intimate friends ; but those to whom he opened his mind on such matters knew his deep admiration for many of the English poets. Shelley, Keats and still more Wordsworth, appealed to him strongly, for he himself was imbued with that love of Nature which inspired them. He was probably never happier than when living in the beautiful Alvechurch district of Worcestershire, where he found an unfailing source of pleasure in the rolling landscape which stretched away into the distance Viii A PERSONAL NOTE around his upland home. He would walk in the fields with some friend, here pointing out the haunt of a rare wildflower, there showing, with all the interest of a schoolboy, the nest of an uncommon bird, or discussing the effect of atmosphere on the landscape. The conversation would wander from the botanical afiinities of the tway-blade to the mechanism whereby the grasshopper performs his prodigious leaps, or from the theory of the grinding action of a cider-press to the causes and prevention of crime in large cities. To such occasions his equable temper and ready sense of humour lent a rare charm. Among his outdoor recreations cycling through the country lanes held a high place. Indoors, in his later years, he derived much pleasure from a piano-player ; and in the evenings, when too tired to read, he would often amuse himself with a game of "patience." In politics he was a liberal, and perhaps the thing for which he stood more strongly than anything else was freedom and liberty of thought; the thing of which he was most intolerant was bigotry, and indeed in an intimate aquaintance of 20 years' duration the only time T ever heard him speak of any man with bitterness was in reference to a case of religious intolerance. The soundness of his judgment being well known, his advice was often sought ; and, whether the matter were small or great, he always gave of his best, earning thereby the gratitude of many. He had a great sympathy with humanity in general and especially with the poorer classes. As a magistrate he tempered judgment with mercy and his experience in this capacity confirmed him in the opinion that delinquents in general were as often sinned against as sinning. The comradeship of his home-life was ideal, while to his students he was an object of admiration and of affectionate regard in a degree which perhaps they alone can appreciate. His ability may to some extent be judged from his pubhshed work, but his personal worth and charm are truly measured by the affection with which he inspired those who knew him best. * G. A. S. OBITUARY NOTICES [From Nature, vol. xciii, p. 138, with additions.] On the evening of Monday, March 30, 1914, surrounded by his family John Henry Poynting passed quietly away. A memorial service was held in Birmingham on the Thursday following, and was attended by representatives of many universities and learned societies, including Sir J. J. Thomson, Sir Joseph Larmor, Dr Glazebrook, Sir William Tilden, Prof. W. M. Hicks, Dr W. N. Shaw, and of course by many colleagues and councillors of the University in which he occupied a chair, as well as by a large number of private citizens and friends. For he was a man universally beloved. At the memorial service, the following true words concerning him were spoken by the Rev. Henry Gow, who knew him well : "We remember that he did work to make him famous throughout the world of science which gave him a high place amongst the discoverers of truth ; but we remember much more than that. We remember how he loved life, how interested he was in little things, how he delighted in children, in flowers, and in birds; what confidence and affection he inspired, how free he was from claims of self and from uneasy egotism; how much happiness he felt and gave. We remember his wise judgments, strong character, cheerful courage, his delightful humour, and a certain peaceful beauty and childlike joyousness of spirit behind all his multifarious gifts. He rejoiced to be the friend as well as the teacher of the young. He kept his heart free from all bitterness and disillusion which come so often to us in our later years. He knew and felt always how beautiful and great a thing it was to be alive." He was born on September 9, 1852, at Monton, near Manchester, son of the unitarian minister of that place. His first education was at home, but the years 1867 to 1872 he passed at Owens College, Manchester, graduating B.Sc. at the London University, and proceeding, in 1872, to Trinity College, Cambridge, where he was bracketed third wrangler in 1876. He was then appointed demonstrator at Owens College by Balfour Stewart, and began a life-long friendship with Sir J. J. Thomson, who was at that time a student. In due time Poynting became a fellow of Trinity, and in 1880 was appointed to the professorship of physics at Birmingham, which he held to the day of his death. The four first professors of the Mason College, which was opened by Huxley in 1880 (who delivered, on this occasion, a notable address, reprinted X OBITUARY NOTICES as the first of his collected essays), were Sir Wm. Tilden, Prof. M. J. M. Hill, Dr T. W. Bridge, who died a few years ago, and Poynting. In this same year Poynting married Miss M. A. Cropper, daughter of the Rev. J. Cropper, of Stand, near Manchester. In 1887 he received the Sc.D. of Cambridge, and in 1888 the fellowship of the Royal Society. In 1893 the Adams prize was awarded to him, and in 1899 he presided over Section A of the British Association at Dover. This meeting was memorable for the clear discovery of the separate existence of electrons, which was announced to Section A bv Sir J. J. Thomson on an occasion when many members of the French Association, meeting simultaneously at Boulogne, had come over for friendly fraternisation. In 1905 Poynting became president of the Physical Society, and was awarded a Royal medal by the Royal Society " for his researches in physical science, especially in connection with the constant of gravitation and the theories of electro-dynamics and radiation." In this brief summary an immense amount of work is referred to. The work for which he is locally best known was his determination of the Newtonian constant of gravitation by the very accurate use of an ordinary balance with an adjustable mass under one or other of the arms — a determination which is popularly called " weighing the earth." His account of it appears in the Phil. Trans, for 1891. It is a classical memoir of its kind, and very instructive to the physical student, but the papers on electro-dynamics eclipse it in value. These were ''communicated" to the Royal Society in 1884 and 1885 respectively, their titles being "On the Transfer of Energy in the Electromagnetic Field," and "On the Connection between Electric Current and the Electric and Magnetic Inductions in the Surrounding Field." The memoir on the transfer of energy aroused universal attention. The paths by which energy travels from an electromotive source to various parts of a circuit were displayed, and their intricacies unravelled, for the first time ; idottihj of energy might legitimately be urged as a supplement to conservation (see a paper by the present writer in Phil. Mag., June, 1885) ; and it is to these papers that we owe that fundamental generalisation, connecting mechanical motion with electric and magnetic forces, which is known all over the world as "Poynting's Theorem." The following letter from Sir Joseph Larmor to the writer expresses a mathematician's view of the importance of this subject: St John's College, Cambridge, 10th May, 1915. •'Nobody before Poynting seems to have thought of tracing the flux of energy in a medium elasticaUy transmitting it, and where the whole process is therefore exposed to view. The line of flow is a ray in optics: thus it includes a dynamical aspect of that conception added on to and of course consistent with the Huygenian or rather Young-Fresnelian one. The electric OBITUARY NOTICES XX and optical ray is implicitly in Maxwell's equations, and is only a corollary to them. But in any other kind of elastic transmission, e.g. waves in an elastic-solid medium, a corresponding theory can be worked out. I take it this idea is Poynting's main contribution, and it clarified many things, especially electrical." A great expansion of this note is contained in a remarkable paper On the Dynamics of Radiation which Sir Joseph Larmor communicated to the Inter- national Congress of Mathematicians meeting at Cambridge in August, 1912. This paper is so intimately associated with Poynting's work, and so pleased him when he saw it, that I have asked and obtained permission to include extracts from it in this volume; they will be found at the end of the Section dealing with the Pressure of Light. " The essential characteristic of an electrodynamic system is the existence of the correlated fields, electric and magnetic, which occupy the space sur- rounding the central body, and which are an essential part of the system; to the presence of this pervading aethereal field, intrinsic to the system, all other systems situated in that space have to adapt themselves. When a material electric system is disturbed, its electrodynamic field becomes modified, by a process which consists in propagation of change outward, after the manner of radiation, from the disturbance of electrons that is occurring in the core. When however we are dealing with electric changes which are, in duration, slow compared with the time that radiation would require to travel across a distance of the order of the greatest diameter of the system — in fact in all electric manifestations except those bearing directly on optical or radiant phenomena — complexities arising from the finite rate of propagation of the fields of force across space are not sensibly involved : the adjustment of the field surrounding the interacting systems can be taken as virtually instantaneous, so that the operative fields of force, though in essence propagated, are sensibly statical fields. The practical problems of electrodynamics are of this nature — how does the modified field of force, transmitted through the aether from a disturbed electric system, and thus established in the space around and alongside the neighbouring conductors which alone are amenable to our observation, penetrate into these conductors and thereby set up electric disturbance in them also? and how does the field emitted in turn by these new disturbances interact with the original exciting field and with its core ? For example, if we are dealing with a circuit of good conducting quality and finite cross section, situated in an alternating field of fairly rapid frequency, we know that the penetration of the arriving field into the conductor is counteracted by the mobility of its electrons, whose motion, by obeying the force, in so far annuls it by Newtonian kinetic reaction ; so that instead of being propagated, the field soaks in by diffusion, and it does not get very deep even when adjustment is delayed by the friction of the vast numbers of ions which it starts into motion, and which have to push their way through the crowd of material molecules; and the phenomena of surface currents thus arise. If (by a figure of speech) we abolish the aether in which both the generating circuit and the secondary circuit which it excites are immersed, in which they in fact subsist, the changing phases of the generator could not thus establish, from instant to instant, by almost instantaneous Xii OBITUARY NOTICES radiant transmission, their changing fields of force in the ambient region extending across to the secondary circuit, and the ions in and along that circuit would remain undisturbed, having no stimulus to respond to. The aethereal phenomenon, viz., the radiant propagation of the fields of force, and the material phenomenon, viz., the response of the ions of material bodies to those fields, involving the establishment of currents with new fields of their own, are the two interacting factors. The excitation of an alternating current in a wire, and the mode of distribution of the current across its section, depend on the continued establishment in the region around the wire, by processes of the nature of radiation, of the changing electromagnetic field that seizes hold on the ions and so excites the current ; and the question how deep this influence can soak into the wire is the object of investigation. The aspect of the subject which is thus illustrated, finds in the surrounding region, in the aether, the seat of all electrodynamic action, and in the motions of electrons its exciting cause. The energies required to propel the ions, and so establish an induced current, are radiant energies which penetrate into the conductor from its sides, being transmitted there elastically through the aether; and these energies are thereby ultimately in part degraded into the heat arising from fortuitous ionic motions, and in part transformed to available energy of mechanical forces between the conductors. The idea — introduced by Faraday, developed into precision by Maxwell, expounded and illustrated in various ways by Heaviside, Poynting, Hertz — of radiant fields of force, in which all the material electric circuits are immersed, and by which all currents and electric distributions are dominated, is the root of the modern exact analysis of all electric activity." Poynting's work on radiation appeared partly in the Phil. Trans, for 1904 and partly in the Phil. Mag. for 1905. In these memoirs the tangential pressure of radiation is analysed and demonstrated; and it is shown, both theoretically and experimentally, that a beam of light behaves essentially as a stream of momentum, and gives all the mechanical results which may thus be expected, though of a magnitude exceedingly minute. Nevertheless, he goes on to show that these radiation-pressures, however small, are of much consequence in astronomy, and have many interesting and some conspicuous results. A noteworthy part of the radiation memoirs, however, is independent of considerations of pressure or momentum, and gives a means of determining the absolute temperatures of sun and planets, and of other masses in space, in a singularly clear and conclusive manner. A complete list of his publications is given below, but special mention must be made here of the important series of text-books on physics, written in conjunction with his friend. Sir J. J. Thomson. He took great interest also in the philosophical aspects of physical science, and his help is acknowledged by Prof. James Ward in connection with the publication of that notable series of Gifford Lectures entitled Naturalism, and Agnosticism. Poynting was strongly inclined, almost unduly, to limit the province of science to description, and to regard a law of nature as nothing but a formulation of observed correspondences. He wished to abolish the OBITUARY NOTICES XUl idea of cause in physics. In some of this he may have gone too far ; but his rebellion against an excessive anthropomorphism which had begun to cling around the notion of natural laws, as if they were really legal enactments to be obeyed or disobeyed by inert matter almost as if it possessed will-power and could exercise choice, some substances being praised as good radiators while others are stigmatised as bad — most gases being admittedly unable to reach a standard of perfection held out to them as Boyle's law, though a few of excessive merit might surpass it, — Poynting's revolt against this kind of attitude to laws of nature, though doubtless more than half humorous, was in itself wholesome. Some of his philosophic views may be read, as a Presidential Address to Section A of the British Association for 1899 (infra p. 599) ; but I think it useful and legitimate to extract a few sentences from that address and quote them here, as an illustration of his mode of approaching the misty region where physics and metaphysics intertwine: " To take an old but never- worn-out metaphor, the physicist is examining the garment of Nature, learning of how many, or rather of how few different kinds of thread it is woven, finding how each separate thread enters into the pattern, and seeking from the pattern woven in the past to know the pattern yet to come.... So, as we watch the weaving of the garment of Nature, we resolve it in imagination into threads of ether spangled over with beads of matter. We look still closer, and the beads of matter vanish; they are mere knots and loops in the threads of ether." And then, a few pages further on, when dealing with the interaction of Matter and Mind : "Do we, or do we not, as a matter of fact, make any attempt to apply the physical method to describe and explain those motions of matter which on the psychical view we term voluntary? Any commonplace example, and the more commonplace the more is it to the point, will at once tell us our practice, whatever may be our theory. For instance, a steamer is going across the Channel. We can give a fairly good physical account of the motion of the steamer. We can describe how the energy stored in the coal passes out through the boiler into the machinery, and how it is ultimately absorbed by the sea. And the machinery once started, we can give an account of the actions and reactions between its various parts and the water, and if only outsiders will not interfere, we can predict with some approach to correctness how the vessel will run. All these processes can be likened to processes already studied — perhaps on another scale — in our laboratories, and from the similarities prediction is possible. But now think of a passenger on board who has received an invitation to take the journey. It is simply a matter of fact that we make no attempt at a complete physical account and explanation of those actions which he takes to accomplish his purpose. We trace no lines of induction in the ether connecting him with his friends across the Channel, we seek no law of force under which he moves. In practice the strictest physicist abandons the physical view, and replaces it by the psychical. He admits the study of purpose as well as the study of motion." Xiv OBITUARY NOTICES In other words he recognises Mind and Purpose as dominant over and in a different category from Matter and Mechanism. In psychical phenomena Poynting was, I judge, an agnostic, but on the question of a materialistic or naturalistic explanation of mental phenomena he expresses himself thus, in the Dover Section A address above referred to : "It appears to me that the assumption that our methods do apply, and that purely physical explanation will suffice to predict all motions and changes, voluntary and involuntary, is at present simply a gigantic extra-polation, which we should unhesitatingly reject if it were merely a case of ordinary physical investigation. The physicist when thus extending his range is ceasing to be a physicist, ceasing to be content with his descriptive methods in his intense desire to show that he is a physicist throughout." But I must not delay further on his scientific work ; the man himself was even more than his work. When the Mason College became the University of Birmingham Poynting was elected Dean of the Faculty of Science; in that capacity his quiet wisdom and efficiency were very manifest, and keen was the regret of all his colleagues when, some twelve years later, failing health necessitated his yielding this office to another. His judgment was as sound as his knowledge, and his conspicuous fairness endeared him to colleagues and the members of his staff. By the latter it is not too much to say that he was regarded with affectionate veneration ; one of them writes to me as follows : " As to his character it is impossible to give the right impression to those who did not know him well. I consider him a man of very extraordinary ability, which might have carried him much farther if it had been associated with more self-assertion. But it was largely this modesty and self-suppression which created a very unusual degree of affection in those who had the privilege of knowing him intimately. I always associate him in my mind with Faraday and Stokes." As a lecturer and teacher he was admirable, and the respect in which he was held by his peers was noteworthy. I am glad to remember that so recently as the 1913 meeting of the British Association, some of the greatest physicists in the world, who were staying with me — Prof. H. A. Lorentz, Lord Rayleigh, and Sir Joseph Larmor — went to his house one evening, and met there in his study Sir J. J. Thomson and Dr Glazebrook, who were staying with him; thus constituting an appropriate and representative gathering, and giving him a pleasure which he remembered to the end of his life. There is much more that might be said ; but let his position in the world of science be what it may, we in the University of his mature life knew him well, and know him best as an admirable colleague, a staunch friend, and a good man. 0. J. L. OBITUARY NOTICES XV [From the Proceedings of the Royal Society, A, vol. xcii, 1914.] John Henry Poynting, the youngest son of the Rev. T. E. Poynting, Unitarian Minister at Monton, near Manchester, was born there on September 9, 1852. He received his earlier education at the school kept by his father and then went, in 1867, to the Owens College, which his elder brother, C. T. Poynting, who was for many years Unitarian Minister at Fallowfield, near Manchester, had just left. Poynting must have received a good grounding in Mathematics at his father's school, as he gained a Dalton Entrance Exhibi- tion in Mathematics before entering the College. Owens College in those days was in a modest building, once the residence of Richard Cobden, in Quay Street, Deansgate. Neither the amenities of the locality nor the accommodation in the building were anything to boast about, but few educa- tional institutions before or since, whatever their equipment or surroundings, have had a more efficient staff than Owens College in the old Quay Street days. As an old Quay Streeter, the writer can speak from personal experience. The cramped space was not an unmixed disadvantage. We were so closely packed that it was very easy for us to get to know each other. Arts students and Science students jostled against each other continually; a crowd of Mathematicians would be waiting outside the doors of a lecture room for it to discharge a Latin or Greek class, and thus one of the chief difficulties of non-residential colleges, the lack of social intercourse between the students, was almost absent. The professors at Owens in Poynting's time were : Barker for Mathematics, of whom Poynting always spoke in terms of the highest appreciation, a feeling shared by all his pupils, for no abler or more conscientious teacher of Mathe- matics than Thomas Barker ever lived; Jack, another great teacher, was Professor of Natural Philosophy; Roscoe of Chemistry, and Williamson of Natural History; on the literary side. Greenwood, the Principal, was Pro- fessor of Classics, Ward of English History and Literature, Jevons of Logic, and that very lovable man, Theodores, lecturer on Modern Languages. At that time Owens had not the power of granting degrees, and most of the students prepared for the examinations of the University of London. In those days these covered a very wide range of subjects, and Poynting, who took the London degree, must have attended the lectures of all these professors. He was second at the London Matriculation in 1869, obtained Second Class Honours in both Physics and Mathematics in the First B.Sc. examination in 1871, and took the B.Sc. degree in 1872. In the spring of 1872 he obtained an entrance scholarship at Trinity College, Cambridge, and came into residence at Cambridge in October. At Cambridge he pursued the normal course of one destined for high honours in the Mathematical Tripos. He read with Xvi OBITUARY NOTICES Routh, he obtained his Major Scholarship in due course, like many of the reading men of his time at Trinity he joined the Second Trinity Boat Club and rowed in the first boat in 1875 ; the fortunes of that once famous club were, however, then declining and it came to an end in 1876. He took his degree in the Mathematical Tripos of 1876 as Third Wrangler, bracketed with Mr Trimmer, of Trinity College, a very brilliant man who suffered from persistent ill-health and died within a few months of taking his degree. As Dr Glazebrook and Dr Shaw both graduated in the same Tripos and Lord Rayleigh was the additional Examiner, Physics was well represented on this occasion. After taking his degree Poynting came back for a short time to the Owens College, which was now in the buildings it at present occupies, and demon- strated in the Physical Laboratory under Prof. Balfour Stewart, who had succeeded Jack as Professor of Natural Philosophy shortly before Poynting's departure for Cambridge. On his election to a Fellowship at Trinity College in 1878, Poynting returned to Cambridge and began, in the Cavendish Laboratory under Clerk Maxwell, those experiments on the mean density of the earth which were destined to occupy so much of his time for the next 10 years. He remained at Cambridge until 1880, when he was elected to the Chair of Physics in Mason College, Birmingham (now the University of Birmingham), which had just been founded; this post he held until his death. The year that he went to Birmingham, he married the daughter of the late Rev. J. Cropper, of Stand, near Manchester. He threw himself whole-heartedly into the arduous duties connected with the starting of a new University College, the preparation of his lectures and the equipment of the physical laboratory, and, as was his wont, without any bustle or hurry he soon had things working efficiently. And so in the efficient discharge of his duties as a Professor, in successful original research, in the fulfilment of municipal duties, the time passed placidly on, the only cloud on an almost idyllic domestic life being his somewhat indifierent health, the first threatenings of the disease from which he ultimately died. To see if a country life would suit his health better than a town one, the Poyn tings moved from Edgbaston to Fox Hill, Alvechurch, a house about 12 miles out of Birmingham. There was a small farm attached to the house and Poynting entered into farming most heartily, though I am afraid he did not derive much pecuniary profit from it. But even farming when the agricultural depression was most acute could not impair his good temper or ruffle his equanimity. If the farm did not yield money, it gave new interests and experiences, and if something was always going wrong, at any rate it drove away monotony. The quietness and simplicity of the life were thoroughly to the taste of Mrs. Poynting and himself. Life in the country OBITUARY NOTICES XVll too gave free scope to his taste for Natural History, in which he always took great interest ; he was a keen and excellent observer, and a favourite conten- tion of his was that physicists were somewhat too much inclined to confine their observations to experiments made in the laboratory and did not sufficiently avail themselves of the opportunities of studying the physical phenomena going on in the sky, the sea, and the earth. The taste for Natural History was a family one; his brother, the late Mr. F. Poynting, was an excellent ornithologist, devoting himself especially to the study of the eggs of British birds, of which he made most careful and accurate water-colour drawings — some of these have been reproduced in his book The Eggs of British Birds. The Poyntings stayed at Foxhill until 1901, when, his health much improved, they returned to Edgbaston. His life at this time was a busy one, for in addition to the work demanded from him as the head of a large and successful School of Physics, he acted as the Dean of the Faculty of Science, was a Justice of the Peace, and for some time Chairman of the Birmingham Horticultural Society. He had also to plan and superintend the erection of a new physical laboratory when his department was transferred from its old quarters to the new buildings of the University of Birmingham. He went with the British Association to Canada in 1909, when it met at Winnipeg, and gave one of the evening lectures ; his subject was the Pressure of Light, on which he had been experimenting for several years. He went the trip to Vancouver and back and seemed thoroughly to enjoy the visit. The pressure of light was also the subject of a lecture which he gave in French at Paris before the French Physical Society at Easter, 1911. In the spring of 1912 a severe attack of influenza was followed by a recrudescence of diabetes, a disease from which he had suffered for some time, and he was ordered to take a long rest; he was, in consequence, away from Birmingham for two terms. On his return to Birmingham he seemed much better, he took an active part in the meeting of the British Association held there in September, 1913, and he and Mrs. Poynting entertained a large party of physicists at their house in Ampton Road, and it then seemed as if he might hope to enjoy many years of useful work. Another attack of influenza in the spring of 1914 brought on a very severe attack of diabetes, and he died on March 30, 1914. It is difficult to attempt to say what Poynting was to his friends without using terms which must appear exaggerated to those who did not know him. He had a genius for friendship, and a sympathy so delicate and acute that whether you were well or ill, in high spirits or low, his presence was a comfort and a delight. During a friendship which lasted for more than thirty years, I never saw him angry or impatient and never heard him say a bitter or unkind thing about man, woman or child. p. c. w. h Xviii OBITUARY NOTICES He took pleasure in many things, in music, in literature, for he was a lover of books and a collector in a modest way, in novels of all kinds, good and bad. He was fond of the country, and especially of North Wales, where he spent most of his vacations, but happiest of all when at home with his family. Throughout his life he took considerable interest in Philosophy, and a discussion of the philosophical basis of Physics formed part of his Presidential Address to Section A at the Dover Meeting of the British Association. Views similar to those he there expressed are now held by many ; he had formed his years before, when but few in this country agreed with them. The excellence of his work received many recognitions, though not in my opinion so many as it deserved. He was elected a Fellow of the Royal Society in 1888, received a Royal Medal in 1905, served on the Council from 1909 to 1911 and was Vice-President in 1910-11. He received the Adams Prize from the University of Cambridge in 1893, the Hopkins Prize from the Cambridge Philosophical Society in 1903. He was President of Section A when the British Association met at Dover in 1899 and was President of the Physical Society in 1909-11. He was in great request as an Examiner in Physics and no one excelled him at this work, his long experience of students, his judgment and common sense, the charitable view he took of the limitations of a student's knowledge, and the fact that he was never afraid of setting easy papers, made him an eminently fair and discriminating examiner. He was very successful as a teacher of students of all kinds, those who only took Physics as a subsidiary subject as well as those who made it their life's work, the latter he inspired with an enthusiasm for research, with some of his own skill in accuracy of measurement and with the desire for thoroughness in their work. Poynting's Scientific Work. This may be divided into four groups : {a) studies on gravitational attrac- tion, {b) on the change of state, (c) on the transfer of energy in the electro- magnetic field, and (d) on the pressure of light. Gravitational Attraction. His experiments on the mean density of the earth were commenced in Cambridge in 1878 but it took twelve years' steady work before he obtained a result with which he was satisfied. The method used was to measure the attraction between two known masses A and B by suspending A from one of the arms of a balance of the ordinary type and finding the increase in weight produced when B was brought underneath it. The balance used in the later experiments was one built specially for the experiment by Oertling and had a beam 123 cm. long. With a balance of this size the difficulties arising from air currents proved very formidable. Poynting fully recognised OBITUARY NOTICES XIX the advantage of Boys' short torsion balance method in this respect and said that if he were designing the apparatus again, instead of using an exceptionally large balance for the sake of being able to suspend large masses, he should go to the other extreme and make the apparatus as small as possible. At the same time, as he points out, the magnitude of the effects produced by the air currents made their detection easy, whereas they might have been over- looked and not allowed for had they been smaller. The final results (Phil. Trans., A, vol. clxxxii, p. 565, 1891) he obtained for A, the mean density of the earth, and G, the gravitational constant, were A = 54934, G = 6-6984 X 10-8. Poynting's long investigation incidentally added considerably to our know- ledge of the technique of accurate weighings. With the co-operation of Gray he made a series of most interesting experi- ments (Phil. Trans., A, vol. cxcii, p. 245, 1899) to see if the attraction between two quartz crystals was the same when the axes of the crystals were parallel as when they were crossed. The method he used was a very ingenious application of the principle of forced oscillations, which was so effective that, though one sphere was only about 1 cm. in diameter and the other about 6 cm., the experiments showed that the attractions in the two positions could not differ by as much as one part in 10,000. Later he made with Phillips a series of experiments to see if weight depended on temperature, using as in his first experiments a balance of the ordinary type; the result of these was (Proc. Roy. Soc, A, vol. Lxxvi, p. 445, 1905) that between 15° C. and 100° C. the change is not greater than 1 in 10^ and between 16*6° C. and - 186° C. it is not so great as 1 in 10i« per 1° C. Change of State. The problem of the change of state was one in which he took especial interest, and it was the subject of one of his earliest papers (Phil. Mag. (5), vol. XI, p. 32, 1887). His way of picturing this change was to suppose that from the surface of a liquid or solid particles were continually breaking free, so that through each unit of area of the surface there was a constant escape of molecules. This loss was balanced by the passage from the vapour above the solid of some of the gaseous particles which struck against its surface, so that when there was equilibrium the flow out from the liquid or solid was balanced by the flow inward from the gas. The proportion of gaseous mole- cules which after striking the surface passed across to the solid or liquid state he assumed to be the same for a solid as for a liquid and to be independent of the temperature, so that it could be measured by the vapour pressure. Thus at the same temperature the flow across water would be proportional to the vapour pressure of water, that across ice to the vapour pressure of 62 XX OBITUARY NOTICES ice, thus ice could only be in equilibrium with water when the vapour pressure over ice is equal to that over water. Poynting supposed that the mobility of the molecules in liquids and solids is increased by pressure — the pressure as it were squeezing the molecules out : the amount of the increase depending on the density of the substance, diminishing as the density increases. Thus, if pressure increases the escape of the molecules from a liquid, a liquid under pressure will evaporate more freely, and so for it to be in equilibrium with its vapour the vapour pressure must be higher than that over the normal liquid; from the equilibrium between water and its vapour in a capillary tube, he found that if Bp is the in- crease in the vapour pressure produced by applying a pressure P to the liquid, Sp =- Po/p, where o- is the density of the vapour and p that of the liquid. Poynting applied this conception of mobility to the case of solutions, taking the view that the molecules of the salt formed aggregates with some of the water molecules and thus diminished their mobility thereby diminishing the number of water molecules which passed from the liquid state through each unit of area of surface per second. The mobility of pure water is thus greater than that of the solution, so that if the two are separated by a semi- permeable membrane more molecules will pass from the water to the solution than from the solution to the water, and the water will flow into the solution. To prevent this flow the mobility of the molecules of water in the solution must be increased by the application of a pressure that will make the mobility of the solution equal to that of pure water; this pressure is the osmotic pressure. Since under this pressure the mobility of the solution is equal to that of pure water the vapour pressure in equilibrium with the pressed solution will be the vapour pressure over pure water, so that another definition of osmotic pressure would be the pressure required to raise the vapour pressure over the solution to that over pure water. On the assumption that the presence of one molecule of salt to n of water would diminish the mobihty of the water in the proportion of (n — l)/w, which would be the case if a molecule of salt imprisoned one and only one molecule of water, Poynting showed that the osmotic pressure on his theory would be the pressure exerted by the salt molecules if they were in the gaseous state and occupying the volume of the solution. Though this theory does not connect the electrical properties of solutions with the properties associated with osmotic pressure so readily as the dissociation theory, it is so simple and fundamental that it helps to give vividness and definiteness to our picture of the processes operative in solutions. Transfer of Energy. The researches by which Poynting is most widely known are those published in the papers " On the Transfer of Energy in the Electromagnetic Field" (Phil Trans., A, 1884), and "On Electric Currents and the Electric OBITUARY NOTICES * XXI and Magnetic Induction in the Surrounding Field" (Phil. Trans., A, 1888). He says in the first paper, " The aim of this paper is to prove that there is a general law for the transfer of energy, according to which it moves at any point perpendicularly to the plane containing the lines of electric and magnetic force, and that the amount crossing unit of area per second of this plane is equal to the product of the two forces multiplied by the sine of the angle between them divided by 47r, while the direction of the flow of energy is that in which a right-handed screw would move if turned round from the positive direction of the electromotive to the positive direction of the magnetic intensity." He shows from the equation of the electromagnetic field that the rate of increase in the energy inside a closed surface is equal to 1 1|[? (R^ -Qy) + m (Py - Ra) + n (Qa - PjS)] dS, where dS is an element of the closed surface, I, m, n the direction cosines of the normal to the surface, P, Q, R the components of the electromotive intensity, and a, jS, y those of the magnetic force. This expression may be regarded as showing that the energy flows across the surface, the components of flux being ^(R^-Qy), ^{Py-Ra). ^(Oa-P^); the vector which has those components is now universally known as Poynting's vector; it is at right angles to both the electric and magnetic forces and is proportional to the product of these forces and the sine of the angle between them. Thus when we can draw equipotential surfaces for both the electric and magnetic forces the energy flows along the lines of intersection of the two sets of surfaces. Poynting illustrates this theorem by applying it to the following cases : a constant current flowing along a straight wire, a con- denser discharged by short-circuiting the plates by a wire of great resistance, a voltaic battery, a thermo-electric circuit. The magnitude of the change in the point of view consequent on the principles brought forward in this paper is perhaps shown most clearly in the case of the discharge of the condenser and the transference of the energy which before the discharge was distributed between its plates into heat in the discharging circuit. Before the publication of this paper the general opinion was that the energy was transferred along the wire much in the same way as hydraulic power is carried through a pipe. On Poynting's yiew the energy flows out from the space between the plates and then converges sideways into the wire, where it is converted into heat, the paths of the energy being those represented in the figure. As shown in this figure the paths of energy near the wire are at right angles to it. This is not so unless the wire is such a bad conductor that the xxu OBITUARY NOTICES lines of electric force in its neighbourhood run parallel to it; if for example the current through the wire were an alternating one with very high frequency the electric force near the wire would be at right angles to it. In this case the energy would flow parallel to the wire but outside it. In the second paper Poynting, taking the view that the electromagnetic field consists of distributions of lines of electric and magnetic force, discusses the question of the transfer of energy from the point of view of the movement of these lines. He applies the same considerations to the question of the residual charge in Leyden jars in his fascinating and instructive paper on "Discharge of Electricity in an Imperfect Insulator"' {Phil. Mag., vol. v, 1886, p. 419). Poynting's vector occurs as a quantity of fundamental importance in many theories of electromagnetic action in which the subject is approached from a point of view somewhat different from the one he adopted. It appears, for example, as a measure of the momentum per unit volume when the electro- magnetic field is regarded as a mechanical system and the properties of the field as the result of the laws of motion of such a system. It appears, too, when we regard magnetic force as the result of the motion of tubes of electric force, the direction of motion of these tubes being parallel to Poynting's vector. Pressure of Light. For some years before his death Poynting devoted much attention to the question of radiation and the pressure of light. On the theory of this subject he published {Phil. Trans., A, vol. ecu) a very valuable paper, in the first part of which he discusses the application of the fourth-power law of radiation to determine the temperature of planets (in this he found afterwards he had been anticipated by Christiansen). Among other interesting results he arrived at the conclusion that the temperature of Mars must be so low that life, as we know it, would be impossible on its surface, this result was criticised OBITUARY NOTICES XXUl by Lowell, but Poynting maintained his ground in a paper published in the Philosophical Magazine, December, 1907. The second part of the paper in the Philosophical Transactions contains investigations of the repulsive force between two hot spheres which arises from the radiation from the one tending to repel the other. He showed that if the bodies are in radiation equilibrium with the Sun at the distance of the Earth from it, the repulsive effect will be greater than the gravitational attraction between them if their radii are less than 19-6 cm., if their density were that of water; if they were made of lead the corresponding radius would be 1-78 cm. Thus if Saturn's rings consisted of very small particles it is possible that the effect of radiation might make them repel instead of attract each other. He considers at the end of the paper the effect produced by radiation on the orbits of small bodies round the Sun and shows that this would ultimately cause them to fall into that body. To quote his own words : " The Sun cannot tolerate dust. With the pressure of his light he drives the finest particles altogether away from his system. With his heat he warms the larger particles. They give out this heat again and with it some of that energy which enables them to withstand his attraction. Slowly he draws them to himself and at last they unite with him and end their separate existence." {Pressure of Light, "Romance of Science" Series.) He made important contributions to the experimental side of the subject, thus with Dr. Barlow he established the existence of the tangential force produced when light is reflected from a surface at which there is some absorp- tion, and also the existence of a torque when light passes through a prism. They also succeeded in demonstrating the existence of the recoil from light of a surface giving out radiation : an account of these experiments was given in the Bakerian Lecture for 1910 {Proc. Roy. Soc, A, vol. lxxxiii, p. 534, 1910). These investigations involved the detection of exceeding minute forces and gave ample scope for Poynting's skill in devising methods and apparatus. He had exceptionally good mechanical instincts and an excellent knowledge of the capabilities of instruments; the result was that the apparatus he designed was always simple and effective. In addition to papers published in scientific journals and the Transactions of Societies he wrote The Mean Density of the Earth : the Adams Prize Essay for 1893, The Pressure of Light ("Romance of Science" Series) and The Earth (Cambridge University Press). Of the Text Book of Physics written in con- junction with J. J. Thomson he wrote the whole of the volumes on Sound and Heat and of the first volume of Electricity and Magnetism and the chapters on Gravitation in the Properties of Matter. His writings exhibit to the full the clearness, simplicity and thoroughness which was characteristic of all his work. J. J. T. Xxiv OBITUARY NOTICES [From the Philosophical Magazine for May, 1914, with additions.] Although Prof. Poynting, whose loss will be universally deplored, graduated with high distinction in mathematics at Cambridge, coming out as third Wrangler in the Tripos of 1876, his interest seems always to have lain in the direct elucidation of physical laws and principles rather than in the evolution and exposition of their consequences by analysis. When he came to Cambridge, in 1872, he was already largely trained in the niceties of refined experimentation ; and after graduation he embraced an early chance to resume experimental work at Manchester. The founding of the Mason University College gave him the opportunity of organising a laboratory of his own. Much work about this time was concerned with instrumental improvements, such as the design of polarimeters and other optical apparatus ; and to the same period belong studies in chemical physics, such as the eluci- dation of osmotic pressure, theories to which in later years he returned with conviction, and which, though perhaps not yet fully appreciated, should not be lost sight of, in view of his proved insight into fundamental problems in other domains. An exami^le of the latter class is the memoir on the transfer of energy in the electromagnetic field, Phil Trans., 1884, culminating in the famous result that will go down to posterity as Poynting's Theorem, which not only specifies the path of transfer of electric energy from one material system to another through the aether, but also as a very special case gives for the first time (strange to say) the dynamical specification of a ray of light. At about the same time 0. Heaviside, and a little later Hertz, were engaged upon this aspect of electric transmission as an elastic effect propagated from body to body across the aether, in place of the older aspect of electric charges in movement, each carrying its field of disturbance along with it, — which latter, rejuvenated ten years later by exact conceptions of the agency of electrons, and duly modified for chano^e of acceleration, now includes the whole field of view. But times not being yet ripe for that, he pursued his subject in 1885 in another Phil. Trans. Memoir "On the connexion between electric current and the electric and magnetic inductions in the surrounding field," which tracks out the relations of a current circuit by the graphical device of the motion of what are now known as Faraday tubes of force, a type of visualisation of the phenomena which is at the present time once more widely in favour. Afterwards he broke new ground in the experimental determination of the constant of gravitation — the problem of weighing the earth — which had been solved by Cavendish with his accustomed genius by aid of Michell's principle of balancing by torsion. To Poynting's mind an ordinary balance with lever and scale-pans gave at least equal promise of practical accuracy, and his long continued experimental investigations, which were summed OBITUARY NOTICES XXV up in a Phil, Trans. Memoir of 1892 and a Cambridge Adams Prize Essay of 1893, were the starting-point of a new interest in this subject which opened up into many methods more or less cognate to his own. By this time, however, the torsion method had renewed its power through the discovery of the pro- duction and properties of quartz fibres by C. V. Boys, whose remarkable subsequent investigation with small-scale apparatus was generously acknow- ledged by Poynting as the last word on the subject. The resource thus acquired in refined dynamical experimenting was to reap further successes in a more untilled field. The ancient idea of a pressure exerted by light, so obvious on the corpuscular view of optics, had been revived by Maxwell on a foundation of an accompanying electric stress in the trans- mitting medium. Its mere existence, as distinct from an analysis of its propagation to the place where it is in evidence, was already indubitably involved in the Amperean forces on the electric currents induced in the surface of a reflector, once the principles of the electric theory of light are admitted. It had assumed some importance in its application, notably by FitzGerald, to the elucidation of the mysterious phenomena of comets' tails. To Poynting, this pressure exerted by a ray coming say from a distant star, far out of reach of direct dynamical effect, involved that the ray carried momentum along with it, and that the pressure effect was of the nature of a thrust exerted along the ray arising from the transfer of this momentum. After long efforts, the disturbances arising from gaseous convection as a whole, and from the radiometric molecular effect, were eliminated by Lebedew, and were com- pensated by balance against each other by Nicholls and Hull, about the same time, and the Maxwell value of the normal component of the pressure was fully verified. But Poynting's line of thought led him straight to a tangential component of the thrust as well as a direct component, and he noted that the former could be investigated without much trouble from the gas-effect. This idea led to many beautiful determinations in conjunction with his assistants and students. His idea of convected momentum also led him in another direction to the conclusion that the pressure on a receding surface must be less than on one at rest; it also suggested that by reaction a moving radiating body would be accelerated by its own radiation— an impossible result which is corrected by recognising that its radiation is greater towards the direction of its motion than towards other directions, which leads to retardation on the whole. As this effect depends on extent of surface, it is greater in proportion for small bodies. Thus he was led to consider clouds of cosmic dust revolving orbitally round the sun, each particle heated by his rays to an equilibrium temperature of the space where it is, retarded in its motion by the reaction of its own exchanges of radiation, and thus gradually sucked into the sun. This clearance of solar spaces from dust must be a prominent feature in views of stellar cosmogony ; the calculation of the time XXvi OBITUARY NOTICES that would be required aptly illustrates his latent mathematical power, which was never unduly obtruded ; and the whole Memoir is an example of that simplification of reasoning and reduction to its lowest terms which is- suggestive of the depth of vision that belongs to genius. When the theory of electrons came to be developed into Maxwell's channels by Lorentz and others, it appeared at once that the stress argument, on which he had based radiation pressure, was in default, and the natural first conclusion was against the objective existence of the stress as thus specified in favour of some type too complex for simple expression. But later Poincare and Abraham introduced the idea of grouping the refractory outstanding terms as a distribution of electric momentum, specified very simply as the vector product of the aethereal and magnetic inductions. The stress in the medium is thus taken to be the sole operating cause : it is unbalanced, and so reveals itself partly in a distribution of mechanical forces exerted on the material bodies that are present, and partly in storage and expenditure of mechanical momentum of some latent type throughout the aether. This latter agrees precisely with the momentum of radiation elucidated on very simple inde- pendent grounds by Poynting. Interest in the subject is thus stimulated, and the problems now under discussion as to whether the effect is in all cases simply momentum, and whence arises the subsidiary travelling inertia which is implied in it, become of pressing interest. The application of the stress method to calculation over a boundary surrounding a material system' leads in fact to an additional result — that when the system gains energy SE of electric type, its effective mass increases by SE/c^, where c is the velocity of radiation : but this is less important practically, and would not for example affect sensibly the clearance of cosmical dust above mentioned,— though the idea that energy possesses inertia naturally assumes prominence in general relativity theory. An experimental and theoretical incursion into the different field of the elongation of a wire due to its torsion was probably prompted primarily by these problems ; tiiough not perhaps strictly pertinent to them, it opened up new views in the theory of elastic solids under stresses so great that mere superposition of strains no longer holds good. The formulation of an exact notion of the temperature of space, above indicated, is but one phase of his interest in the theory of natural radiation ; and It seems but yesterday that he was discussing, in private correspondence, with his usual acuteness and judgment and no sign of failure of powers, the theory of Stefan's law and its other fundamental relations. J. L. CONTENTS PAGE PREFACE V A PERSONAL NOTE vii OBITUARY NOTICES ix PART I THE BALANCE AND GRAVITATION ART. 1. On the Estimation of Small Excesses of Weight by the Balance from the Time of Vibration and the Angular Deflection of the Beam . 1 [Manchester Lit. Phil. Soc. Proc. 18, 1879, pp. 33-38. Read December 10, 1878.] 2. On a Method of using the Balance with great delicacy, and on its employment to determine the Mean Density of the Earth . 7 [Roy. Soc. Proc. 28, 1879, pp. 2-35. Received June 21, 1878.] 3. On a Determination of the Mean Density of the Earth and the Gravitation Constant by means of the Common Balance . 43 [Phil. Trans. A, 182, 1892, pp. 565-656. Received May 13. Read June 4, 1891.] 4. An Experiment in Search of a Directive Action of one Quartz Crystal on another. By J. H. Poynting and P. L. Gray, B.Sc. . 137 [Phil. Trans. A, 192, 1899, pp. 245-256. Received September 27. Read November 17, 1898.] 5. An Experiment with the Balance to Find if Change of Temperature has any Effect upon Weight. By J. H. Poynting and Percy Phillips, M.Sc 149 [Roy. Soc. Proc. A, 76, 1905, pp. 445-457. Received July 12, 1905.] 6. On a Method of Determining the Sensibility of a Balance. By J. H. Poynting and G. W. Todd, M.Sc 162 [Phil. Mag. 18, 1909, pp. 132-135. Read June 25, 1909.] PART II ELECTRICITY 7. On the Law of Force when a Thin, Homogeneous, Spherical Shell exerts no Attraction on a Particle within it . . . .165 [Manchester Lit. Phil. Soc. Proc. 16, 1877, pp. 168-171. Read March 6, 1877.] 8. Arrangement of a Tangent Galvanometer for lecture room purposes to illustrate the Laws of the Action of Currents on Magnets, and of the Resistance of Wires 168 [Manchester Lit. Phil. Soc. Proc. 18, 1879, pp. 85-88. Read April 1, 1879.] XXViii CONTENTS ART. PAGE 9. On the Graduation of the Sonometer ... . . 170 [Phil. Mag. 9, 1880, pp. 59-64. Read before the Physical Society, December 13, 1879.] 10. On the Transfer of Energy in the Electromagnetic Field . . 175 [Phil. Trans, lib, 1884, pp. 343-361. Received December 17, 1883. Read January 10, 1884.] 11. On the Connection between Electric Current and the Electric and Magnetic Inductions in the Surrounding Field . . .194 [Phil. Trans. 176, 1885, pp. 277-306. Received January 31. Read February 12, 1885.] 12. Discharge of Electricity in an Imperfect Insulator . . . 224 [Birmingham Phil. Soc. Proc. 5, (1885), pp. 68-82. Read December 10, 1885.] 13. On the Proof by Cavendish's Method that Electrical Action varies Inversely as the Square of the Distance 235 [British Association Report, 1886, pp. 523-524.] 14. On a Form of Solenoid-Galvanometer 237 [Birmingham Phil. Soc. Proc. 6, (1888), pp. 162-167. Read May 10, 1888.] 15. On a Mechanical Model, illustrating the Residual Charge in a Dielectric 242 [Birmingha?n Phil. Soc. Proc. 6, (1888), pp. 314-317. Read November 8, 1888.] 16. Electrical Theory. Letters to Dr Lodge 245 [Electrician, 21, 1888, pp. 829-831.] 17. An Examination of Prof. Lodge's Electromagnetic Hypothesis . 250 [Electrician, 31, 1893, pp. 575-577, 606-608, 635-636.] 18. Molecular Electricity ....... 269 [Electrician, 35, 1895, pp. 644-647, 668-671, 708-712, 741-743.] PART III WAVE PROPAGATION— RADIATION— PRESSURE OF LIGHT— AND RELATED SUBJECTS 19. Note on an Elementary Method of Calculating the Velocity of Pro- pagation of Waves of Longitudinal and Transverse Disturbances by the Rate of Transfer of Energy 299 [Birmingham Phil. Soc. Proc. 4, (1885), pp. 55-60. Read November 8, 1883.] Radiation in the Solar System : its Effect on Temperature and its Pressure on Small Bodies 3Q4 [Phil. Trans. A, 202, 1903, pp. 525-552. Received June 16. Read June 18, 1903.] 20 CONTENTS XXIX ART. PAGE 21. Note on the Tangential Stress due to Light incident obliquely on an Absorbing Surface 332 [Phil. Mag. 9, 1905, pp. 169-171. Read at Section A, British Association, Cambridge, August, 1904.] 22. Radiation-Pressure 335 [Phil. Mag. 9, 1905, pp. 393-406. Presidential Address, delivered at the Annual General Meeting of the Physical Society, February 10, 1905.] 23. On Prof. Lowell's Method for Evaluating the Surface-Temperatures of the Planets ; with an Attempt to Represent the Effect of Day and Night on the Temperature of the Earth .... 347 [Phil. Mag. 14, 1907, pp. 749-760.] 24. The IVEomentum of a Beam of Light 357 [Atti del IV Congresso internazionale dei Matematici (Rome), 3, 1909, pp. 169-174.] 25. On Pressure Perpendicular to the Shear-Planes in Finite Pure , Shears, and on the Lengthening of Loaded Wires when Twisted 358 [Roy. Soc. Proc. A, 82, 1909, pp. 546-559. Read June 24, ]909.] 26. The Wave-Motion of a Revolving Shaft, and a Suggestion as to the Angular Momentum in a Beam of Circularly Polarised Light . 372 [Boy. Soc. Proc. A, 82, 1909, pp. 560-567. Read June 24, 1909.] 27. Preliminary Note on the Pressure of Radiation against the Source : The Recoil from Light. By J. H. Poynting and Guy Barlow, D.Sc 380 [British Association Beport, 1909, p. 385.] 28. Bakerian Lecture. The Pressure of Light against the Source : The Recoil from Light. By J. H. Poynting and Guy Barlow, D.Sc. 381 [Boy. Soc. Proc. A, 83, 1910, pp. 534-546. Read March 17, 1910.] 29. On Small Longitudinal Material Waves accompanying Light- Waves 394 [Boy. Soc. Proc. A, 85, 1911, pp. 474-476.] 30. On the Changes in the Dimensions of a Steel Wire when Twisted, -^ and on the Pressure of Distortional Waves in Steel . . . 397 [Boy. Soc. Proc. A, 86, 1912, pp. 534-561. Read March 21, 1912.] 31. The Changes in the Length and Volume of an India-Rubber Cord when Twisted 424 [The India-Bubber Journal, October 4, 1913.] Appendix by Sir J. Larmor on the Momentum of Radiation . 426 XXX CONTENTS PART IV LIGHT ART. PAGE 32. On a Simple Form of Saccharimeter 435 [Phil. Mag. 10, 1880, pp. 18-21.] 33. On the Law of the Propagation of Light. By J. H. Poynting and E. F. J. Love, B.A. 438 [Birmingham Phil. Soc. Proc. 5 (1887), pp. 354-363. Read March 31, 1887,] 34. Haze 446 [Nature, 39, 1889, pp. 323-324.] 35. A Graphical Method of Explaining the Diffraction Bands at the Edge of a Shadow ......... 449 [Birmingham Phil. Soc. Proc. 7 (1890), pp. 210-219. Read November 5, 1890.] 36. On a Parallel -Plate Double-Image Micrometer .... 455 [Roy. Astr. Soc. Monthly Notices, 52, 1892, pp. 556-560.] 37. Historical Note on the Parallel-Plate Double-Image Micrometer 460 [Eoyal Astr. Soc. Monthly Notices, 53, 1893, p. 330.] 38. A Method of Making a Half -Shadow Field in a Polarimeter by two inclined Glass Plates ......... 462 [British Association Report, 1899, pp. 662-663.] PART V MISCELLANEOUS 39. Change of State: Solid-Liquid 454 [Phil. Mag. 12, 1881, pp. 32-48, 232.] 40. Note on a Method of Determining Specific Heat by Mixture . 481 [Birmingham Phil. Soc. Proc. 4 (1883), pp. 47-54. Read November 8, 1883.] 41. Osmotic Pressure 4gg [Phil. Mag. 42, 1896, pp. 289-300.] 42. Musical Sands 4gg [Nature, 77, 1908, p. 248.] PART VI STATISTICS 43. The Drunkenness Statistics of the Large Towns in England and Wales ^g^ [Manchester Lit. and Phil. Soc. Proc. 16, 1877, pp. 211-218. Read April 3, 1877.] CONTENTS XXXI ART. PAGE 44. The Geographical Distribution of Drunkenness in England and Wales. By J. H. Poynting and John Dendy, Jun. . . 504 {^Fourth Report from the Select Committee of the House of Lords on Irdemperance 1878. Appendix R, pp. 580-591.] 45. A Comparison of the Fluctuations in the Price of Wheat and in the Cotton and Silk Imports into Great Britain .... 506 [Statistical Society Journal, 1884. Read before the Statistical Society, 15 January, 1884.] PART VII ADDRESSES AND GENERAL ARTICLES 46. Change of State: Fusion and Solidification .... 538 [Birmingham, Phil. Soc. Proc. 2 (1881), pp. 354-372. Read May 12, 1881.] 47. Overtaking the Rays of Light 552 [Mason College Magazine, 1, 1883, pp. 107-111.] 48. University Training in our Provincial Colleges. An Address delivered at the Mason Science College, Birmingham, Oct. 2, 1883 557 49. The Growth of the Modern Doctrine of Energy. Address to the Mason College Physical Society, March 26, 1884 ... 565 50. The Electric Current and its Connection with the Surrounding Field 576 [Birmingham Phil. Soc. Proc. 5 (1887), pp. 337-353. Read March 10, 1887.] 51. The Foundations of our Belief in the Indestructibility of Matter and the Conservation of Energy. A Criticism of Spencer's 'First Principles,' Part II, Chaps. IV, V, and VI . . . 588 [Midland Naturalist, 12, 1889. Read before the Sociological Section of the Birmingham Natural History and Microscopical Society, November 22, 1888.] 52. Presidential Address to the Mathematical and Physical Section of the British Association (Dover), 1899 .... 599 [British Association Report, 1899, pp. 615-624.] 53. A History of the Methods of Weighing the Earth. Presidential Address delivered to the Birmingham Philosophical Society, October 19, 1893 613 [Birmingham Phil. Soc. Proc. 9 (1894), pp. 1-23.] 54. The Mean Density of the Earth [Letter] 628 [Nature, 48, 1893, p. 370.] 55. Recent Studies in Gravitation. Address: Royal Institution of Great Britain, February 23, 1900 629 [Roy. Inst. Proc. 16, 1900-02, pp. 278-294.] XXXll CONTENTS ART. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 284-300. Le mode de propagation de I'Energie et de la tension Electrique dans le champ Electromagnetique .... [Rapport presente au Congres International de Physique de 1900, 3, pp. Paris, Gauthier-Villars.] The Transformation and Dissipation of Energy . [The Inquirer, 1902, pp. 627-628.] Molecules, Atoms and Corpuscles .... [The Inquirer, 1902, pp. 740-741, 772-773.] The Pressure of Light [The Inquirer, 1903, pp. 195-196.] Mysteries of Matter. Radium at the British Association [The Inquirer, 1903, pp. 635-636.] A City University [Letter] [The Inquirer, 1903, p. 660.] The Universities and the State [The Inquirer, 1903, p. 779.] Physical Law and Life [Hibhert Journal, 1, 1903, pp. 728-746.] Radiation in the Solar System. Afternoon address delivered at the Cambridge meeting of the British Association, August 23, 1904 [Nature, 70, 1904, pp. 512-515.] Radiation-Pressure [Letter in correction to above address] [Nature, 71, 1904, pp. 200-201.] Radiation-Pressure. Presidential Address to the Physical Society of London, February 1905. See Part III, Art. 22 . . . Some Astronomical Consequences of the Pressure of Light. Dis- course delivered at the Royal Institution on May 11, 1906 [Nature, 75, 1906, pp. 90-93.] George Gore, 1826—1908 [Roy. Soc. Proc. 84, 1911, pp. xxi-xxii.] Atomic Theory (Mediaeval and Modern) [Encyclopaedia of Religion and Ethics, 2, 1909, pp. 203-210.] Quelques experiences sur la Pression de la lumiere. Address to the French Physical Society, March 31, 1910 .... [Bulletin des seances de la Societe frangaise de Physique, 1, 1910.] POSTSCRIPT (1918). Retardation by Radiation Pressure: A correction. By Sir Joseph Larmor, F.R.S BIBLIOGRAPHY INDEX PAGE 645 658 664 673 677 682 683 686 699 708 711 712 722 724 742 754 758 764 PART I. THE BALANCE AND GRAVITATION. 1. ON THE ESTIMATION OF SMALL EXCESSES OF WEIGHT BY THE BALANCE FROM THE TIME OF VIBRATION AND THE ANGULAR DEFLECTION OF THE BEAM. [Manchester Lit. Phil Soc. Proc. 18, 1879, pp. 33-38.] [Read Dec. 10, 1878.] While working last year on an experiment to determine the mean density of the earth by the balance, I had to measure such an exceedingly small difference of weight that I could not at that time estimate it by means of a rider, but was obliged to adopt the method described in this paper. Stated generally, it consists in treating the balance as a pendulum. Knowing the nature of the pendulum (that is, its moment of inertia) and its time of vibration, we can calculate what force acting at the end of one arm of the beam will produce a given angular deflection. It is, in fact, an application to the common balance of the method which has always been used with the torsion-balance when it has been necessary to calculate the forces measured in absolute measure. I cannot find any record of a previous application of the method ; and as it might be of use in very delicate weighings or in verifying the small weights in a laboratory, I have thought it worth while to give a full account of it. When small quantities of the second order are neglected and the oscillations are of the first order, it will easily be found that the equation of motion of the beam of the balance is (mP + ^^) 6 + (2Ph + Mgk) d = ap, (1) p. o. w. 1 2 ON THE ESTIMATION OF SMALL EXCESSES OF WEIGHT where MP = moment of inertia of beam about central knife-edge, M = mass of beam, a = half length of beam, p = weight of either pan and the mass in it, h = distance of Hne joining terminal knife-edges below the central knife-edge, k = distance of centre of gravity of beam below central knife-edge, p = small excess in one pan, 6 = angular deflection in circular measure produced by p, 9 = gravity. If 6* = 0, we have the position of equihbrium given by ft- ^P (2) 2Ph + Mgk ^ ^ The semiperiodic time is MI ^^^\/ mrrwk ^^^ From equations (2) and (3) we can eliminate 2Ph -f- Mgk, obtaining ^MgP + 2Pa^d ... P^TT^ -^ -^ (4) ■^ ag t^ ^ From this expression it appears that, if we know the moment of inertia of the beam, its length, and the weight at each end, we can find the excess p from the time of vibration and deflection. The results given in this paper were obtained with a 16-inch chemical balance by OertUng. The exact length of the half beam {a) measured by a dividing-engine is 20-2484 centimetres. To find the Moment of Inertia MP of the Beam. The simplest way theoretically would appear to be this. Find the times of vibration t^, t^, and the deflections 6^, d^, due to the same excess p with two different loads Pj, P2 in each pan. Equating the values of p given for the two by equation (4) we have MgP + 2P^a^ d^ti" MgP + 2P2a2 d^t^^ ' an equation which will give MgP in terms of known quantities ; but on trial it was found that a very small proportional error in the observed time made a large error in the value of MgP ; and the following method, that usually adopted in magnetic observations, was employed in preference. A stirrup was suspended by a platinum wire, and its time of vibration (^1) against the force of torsion (/x) of the wire was observed. The moment of inertia of the stirrup being S, we have tj fX ON THE ESTIMATION OF SMALL EXCESSES OF WEIGHT 3 The time of vibration (^2) was then observed when a cylindrical brass bar of known moment of inertia (B) was inserted in the stirrup. We now have The bar was then removed and the balance-beam inserted in its place ; and the time of vibration (/^) gives t^^=-(S+MI^). From these three equations, eliminating S and /j,, we obtain 3JP Bjh'-t^) Now Eg was calculated from the weight and dimensions of the bar to be 6332-83 (in centimetres and grammes). The observed times were t^ = 3-6792 sees., t^ = 4-495 sees., ^3 - 7-1483 sees. From these values we find M^/2 = 35651-6*. To measure 9. The angle of deflection was measured by the number of divisions of the scale which the pointer moved over. As the length of the pointer is 32-1006 centimetres, while 20 divisions of the scale measure 2-5658 centimetres, a tenth of a division, in terms of which the deflection was measured, corresponds to an angle of 0-0003996. The oscillations were observed from a distance of six or eight feet by a telescope. The resting- point (i.e. the point where the balance would be in equilibrium) was found in the usual way by observing three successive extremities of two swings and taking the mean of the second and the mean of the first and third. Five determinations of the resting-point were usually made with the excess to be measured alternately added and removed. From these five, three values of the deflection {n) due to the excess were calculated in a manner which will be seen from the example below. The Time of Vibration. This was found from several determinations of the time of ten oscillations. The method will be seen from the example. No correction was needed for the resistance of the air as long as the vibrations did not exceed two divisions of the scale. When, however, they were much more than that, the time of vibration was found to increase with the arc. As the time of vibration frequently changes slightly, probably through variations of temperature, it was usually observed before- and after the determination of the deflection (n) and the mean of the two taken as the true time. * To this a small correction should be added if the adjusting-bob is not in its lowest position. This amounts to 7-6 for each turn of the screw, and may therefore in general be neglected. 1—2 4 ON THE ESTIMATION OF SMALL EXCESSES OF WEIGHT The following example of the determination of the value of a centigramme- rider by placing it halfway along the beam will sufficiently explain the details of the method. Time of Vibration at Commencement. No. of vibration Observed time of passage of pointer through resting- point No. of vibration Observed time of passage of pointer through resting- point 1 Time of 10 vibrations Pointer apparently moving from left to right 2 4 6 h. m. s. 11 15 36 11 16 1 11 16 26-5 11 16 52 10 12 14 16 h. m. s. 11 17 43 11 18 8 11 18 33-5 11 18 59 s. 127 127 127 127 Mean value of 10 vibrations 127 Pointer ajiparently moving from right to left 1 3 5 7 11 15 49 11 16 14 11 16 39-5 11 17 5 11 13 15 17 11 17 56 11 18 21 11 18 46 11 19 11-5 127 127 126-5 126-5 Mean value of 10 vibrations 126-75 Mean of means = 126-875 ; t^ = 12-6875 sees. Determination of Deflection (n). Excess weight Extremities of oscillation Resting- point Mean of preceding and succeeding resting-points Deflection due to excess Added 109 96 109 102-5 Removed 93 40 92 6625 102-25 35-25 Added 152 53 150 102 66-75 36 Removed 80 67-25 102-5 35-25 Added 147 60 145 103 Mean value of n = 35-83. ON THE ESTIMATION OP SMALL EXCESSES OF WEIGHT Time of Vibration at End. I No. of vibration Observed time of passage of pointer through resting- point No. of vibration Observed time of passage of pointer through resting- point Time of 10 i vibrations Pointer apparently moving from left to right 2 4 6 h. m. s. 11 26 19 11 26 44-5 11 27 10 11 27 35-5 10 12 14 16 h. m. s. 11 28 27 11 28 53 11 29 18 11 29 44 s. 128 128-5 128 128-5 Mean value of 10 vibrations 128-25 Pointer apparently moving from- right to left 1 3 5 7 11 26 32-5 ; 11 26 58 i 11 27 23-5 11 27 49 11 13 15 17 11 28 39 11 29 5 11 29 30-5 11 29 56-5 126-5 127 127 127-5 Mean value of 10 vibrations . .. 127 Mean of means = 127-625 ; ^2 = 12-7625 sees. Remembering that one-tenth of a division of the scale is an angle of •0003996 in circular measure, formula (4), expressed in milligrammes, becomes f = '% 0-3996 — {Mgl^ + 2Pa2). ^ t^ ag ^ In our present example' n = 35-83, ^ = ^^t^2= 12-725 sees., Mgl^ = 35651, 2Pa^ = 94704, p = 5-724 milligrammes. The length of time occupied in this determination was not quite a quarter of an hour. * For this, as for several other cases, I removed the pans and hung the weights directly by fine wires from the suspending-pieces. By this means the resistance of the air was very much diminished. 6 ON THE ESTIMATION OF SMALL EXCESSES OF WEIGHT The following table contains a series of results which I have obtained of the weight of two centigramme-riders, the first of which was accidentally destroyed after the conclusion of the fourth determination. As the rider was always placed at division 5 on the beam, the values given in the table are double those actually obtained. No. of experi- ment Mgl^ + 2Pa2 t in seconds n Weight of rider in milli- grammes Mean value 1 145364 8-921 13-458 9-78^ 2 3 309356 519769 17-65 20-435 25-49 19-12 10-05 1 9-55 1 9-96 milligrammes 4 130355 1310 34-71 10-47 j 5 130355 12-87 36-6 11-44) 6 130355 12-72 35-5 11-35 7 8 130355 130355 12-725 1281 35-83 35-5 11-45 11-20 11-35 milligrammes 9 130355 12-903 36-37 11-31 10 454405 19-406 22-08 10-58J 2. ON A METHOD OF USING THE BALANCE WITH GREAT DELICACY, AND ON ITS EMPLOYMENT TO DETERMINE THE MEAN DENSITY OF THE EARTH. [Roy. Soc. Proc. 28, 1879, pp. 2-35.] [Received June 21, 1878.] In the ease and certainty with which we can determine by the balance a relatively small difference between two large quantities, it probably excels all other scientific instruments. By the use of agate knife-edges and planes, even ordinary chemical balances have been brought to such perfection that they will indicate one-milHonth part of the weight in either pan, while the best bullion-balances are still more accurate. The greatest degree of accuracy which has yet been attained was probably in Professor Miller's weighings for the construction of the standard pound, and its comparison with the kilogramme, in which he found that the probable error of a single comparison of two kilogrammes, by Gauss's method, was TiWWoo*^ P^^^ ^^ ^ kilogramme*. (Phil. Trans. 1856.) But, though the balance is peculiarly well fitted to detect the relatively small differences between large quantities, it has not hitherto been considered so well able to measure absolutely small quantities as the torsion-balance. The latter, for instance, was used in the Cavendish experiment, when the force measured by Cavendish was the attraction of a large lead sphere upon a smaller sphere, weighing about 1 Jibs., the force only amounting to 5 000^0000" ^^ part of this weight, or about 50^0^^ P^^^ ^^ ^ grain. The two great sources of error, which render the balance inferior to the torsion-balance in the measurement of small forces, are : 1. Greater disturbing effects produced by change of temperature, such as convection-currents and an unequal expansion of the two arms of the balance. 2. The errors arising from the raising of the beam on the supporting frame between each weighing, consisting of varying flexure of the beam and inconstancy of the points of contact of the knife-edges and planes. * Even so far back as 1787, Count Rumford used a balance which would indicate one in a million and measure one in seven hundred thousand. {Phil. Trans. 1799.) 8 ON A METHOD OF USING THE BAIjANCE The disturbances due to convection-currents interfere with the torsion- balance as well as with the ordinary balance, though they are more easily guarded against with the former, by reason of the nature of the experiments usually performed with it. They might, perhaps, as has been suggested by Mr. Crookes, be removed from both by using the instruments in a partial vacuum, in which the pressure is lowered to the 'neutral point,' where the convection- currents cease, but the radiometer- effects have not yet begun. But a vacuum-balance requires such comphcated apparatus to work it, that it is perhaps better to follow the course which Baily adopted in the Cavendish experiment. He sought to remove the disturbing forces as much as possible, and to render those remaining as nearly uniform as possible in their action during a series of experiments, so that they might be detected and eliminated. For this purpose the instrument was placed in a darkened draughtless room, and was protected by a thick wooden casing gilded on its outer surface. Most of the heat radiated from the surrounding bodies was reflected from the surface of the case by the gilding. The heat absorbed only slowly penetrated to the interior, and was so gradual in its action, that, for a considerable time, the effect might be supposed nearly uniform. Under this supposition it was then eliminated by the following method of taking the observations. The resting-point (that is, the central position of equilibrium, about which the oscillations were taking place) of the torsion-rod, at the ends of which were the small attracted weights, was first observed when the two large masses pulled it in one direction. The masses were then moved round to the opposite side, when they pulled the rod in the opposite direction and the resting-point was again observed. The masses were then replaced in their original position and the resting-point was observed a third time. These three observations were made at equal intervals of time; if, then, the disturbing effect was uniform during the time, the mean of the first and third observations gave what the resting-point would have been, had the rod been pulled in that one direction at the same time that it was actually observed when pulled in the opposite direction. The difference between the second resting-point and the means of the first and third might, therefore, be considered as due to the attractions of the masses alone. In the experiments of which this paper contains an account, I have en- deavoured to apply this method of introducing time as an element to the ordinary balance. But, before it could be properly applied, it was necessary to remove the errors due to the raising of the beam between successive weighings, as they could not be considered to vary in any uniform way with the time. I think I have effected this satisfactorily, by doing away altogether with the raising of the beam by the supporting frame, between the weighings. For this purpose I have introduced a clamp underneath one of the pans, which the observer can bring into action at any time, to fix that pan in whatever position it may be. The weight can then be removed from WITH GREAT DELICACY, ETC. 9 the pan, and another, which is to be compared with it, can be inserted in its place without altering the relative positions of the planes and knife-edges. The counterpoise in the other pan, meanwhile, keeps the beam in the same state of flexure. The pan is then undamped and the new position about which it oscillates is observed. The only changes are due to the change in the weight and the effect of the external disturbing forces ; the latter we may- consider as proportional to the time, if sufficient precautions have been taken, and by again changing the weights and again observing the position of the balance, we may eliminate their effects. Though the method when applied to the balance does not yet give such good results as Baily obtained from the torsion-balance — partly, I believe, because I have not yet been able to apply all his precautions to remove external disturbing forces — ^it still gives better results than would have been obtained without it. This may be seen by the numbers recorded in the tables, where a progressive motion of the resting-point may be noticed, in most cases in the same direction, during a series of experiments. Even when this is not the case, the method at once shows when the disturbing forces are irregular, and when we are justified in rejecting an observation on that account. I give in this paper two applications of the method, one to the comparison of two weights, the other to the determination of the mean density of the earth. The latter is given only as an example of the method, but I hope shortly to continue the experiments with a large bullion-balance, for the construction of which I have had the honour to obtain a grant from the Society. The balance is now in course of construction, by Mr. Oertling, of London. Description of the Apparatus. The balance which I have employed is one of Oertling's chemical balances, with a beam of nearly 16 inches, and fitted with agate planes and knife-edges. It will weigh up to a little more than 1 lb. To protect it from sudden changes of temperature, the glass panes of the case are covered with flannel, on both sides of which is pasted gilt paper, with the metallic surface outwards. This case is enclosed in another outer case, a large box of inch deal, lined inside and out with gilt paper. The experiments have been conducted in a darkened cellar under the chemical laboratory at Owens College, which was kindly placed at my disposal by Professor Roscoe. As the ceilings and floors of the building are of concrete, any movement near the room causes a considerable vibration of the floor and walls. It was necessary, therefore, to support the balance independently of the floor. For this purpose, six wooden posts (A, B, C, D, Ey F, Fig. 1) were erected resting on the ground underneath and passing freely through the floor to a height of 6 feet 6 inches above it. They are connected at the top by a frame like that of the table, and stayed against each other to give firmness. The wider part of the frame, near the posts 10 ON A METHOD OF USING THE BALANCE E and F, is boarded over to form a table for the telescope {t, Fig. 1) and scale (s), by which the oscillations of the balance are observed. The box con- taining the balance rests on two cross-pieces, on the narrower part, ABCD, of the frame, with the beam parallel to AD, and its right end towards the telescope. In order to observe the position of the beam, a mirror, 1 J inches by f inch, is fixed in the centre of the beam, and the reflection of a vertical scale (s, Fig. 1) in this is viewed with a telescope (t) placed close to the scale. The light from the scale passes through two small windows cut in each of the cases of the balance and glazed with plate glass. The position of the beam Fig. 1. is given by the division of the scale upon the cross-line on the eyepiece of the telescope. The scale, which was photographed on glass, and reduced from a large scale, drawn very carefully, has 50 divisions to the inch. These are ruled diagonally with ten vertical cross-lines. It is possible to read, with almost certainty, to a tenth of a division, or ^i^th of an inch. Since the mirror is about 6 feet from the scale, a tenth of a division means an angular deflection of the beam of about 3"*. The scale is illuminated from behind by a mirror (m), several inches in diameter, which reflects through it a parallel beam from a paraffin lamp (Q. * The numbers on the scale run from below upwards, so that an increase in the weight in the right-hand pan is indicated by a lower number on the scale. WITH GREAT DELICACY, ETC. 11 A plate of ground glass between the scale and mirror diffuses the light evenly over the scale and, by altering the position of the mirror, any desired degree of brilliancy may be given to the illumination of the scale. A screen (not shown in Fig. 1) prevents stray light from striking the balance-case. This method of reading — which, of course, doubles the deflection — has been so far sufficiently accurate for my purpose; that is to say, the errors arising from other sources are far greater than those arising from imperfections of reading. But in a long series of preliminary experiments I used the following plan to multiply the deflection still further. A rather smaller fixed mirror, ah, is placed opposite to and facing the beam-mirror, AB, fixed on the beam, and a few inches from it. Suppose the beam-mirror to be deflected from the position BL, parallel to ah, through an angle, 6, to the position AB. If a ray, PQ, perpendicular to ah strikes AB at Q, it will make an angle 6 with QM, the normal at Q, and will be reflected along QR, making an angle 29 with its original direction, and therefore with the normal RO, at R, when it strikes it. If it be reflected again to AB at S, it will make an angle 3^ with the normal SN, and the reflected ray, ST, will make an angle 4^ with the original direction, PQ, of the ray. It may be still further reflected between the two mirrors, if desirable, each reflection at the mirror, AB, adding 2^ to the deflection of the ray. I have, for instance, employed three reflections from the beam-mirror, so multiplying the deflection six times. In this case, one division of my scale, at the distance at which it was placed from the beam, corresponded to a deflection of 7" in the beam, and this could be subdivided to tenths by the eye. The only limit to the multiplication arises from the imperfection of the mirrors and the decrease in the illumination of the successive reflections*. The chair of the observer is placed on a raised platform, and a small table rising from the platform and free from the frame on which the instruments rest, is between the observer and the telescope. On this he can rest his note- book during an experiment. As the differences of weight observed are some- times exceedingly minute, the balance is made very sensitive— usually * This method was used in the seventh and eighth series here recorded. Two reflections from the beam-mirror were employed, giving four times the actual deflection. 12 ON A METHOD OF USING THE BALANCE vibrating in periods between 30 sees, and 50 sees. The value of a division of the scale cannot be determined by adding known small weights to one pan, as the deflection would usually be too great. Any approach of the observer to the case causes great disturbances, so that the ordinary method of moving a rider an observed distance along the beam is inapplicable. In some experi- ments made last year I calculated the force equivalent to the small differences in weight, in absolute measure, by observing the actual angular deflection and the time of vibration. With a knowledge of the moment of inertia of the beam and treating it as a case of small oscillations, it was possible to calculate the value of the scale. But the observations and subsequent calculations were so complicated that the following method of employing riders was ultimately adopted. {a) (h) (c) id) A small bridge about an inch long (Fig. 2 a) is fitted on to the beam. The sides of the bridge are prolonged about half an inch above the arch which fits on to the beam, as shown in the end view (Fig. 2b). In each of these sides are cut two V-shaped notches directly opposite to each other, one of the opposite pairs being 6-654 millims. (about J inch) distant from the other pair. Two equal riders of the shape shown in Fig. 2 c are placed across the bridge, and are of such a size that they will just fit into the bottom of the notches. When one of these rests across the bridge the other is raised up WITH GREAT DELICACY, ETC. 13 from it. The lowering of one rider and the raising of the other corresponds therefore to a transference of a single rider from one pair of notches to the other. The length of the half beam being 202-716 millims. and the distance between the notches 6-654 millims., this transference will be equivalent to the addition to one pair of 0-03282 of the weight of the rider used. As I have generally used a centigramme-rider this means 0*3282 mgm. Two levers I, I' (Fig. 2d), with hooks h, h' are used to raise one rider while the other is lowered. These levers are worked by two cams c, c' on a rod R, which is prolonged out of the balance-case to the observer. By turning this rod round, the one lever is raised while the other is depressed. The hook at the end of the raised lever picks up its rider while the other hook deposits its rider on the bridge, and then sinks down between the raised sides (as shown in Fig. 2d), leaving the rider resting freely on the bridge. The levers are so adjusted that the beam even in its greatest oscillations never comes in contact with the hooks. This arrangement might probably be still further perfected by introducing two small frames for the riders to rest upon, the frames resting on the beam by knife-edges. It would then be certain that the movement of the riders was equivalent to a transference from one knife-edge to the other, whereas the rider at present may not rest exactly over the centre of the notch. But I find that I get fairly consistent results by lowering the rider somewhat suddenly so as to give it sufficient impetus to go to the bottom of the notch, and have not therefore thought it necessary as yet to introduce more com- plicated apparatus. In place of the right-hand pan of the usual shape, another of the shape shown in Fig. 3 a is employed. To the centre of the pan underneath is I ^ n /// (a) (&) 3. (c) attached a vertical brass rod which passes downwards through the bottom of the inner case of the balance. To the under side of this case is attached 14 ON A METHOD OF USING THE BAI^ANCE the clamping arrangement before referred to. This consists of two sliding pieces (Fig. ia, s, s) working horizontally in a slot cut in a thick brass plate which is fastened to the case. Through a circular aperture in this plate (the slot is not cut through the whole thickness of the plate, but only as shown in Fig. 46) and about the middle of the slot hangs the rod r attached to the scale-pan. (a^ By means of right and left handed screws on a rod R, which is prolonged out of the case to the observer, these two sliding- pieces can be made to approach, and clamp the rod, or to recede and free it. By having the opposite surfaces of the sliding- pieces and the rod polished and clean, it is possible to clamp and unclamp without producing any disturbance. The clamp is of great use also to lessen the vibrations when they are too large, as it may be brought into action at any moment, and on releasing carefully the beam will start again from rest without any impetus. It may be used too to increase the vibrations by releasing suddenly, when the beam will have a slight impetus in one direction or the other. The weights which I have compared are two brass pounds avoirdupois, made for me by Mr. Oertling, and marked A and B respectively. They are WITH GREAT DELICACY, ETC. 15 of the usual cylindrical shape with a knob at the top (Fig. 36). Two small brass pans (Fig. 3 c) with a wire arch by which they can be suspended, are used to carry them; these are called respectively X and Y. I found on beginning to use them that there was too great a difference between A and B. I therefore adjusted them by putting a very small piece of wax upon A, the lighter. But the difference between them increased by 0-0782 mgm. in two days, which I thought was probably due to the wax. After the fourth series I therefore removed it and scraped B till it was more nearly equal to A. The weighings I — IV have, however, been retained, for though the differences on different days vary they are fairly constant on the same day. The weights are changed by the following apparatus which has been designed to effect the change as simply and quickly as possible. A horizontal ' side-rod ' or Hnk {ss, Fig. 5) is worked by two cranks (c, c, Fig. 56), which are attached to the axles of two equal toothed wheels {t, t) 16 ON A METHOD OF USING THE BALANCE with a pinion (p) connecting them. A second pinion (q), on a rod prolonged out of the case to the observer, gears with one of the toothed wheels. By turning this rod the toothed wheels are set in motion, both in the same direction, moving the horizontal 'side-rod' from the right say upwards and over to the left. A pin (pn) stops its motion downwards further than is shown in Fig. 6 a. Near each end of the rod is cut a notch, and across these are hung the pans carrying the weights. The apparatus is fastened to the floor of the case between the central upright, supporting the beam, and the scale- pan, *the side-rod being perpendicular to the direction of the beam, and exactly over the centre of the pan. In Fig. 5 a, one of the weights B is sup- posed to be resting on the scale- pan (the wires suspending the pan from the beam not being shown), the side-rod having moved down so far below the wire of the smaller pan carrying the weight that it leaves it quite free. If, now, it is desired to change the weights the rod R is turned, setting the wheels in motion, the side-rod moves up, picks up B — the notch catching the wire — then travels over round to the extreme right, when A will be just over and nearly touching the scale-pan. By continuing the motion slightly A will be gently deposited on the pan, and the side-rod will move slightly down leaving the weight quite free. On the scale-pan are four pins, turned slightly outwards, acting as guides for the small pan, and ensuring that it shall always come into the same position. The wheels and pinions are of such a size that two revolutions of the rod just suffice to change one weight for the other. It will be seen that all the manipulation required from the observer during a series of weighings is the simple turning of three rods, which are prolonged out of the balance case to where he is stationed at the telescope. By turning one of these he can change the position of the rider on the beam by a known amount, and so find the value of his scale. By turning a second he clamps the scale-pan, and so steadies the balance while the weights are changed by turning a third rod. I have made this arrangement not only because it seems as simple as possible to secure the end required, but also because it seemed more applicable to a vacuum-balance (with which I hope ultimately to test it). I take this opportunity of expressing my thanks to Mr. Thomas Foster, mechanician of Owens College, for his aid in the construction of the apparatus, and in the planning of many of its details. Method of conducting a Series of Weighings. After the counterpoise has been adjusted so that the beam swings nearly about its horizontal position, the frame is lowered so that the balance is ready for use. The pan is then clamped and the balance is left to come to a nearly permanent state of flexure if possible, sometimes for the night or even longer. The lamp is lighted usually half-an-hour or more before begin- ning to observe, so that its effect on the balance may attain a more or less WITH GREAT DELICACY, ETC. 17 steady state. It is necessary also to wait some time after coming into the room, for the opening of the door will always cause a considerable and immediate deflection of the beam. When a sufficient time has elapsed, the observations are commenced with a determination of the value of one scale- division by means of the riders. The three extremities of two successive oscillations are observed with one of the riders resting on the beam. These are then combined as follows : The mean of the first and third is taken, and the mean again of this and the second, this constituting the 'resting- point,' that is, the position of equilibrium of the beam at the middle of the time. For instance, in weighing No. I (see tables at the end) the three extremities of successive oscillations were 280-5, 312-0, and 286-0 (column 2). The resting- point was taken as 280- 5 + 286-0 + 2x31 2-0 _ ^97.62 the rider on the beam being the right-hand one denoted by R (column 1). The balance is then clamped, and the other rider is brought on to the beam while the first is taken up. The resting- point is again observed. In No. I it was 270-05. The balance is again clamped, and the first rider again brought on to the beam, and, on unclamping, the resting-point again observed. In the same weighing it was 296-75. These three are sufficient to give one deter- mination of the deflection due to the transference of a rider. This will be the difference between the second resting-point and the mean of the first '^97-62 + 296-75 and third. For instance, '^ ^ 270-05 = 27-13 divisions. This number is found in the fifth column. This process is continued, the resting- points being combined in threes till several values of the deflection due to the rider have been obtained, and the mean of these is taken as the true value. This plan of combining the resting- points requires that the observations should be taken at nearly equal intervals. After a little practice it will always take the observer about the same time to go through the same operations of clamping, changing the riders, unclamping, clamping again to lessen the vibrations about the new resting-point, and then beginning to observe, and I have considered that this was a sufficiently correct method of timing the observations. When a series has been taken it will at once be seen whether they were begun too soon after entering the room, or whether any irregular disturbing force has acted. For instance, in weighing No. II, determination of one scale-division, the first resting-point is so much lower than the succeeding with the same rider that evidently the balance was still affected by my entrance into the room. It was, therefore, rejected. Again, in weighing No. Ill, determination of the difference between the weights, the fourth resting-point was much lower than the others with the same weight in the pan. p. o. w. 2 18 ON A METHOD OF USING THE BALANCE The resting-points, when the other weight was in the pan, showed no similar sudden drop of such magnitude. This observation was, therefore, rejected as being affected by some irregular disturbance. When the value of the deflection is determined, the value of one scale- division is at once found by dividing -3282 mgm. by the number of divisions of the deflection, since the change of the sides is equivalent to the addition of -3282 mgm. to one pan. The determination of the difference between the weights is then begun. This is carried on in a precisely similar manner, the only difference being that the rod changing the weights is now turned round in place of the rod changing the riders. I have usually taken a greater number of observations of the difference between the weights than of the deflection due to the riders, as the former is somewhat more irregular than the latter. This irregularity I believe to arise from slight differences of temperature of the two weights, and perhaps from air currents caused by their motion inside the case. They do not seem to be due to any fault in the clamping arrangement, since that is employed equally in both, and the changing of the weights, if effected gently, does not move the beam at all. When the deflection has been determined, it is multiplied by the number of milligrammes corresponding to one scale-division, and this, of course, gives the difference between the weights. I have interchanged the weights in the two pans X and Y, between the series of weighings, in order to make the experiments like those conducted in the weighings for the standard pound. But my object has not been to show at all that the method gives consistent results day after day, and, in fact, the difference between the weights has varied. For instance, according to weighings I and II, A — B = -0446, while, according to weighings III and YV, A — B = -0116. There is a greater difference between these than can be accounted for by errors of experiment, and it probably arose from the small piece of wax with which I made A nearly equal to B. The difference between the weights when measured to such a degree of accuracy as that which I have attempted, will, no doubt, vary from time to time, partly with deposits of dust, partly with changes in the moisture in the atmosphere, and so on. But I think the numbers which are given in the tables are sufficient to show that the difference between two weights in any one series of weighings can be measured with a greater degree of accuracy than has hitherto been supposed possible. I give in the tables a full account of the weighings, each series containing a determination of the value of one scale-division and a determination of the difference between the weights. The greatest deviation of any one of a series from the mean of that series of differences is always given. This I consider a better test of accuracy of weighing than the probable error. What is wanted in weighing is rather a method which will give at I WITH GREAT DELICACY, ETC. 19 once a good determination of the difference between two weights. But I may state, that if the error of any one of a series be taken as its difference from the mean of that series, the probable error of a single determination of the difference between the weights in the first four series is -4344 of a division, or -0054 mgm., that is, s4-ituuuwu^^ ^^ ^^^ *otal weight, while the greatest error is 1-8 divisions, or -0224 mgm., that is, ^ ooo^oooo ^^ ^^ *^® ^otal weight. It may be remarked that these weighings were all made during peculiarly unfavourable weather when there were frequent heavy showers, causing sudden changes of temperature, and thus seriously affecting the working of the balance. In the series V — VIII the greatest error is only -^ ooo^oooo ^^ the total weight, the weather having improved considerably. On the Employment of the Balance to determine the Mean Density of the Earth. In the Cavendish experiment, the attraction of a large sphere of lead of known mass and dimensions upon another smaller sphere, also of known mass and dimensions, is measured when the two are an observed distance apart. Comparing this attraction with the weight of the small sphere — that is the attraction of the earth upon it — and knowing the dimensions of the earth, we can deduce the mass of the earth in terms of the mass of the large lead sphere, and so obtain its mean density. The torsion-balance, which was invented for the purpose by Mitchell, the original contriver of the experiment, has hitherto been used to determine the force exerted by the mass upon the small sphere. In the arrangement here described, I have replaced the torsion- balance by the ordinary balance, and have so been able to compare the attraction of a lead sphere with that of the earth upon the same mass somewhat more directly. The results which I have obtained have no value in themselves, but they serve as an example of the employment of the balance for more delicate work than any which it has as yet been supposed able to perform. The method is shortly this: A lead weight (called ''the weight') weighing 452-92 grms. (nearly 1 lb.) hangs down by a fine wire from one arm of a balance, from which the pan has been removed, at a distance of about six feet below it, and is accurately counterpoised in the other pan, suspended from the other arm. A large lead mass (called 'the mass') weighing 154,220-6 grms. (340 lbs.) is then introduced directly under the hanging ' weight.' The attraction of this mass increases the weight slightly and the beam is deflected through an angle which is observed. The value of this deflection in milH- grammes is measured by the employment of riders in the manner described above, and so the attraction of the ' mass ' is known. The increase of the weight caused- by the 'mass' has been in my experiments about -01 of a milligramme, or 4500^0000 ^^ ^^ *^® whole weight. The balance which I have used is that which I have described above. It was placed in the same room and in the same position as in the weighing 2—9. 20 ON A METHOD OF USING THE BALANCE experiments. The same method was used to observe the oscillations with a single mirror on the beam. The scale was a simple one etched on glass and not diagonally ruled. It had about 50 divisions to the inch, and the numbers increased from above downwards, so that an increase in the weight hanging from the left arm was indicated by a lower number on the scale. The ' weight' which is suspended by a very fine brass wire from the left arm, passing through a hole in the bottom of the balance-case, hangs in a double tin tube, 4 inches in diameter, to protect it from air-currents. At the bottom of the tube is a window, through which can be seen the bottom of the 'weight' as it hangs. The 'weight' is 4-248 centims. in diameter and is gilded. The ' mass ' is a sphere of an alloy of lead and antimony. It was cast with a ' head ' on and then accurately turned. Its vertical diameter is 30-477 centims. (about 1 foot). The specific gravity of a specimen of the metal was found to be 10-422. Its weight given by a weighing-machine is 340 lbs. about, and this agrees very nearly with the weight calculated from the specific gravity. I am obliged to accept this as the true weight provisionally, until it is found more correctly by the large balance referred to above and now being constructed. This mass (Fig. 1, M) is placed in a shallow wood cup at one end of a 2-inch plank, 8 inches wide and 6 feet 1 1 inches long, mounted on four flanged brass wheels, and serving as a carriage for it (Fig. 1). A plank about 12 feet long nailed to the floor in a direction perpendicular to the beam of the balance, as shown in Fig. 1, pp, acts as a railway for the carriage, and a firm stop at each end prevents the carriage from running off the rail. The distance be- tween the stops is rather less than twice the length of the carriage, and the ' weight ' hangs down from the balance exactly midway between the stops. The 'mass' is placed on the carriage so that it is exactly under the 'weight' when the carriage is at one end of its excursion against one of the stops. An empty cup (c. Fig. 1) of the same dimensions as that in which the ' mass ' rests is placed at the other end of the carriage, and is just under the 'weight' when the carriage is against the other stop. By this arrangement no correction is needed for the attraction of the carriage upon the ' weight ' or counterpoise, and the eflect caused by the removal of the carriage from one end of its excursion to the other is entirely due to the difference of attractions of the 'mass' upon the 'weight' and counterpoise in its two positions. The position of the 'mass' when directly under the 'weight' is called its 'in' position, and that when it is at the other end of its excursion is called the ' out ' position. The length of the excursion is 5 feet 7-3 inches. To draw the carriage along the rail a vertical iron shaft with a wood cyhnder at the lower end pivots on the floor, and is prolonged up to the level of the observer as he sits at the telescope with a handle by which he can turn it. The two ends of a rope which winds round the cylinder pass through pulleys on the stops, and are attached to the ends of the carriage. WITH GREAT DELICACY, ETC. 21 The observer can then move the 'mass' with great ease by turning the handle, even while looking through the telescope. When a series of observations is made, the general method is this. The deflection (r) due to the transference of a rider from one notch to the other on the beam is first observed exactly in the manner before described, the mean of four or five values being taken as the true value. Then the deflection (n) due to the difference of attraction of the 'mass' in its two positions is found in exactly the manner in which the difference between two weights is found, except that now when three successive extremities of oscillations have been observed for a resting-point the ' mass ' is moved from one position to the other where the weights were changed in the former experiments, the clamp not being brought into action. The second extremity of the oscillation which is proceeding while the 'mass' is moved, is observed as the first of the next three. When nine or more resting-points have been observed they are com- bined in threes, and the mean of the resulting values of the deflection n is used in the subsequent calculation. This deflection is, of course, less than that which would be observed were there no attraction on the counterpoise, and were the 'out' position of the A ^ B 'mass' at an infinite distance. To find the factor/ by which the deflection n due to the change of position of the ' mass ' must be multiplied in order to reduce it to the deflection which would be observed under these conditions, let AB be equal and parallel to the beam of the balance at the level of the counterpoise of which B is the centre. Let C be the centre of the ' weight,' D that of the 'mass' in its 'in' position, E that of the 'mass' in its 'out' posi- tion. Draw BF, DF, parallel to AD, AB. Let ju, = the mass of the 'mass.' The vertical attraction of the 'mass' in its 'in' position will be jM fiBF CD^ BD^ ' The vertical attraction in its ' out ' position will be ^CD ixBF CE^ BE^' 22 ON A METHOD OF VSINQ THE BALANCE The difference between these is actually observed, viz. : ^ ( CD^.BF CD^ CD^ . BF C&\ BD^ CE^ BE^ The factor by which we must multiply the observed difference to reduce it to the attraction of the 'mass' on the 'weight' in its 'in' position is therefore CB^ . BF CD^qJDKBF *^~ "^ BD^ '^CE^ BE^ = 1-0185 since CD = 22-13 centimetres. BD = 192-03 BF = 187-70 CE = 172-36 BE = 257-09 The values of r and n being observed, the distance d between the centres of the 'mass' and 'weight' is then measured by adding to 17-362 centims. (the sum of their radii) the distance from the top of the 'mass' to the bottom of the 'weight' as measured by a cathetometer. It now remains to explain the calculation of the mean density A from the observed values of r, n, and d. We have / X increase in weight observed _ Attraction of 'mass' on 'weight' when 'in' Weight of 'weight' Attraction of earth on 'weight' But the increase in weight is ^„^.^.,^ mgms., since the distance between ^ r X 202716 ^ the notches is 6-654 milHms., and the half beam 202-716 millims. The weight of the 'weight' is 453-29 grammes. The attraction of the 'mass' per gramme of the 'weight' _ Volume X density (distance d between centres of 'mass' and 'weight')^' _ Mass in grammes ~ d^ ' _ 154220-6 ~ I^ • The attraction of the earth is similarly A X inR {1 + M- (f M - e) cos2 A}, where A = mean density of the earth, R = earth's polar radius in centimetres, j^ _ centrifugal force at the equator Equatorial gravity ' € = ellipticity, A = latitude. WITH GREAT DELICACY, ETC. 23 The logarithm of the coefficient of A when R is in inches is 9-0209985 (Astron. Soc. Mem., vol. 14, p. 118), or if R is in centimetres it is 94258322. Inserting these values in the equation we obtain 154220-6 453290 A = £ ^ttR {1 + M - (f M - e) cos2 A} fx n 6654 r 202716 ^""^^^ where and C = 154220-6 X 453290 x 202716 iTvR {1 + M - (f M - e) cos2 A} X / X 6654 log C = 1-8951337, .-. log A = 1-8951337 + log r — log n - 2 log d. The following table is an account of an experiment made on May 30th, 1878, and will serve as a specimen of the method of making the observations. It is the best which I have yet made in the closeness with which all the values of n agree with each other. VII. May 30, 1878. Determination of r. Eider on beam Extremities of oscillations Resting - point Mean of pre- ceding and succeeding resting -points Differences R 260-9 250-9 260-6 255-82 — — L "214-6 205-0 212-3 209-22 256-74 47-52 R 271-9 244-7 269-4 257-67 210-31 47-36 L 214-3 209-2 212-9 211-40 257-62 46-22 R 249-7 263-8 253-0 257-57 212-23 45-34 L 204-9 220-7 206-0 213-07 — Mean r = 46-61. 24 ON A METHOD OF USING THE BALANCE Determination of n. Position of mass Extremities of oscillations Resting - point Mean of pre- ceding and succeeding resting-points Differences = n In 216-8 208-1 215-9 210-9 216-1 211-0 212-22 — — Out 213-52 212-27 1-25 In 213-9 210-8 213-8 212-32 213-61 1-29 Out 212-7 214-6 212-9 21370 212-41 1-29 In 211-5 213-6 211-3 212-50 213-78 1-28 Out 215-9 211-8 216-0 213-87 212-60 1-27 In 210-2 2150 210-6 212-70 213-93 1-23 Out 216-7 2116 216-1 214-00 212-73 1-27 In 209-9 215-4 210-4 212-77 214-08 1-31 Out 217-0 211-4 216-9 21417 213-20 212-98 M9 In 209-8 216-2 210-6 — — Mean n = 1 26. WITH GREAT DELICACY, ETC. 25 At the close of the experiment d was found to be 22-226 centimetres. We have therefore log A = log + log 46-61 - log 1-26 -2 log 22-226 = 1-8951337 I ' +1-6684791 I i- 0-1003705 -2-6937226 = 0-7695197. .-. A =5-882. I have made in all eleven experiments with this method. The resulting values of A are 1 May 20 5-393. 2 ... . . . ,, 23 ... ... 5-570. 3 ... 5) 24 ... ... 4-415. 4 ... 5, 28 ... ... 7-172. 5 ... . . 5) 29 ... ... 5-109. 6 ... ,5 29 ... ... 6-075. 7 ... ,, 30 ... ... 5-882. 8 ... ,, 30 ... ... 6-336. 9 ... June 5 ... ... 5-977. 10 ... . . . 5j 5 ... ... 5-580. 11 ... 6 ... ... 5-100. The resulting mean value of the mean density of the earth is 5-69. If the eleven determinations be supposed to have equal weight, the probable error of their value is 0-15. The various determinations differ very much among themselves, but they seem to me sufficiently close to justify the hope that with a large balance and a large weight, which will not be so easily affected by air-currents, and with greater precautions to prevent those air-currents, a good determination of the mean density of the earth may ultimately be obtained by this method. 26 ON A METHOD OP USING THE BALANCE I. June 12. Determination of 1 Scale- Division. Rider on beam Extremities of three successive oscillations Resting- point Mean of pre- ceHing and succeeding resting -points DifiEerence due to R - L L 280-5 312-0 286-0 297-62 — — R 253-1 284-9 257-3 270-05 297-18 27-13 L 306-2 288-5 303-8 296-75 268-68 28-07 R 258-8 275-0 260-5 267-32 294-81 27-49 L 285-8 298-7 288-3 292-87 266-79 26-08 R 272-9 260-6 271-0 266-27 292-88 26-61 L 296-1 290-0 295-5 292-90 — — Mean R - L = 27-08 divisions. 0-3282 1 division = ' - = 0-01212 milligramme. WITH GREAT DELICACY, ETC. 27 Determination of {B -{- X) — (A -\- Y). Weight in pan Extremities of oscillations Resting - point Mean of pre- ceding and succeeding resting -points Difference A+ Y , 254-5 272-9 257-3 264-40 — — B + X 267-5 249-8 264-8 257-97 263-27 5-30 ^ + F 258-5 265-4 259-3 26215 257-04 5-11 B + X 259-3 , 253-6 258-0 256-12 261-87 5-75 A+ Y 253-5 268-5 255-9 261-60 255-63 5-97 B + X 244-6 264-5 247-0 255-15 261-87 6-72 A+Y 273-6 252-0 271-0 262-15 255-85 6-30 B + X 252-7 260-0 253-5 256-55 262-92 6-37 A+Y 275-0 253-7 272-4 263-70 — Mean difiference 593 .-. {B + X) - {A+ Y) = 0-01212 X 5-93 = 0-0718 milligramme. Greatest deviation from the mean = 0-82 division = 0-0099 milligramme. The weather during this series of weighings was very unfavourable, with frequent heavy showers. ON A METHOD OF USING THE BALANCE II. June 13. Determination of 1 Scale-Division. Rider on beam Extremities of oscillations Resting - point Mean of pre- ceding and succeeding resting -points Difference due to R - L L 302-8 299-9 302-9 301-37 This is rejected as it is so much lower than the succeeding R 321-8 335-7 326-8 330-00 — — L 299-6 308-7 301-0 304-50 330-68 26-18 R 322-9 337-5 327-6 331-37 305-10 26-27 L 299-5 310-9 301-5 305-70 331-77 26-07 R 325-2 337-1 329-3 332-17 305-90 26-27 L 290-4 319-4 295-2 306-10 — Mean R - L = 26-20 divisions. .*. 1 division = 0-01252 milligramme. The weather was as unfavourable as on the previous day. The weights were changed shortly before the commencement of this series and the balance then worked so irregularly that for some time I was unable to begin the rider- determination. Even then the first resting-point had to be rejected. WITH GREAT DELICACY, ETC. 29 Determination of (A+ X)— {B+ Y). Weight in pan Extremities of oscillations Resting - point Mean of pre- ceding and succeeding resting -points Diflference B+ Y 307-8 326-0 311-6 317-85 These are all rejected, as the motion was so irregular. The weights had been \ changed a short time be- fore, and had probably not reached an uniform tem- perature A + X 293-7 307-8 295-8 30127 B+ Y 309-6 322-3 312-6 316-70 A + X 295-5 309-6 297-6 303-07 B+ Y 304-1 322-7 308-1 314-40 A + X 289-3 304-4 291-5 297-40 — — B+ Y 305-5 3150 307-5 310-75 297-35 13-40 A + X 294-2 300-1 294-8 297-30 309-95 12-65 B+ Y 304-6 312-7 306-6 309-15 296-51 12-64 A + X 290-0 300-7 291-5 295-72 — — Mean {A + X) - {B + Y) = 12-89 divisions. .-. {A + X) - (B + Y) = 0-1614 milligramme. Greatest deviation from the mean = 0-51 division = 0-0062 milligramme. 30 ON A METHOD OF USING THE BALANCE III. June 13. Determination of 1 Scale- Division. Rider on beam Extremities of oscillations Resting - point Mean of pre- ceding and succeeding resting-points Difference due to R - L R 301-3 290-6 299-7 295-55 — — L 313-7 328-7 318-1 322-30 296-38 25-92 R 293-6 300-4 294-5 297-22 322-47 25-25 L 311-0 331-6 316-4 322-65 297-88 24-77 R 281-6 313-2 286-2 298-55 323-57 2502 I. 310-1 335-8 316-3 324-50 1 298-62 1 25-88 R 287-8 308-1 ' 290-8 298-70 — - Mean R - L = 25-37 divisions. 0-3282 1 division = o^-qy = 0-01293 milligramme. WITH GREAT DELICACY, ETC. 31 Determination of {A-\- X) — {B-^Y). Weight in pan Extremities of oscillations Resting - point Mean of pre- ceding and succeeding resting.points DiflEerence A-\- X 303-3 294-5 301-8 298-52 — B+ Y 303-6 315-8 306-2 310-35 297-07 13-28 A + X 288-9 301-5 290-6 295-62 — — B+ Y 300-2 311-6 302-5 306-47 This is evidently due to some irregular and short disturb- ing cause, and is rejected A^ X 28S-8 304-2 291-1 297-07 — — B+ Y 301-5 319-3 303-9 311-00 297-16 13-84 A + X 291-7 3020 293-2 297-22 310-33 13-11 B+ Y 3010 316-8 304-1 309-67 — — Mean {A + X) - {B + Y) = 13-40 divisions. .-. (A + X) - (B + Y) = 0-1732 milligramme. Greatest deviation from the mean = 0-44 division = 0-0057 milligramme. 32 ON A METHOD OF USING THE BALANCE IV. June 14. Determination of 1 Scale-Division. Rider on beam Extremities of oscillations Resting - point Mean of pre- ceding and succeeding resting-points DifEerence due to R - L R 252-5 253-4 251-7 252-25 This was taken soon after enter- ing the room. It is so much lower than the succeeding that it is rejected L 290-8 273-7 288-8 281-75 — — R 242-4 266-7 245-1 255-22 281-96 26-74 L 272-6 289-7 276-7 282-17 255-89 26-28 R 259-8 253-9 258-7 256-57 282-67 26-10 L 286-2 280-0 286-5 28317 — — Mean R - L = 26-37 divisions. 0-3282 ,-. 1 division = = 0-01244 milligramme. Being interrupted, I could not continue the series of rider- determinations further. WITH GREAT DELICACY, ETC. 33 Determination of (B + X) — (A+ Y). Weight in pan Extremities of oscillations Resting - point Mean of pre- i resting -points 1 A+ Y 291-7 275-0 289-5 282-80 — — B+ X 261-0 280-3 263-4 271-25 282-85 11-60 A+ Y B + X 285-7 280-4 285-1 282-90 271-66 11-24 280-2 265-3 277-5 272-07 283-90 11-83 A+ Y 294-9 276-1 292-5 284-90 273-43 11-47 B + X 276-0 273-9 275-4 274-80 286-40 11-60 A+ Y 267-3 305-4 273-5 287-90 275- n 12-79 B+ X 278-5 272-9 277-4 275-42 289-28 i 13-S6 A+ Y 296-6 285-5 295-1 290-67 — — Mean {B + X) - [A + Y) ^ 12-06 divisions. .-. (5 + Z) - (^ + 7) = 0-1500 milligramme. Greatest deviation from the mean =1-8 divisions = 0-0224 mgm. . The previous determination oi{B+ X)— (A+ Y) was -0718 mgm. The difference is too great, -0782 mgm., to be accounted for by errors of experiment. There must have been some deposit on one of the weights, either of dust or moisture. I therefore took them out, cleaned and adjusted them by scraping B till nearly equal to A, and removing the wax from A. p. c. w. 34 ON A METHOD OF USING THE BALANCE V. June 14. Determination of 1 Scale-Division. Rider on beam Extremities of oscillations Resting - point Mean of pre- ceding and succeeding resting -points Difference due to R - L R 212-6 230-3 214-7 221-97 — 1 L 252-3 242-5 251-2 247-12 222-16 24-96 R L 225-0 220-5 223-6 222-40 246-67 24-27 238-6 252-8 240-7 246-22 221-87 24-35 R 224-9 218-7 223-3 221-40 245-98 1 24-58 L 249-7 242-2 248-9 245-75 221-26 24-49 R 226-8 216-7 224-3 221-12 ! Mean R - L = 24-53 divisions. Q.Q902 1 division = - ^ = 0-01339 milligramme. WITH GREAT DELICACY, ETC. 35 Determination of {B+ Y)— (A+ X). Weight in pan Extremities of oscillations Resting - point Mean of pre- ceding and succeeding resting-points DifiEerenoe A + X 206-3 212-7 206-8 209-62 This was rejected as being so much higher than the rest B+ Y 209-0 206-4 208-6 207-60 — A + X 212-4 204-0 211-3 207-92 20771 21 B+ Y 206-8 208-8 206-9 207-82 207-88 -06 A + X 211-8 204-4 210-8 207-85 207-64 •21 B+ Y 209-1 206-1 208-6 1 207-47 207-86 i 1 -39 A + X 210-7 205-5 209-8 207-87 207- 11 -76 B+ Y 208-5 205-3 207-9 206-75 207-53 •78 A + X 209-7 205-1 208-9 1 207-20 206-45 i -75 B+ Y 203-3 208-7 206-15 203-9 — — Mean {B + Y) - {A + X) = 0-45 division = 0-00602 milligramme. Greatest deviation from the mean = 0-39 division = 0-00522 mgm. A and B had here been cleaned and B readjusted by scraping. A small vessel containing calcium chloride was put inside the balance to dry the air. This improved the action of the clamp, diminishing the cohesion. 3—2 36 ON A METHOD OF USING THE BALANCE VI. June 17. Determination of 1 Scale-Division. Mean R - L = 23-02 divisions. 1 division = -k^^;^ = 0-01425 milligramme. WITH GREAT DELICACY, ETC. 37 Determination of (5+ X)— (A-{- Y). Weight in pan Extremities of oscillations Resting- point Mean of pre- ceding and succeeding resting -points Difference B+X 216-9 214-6 216-6 215-67 — — A+Y 256-7 233-8 253-9 244-55 216-08 28-47 B + X 220-1 213-7 218-5 216-50 244-90 28-40 A+Y 254-8 236-7 252-8 245-25 217-65 217-07 28-18 B + X 221-5 214-5 220-1 245-48 27-83 A+Y 257-5 235-3 254-8 245-72 217-37 28-35 B+X 220-9 214-0 219-5 217-10 245-68 28-58 A+Y 246-8 244-5 246-8 245-65 216-83 28-82 B + X 223-7 210-5 221 -6 216-57 — — Mean {B + X) - {A + Y) = 28-38 divisions = 0-4043 milligramme. Greatest deviation from the mean = 0-55 division = 0-00784 milligramme. 38 ON A METHOD OF USING THE BALANCE VII. June 18. Determination of 1 Scale-Division. Rider on beam R Extremities of oscillations Resting- point Mean of pre- ceding and succeeding resting -points Difference due to R - L 171-6 190-7 172-7 181-42 — — L 2103 225-3 212-4 218-32 180-47 37-85 1 R 189-2 171-7 185-5 179-52 217-54 38-02 L 223-2 210-8 222-3 216-77 179-04 37-73 1 R 1850 173-2 182-9 178-57 216-17 37-60 1 1 L 220-2 211-3 219-5 215-57 178-22 37-35 1 1 - 1 189-1 168-8 184-8 177-87 1 — — Mean R - L = 37-71 divisions. Q.Q2Q9 ■. 1 division ^ 07^ = 0-00870 milligramme. WITH GREAT DELICACY, ETC. Determination of (B-{- Y)— {A-\- X). Weight in pan Extremities of oscillations Resting - point Mean of pre- ceding and succeeding resting -points Difference B+ Y 208-5 221-7 209-5 215-35 — — A + X 246-9 214-4 244-1 229-95 216-22 13-73 B+ Y 219-2 215-6 2180 217-10 230-91 13-81 A + X 240-8 224-0 238-7 231-87 217-12 14-75 B+ Y 226-1 209-4 223-7 217-15 232-24 1.509 A + X 221-8 242-3 224-1 232-62 216-96 15-66 B+ Y 211-5 221-8 212-0 216-77 232-21 15-44 A + X 219-0 243-0 222-2 231-80 — — B+ Y 208-3 2290 209-9 21905 This sudden change of resting - point must be due to some irregular disturbance. It is therefore rejected. It was slowly returning to nearly its former values Mean {B + Y) - {A + X) = 14-75 divisions = 0-12831 milligramme. Greatest deviation from the mean = 1-02 division = 0-0089 mgm. The great difference between the result here and that in series V is probably- due to deposit of dust. The new mirrors had to be fixed up just before the experiment began, and the doors were open for some time. At the conclusion of the weighing I found a good deal of dust on the weights. 40 ON A METHOD OF USING THE BALANCE VIII. June 19. Determination of 1 Scale- Division. Rider on beam Extremities of oscillations Resting- point Mean of pre- ceding and succeeding resting -points Difference due to R - L L 235-8 222-3 233-3 228-42 This is so much higher than the rest, probably through being observed soon after I entered the room, that it is rejected R 197-0 181-2 193-9 188-32 — — L 228-2 222-7 228-0 225-40 188-24 37-16 R 197-4 180-7 193-9 188-17 i 225-46 1 37-29 L 233-6 1 218-0 225-52 232-5 j i 187-63 ' 37-89 i R 191-7 183-3 187-10 190-1 225-28 38-18 L 230-2 220-3 229-4 225-05 187-35 37-70 R 193-8 182-5 191-8 187-60 — Mean R - L = 37-64 divisions. 0*3282 1 division = -^ - = 0-00872 milligramme. WITH GREAT DELICACY, ETC. 41 Determination of {B -\- X) — (A -\- Y). Weight in pa,n Extremities of oscillations Resting, point Mean of pre- ceding and succeeding resting -points Difference A-\- Y 245-3 238-7 244-2 241-72 In one observation not recorded just before this the clamp had been loose, and the scale-pan had slipped, and the resting-point was thereby changed. The disturbance had apparently not subsided when this was taken, it is therefore rejected B + X 193-0 185-7 191-4 188-95 — — A+ Y 226-6 243-3 229-2 235-60 188-25 47-35 B+ X 182-2 192-8 182-4 187-55 235-22 47-67 A+ Y 229-4 239-2 231-6 234-85 187-72 47-13 B+ X 194-4 182-5 192-2 187-90 234-71 46-81 A+ Y 239-8 229-7 2391 234-57 187-75 46-82 B + X 194-7 181-6 192-5 187-60 234-51 46-91 A+ Y 230-6 237-4 232-4 234-45 187-50 46-95 B + X 186-0 1890 185-6 187-40 — Mean (B + X) - {A + Y) = 47-09 divisions = 0-4100 miUigramme. Greatest deviation from the mean = 58 division = 000506 mgm. 42 ON A METHOD OF USING THE BALANCE WITH GREAT DELICACY, ETC. Summary. Greatest deviation Series Mgms. from mean in milligrammes II III IV I (B+Z)-(^+y) = -0n8 -00991 I (A+X)-{B+Y) = -\U4: -0062 (^ ^ ^ - (^ + Z)-(B+r) = -1732 •0057) ^^^^^g (B+X)~(A+Y) = -\m) -0224 ( ^ °^ ^ V (5+y)-M + Z) = -0060 -0052) VI (B+X)-(A+Y) = -4043 -0078 / ^ " ^ + ^051 mgm. VII (B+y)-M + Z) = -1283 -00891 VIII (,B+X)-(A+Y) = -4100 -0051 ( ^ " ^ + ^^^^ ""8™- The greatest error — that is the greatest deviation of any one value from the mean of its series — in the first four series is yoooVttoo^^ ^^ ^ pound. The greatest error in the four series V — VIII is n oWoooo^^ ^^ ^ pound. 3. ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE. [Phil. Trans. A, 182, 1892, pp. 565-656.] [Received May 13. Read June 4, 1891.] I. Account of Apparatus and Method. In a paper printed in the Proceedings of the Royal Society, No. 190, 1878 (vol. 28, pp. 2-35)*, I gave an account of some experiments undertaken in order to test the possibihty of using the Common Balance in place of the Torsion-Balance in the Cavendish Experiment. The success obtained seemed to justify the intention expressed in that paper to continue the work, using a large bullion-balance, instead of the chemical balance with which the pre- liminary experiments were made. As I have had the honour to obtain grants from the Royal Society for the construction of the necessary apparatus, I have been able to carry out the experiment on the larger scale which appeared likely to render the method more satisfactory, and this paper contains an account of the results obtained. At the time I was making the preliminary experiments the late Professor von Jolly was already employing the balance for gravitation investigations (W iedemann'' s Annalen, vol. 5, p. 112), though I was not aware of the fact. Later he published an account {Wied. Ann., vol. 14, p. 331) of a determination of the Mean Density of the Earth by the use of the Balance. Still more recently Drs. Koenig and Richarz have devised a method of using the balance for the same purpose {Nature, vol. 31, pp. 260 and 475), and I believe that their work is still in progress. It might appear useless to add another to the list of determinations, especially when, as Mr. Boys has recently shown, the torsion-balance may be used for the experiment with an accuracy quite unattainable by the common balance. But I think that in the case of such a constant as that of gravitation, where the results have hardly as yet begun * [Collected Papers, Art. 2.] 44 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND to close in on any definite value, and where, indeed, we are hardly assured of the constancy itself, it is important to have as many determinations as possible made by different methods and different instruments, until all the sources of discrepancy are traced and the results agree. The apparatus for the experiments described in this paper was first set up in the Cavendish Laboratory at Cambridge through the kindness of Professor Clerk Maxwell. After spending some months in working at the experiment, but without much success beyond the detection of some sources of error, I left Cambridge, and ultimately the apparatus was again set up at the Mason College, Birmingham. The difficulties in carrying out the work with any approach to exactness have been far greater than were anticipated, and many times work has been begun and results have been obtained, but examination has shown them to be affected by large errors which could be traced and eliminated by further improvements in the apparatus. At the beginning of 1890, however, the apparatus was brought into fair working order, and during the course of the year I made a number of experi- ments with the results recorded in this paper. The Princifle of the Experiment. The object of the experiment, in common with all of its class, may be regarded, primarily, as the determination of the attraction of one known mass M on another known mass M' a known distance d away from it. The law of universal gravitation states that when the masses are spheres with centres d apart this attraction is GMM'/d^, G being a constant — the gravita- tion constant — the same for all masses. Astronomical observations fully justify the law as far as M'jd^ is concerned. They do not, however, give the value of G, but only that of the product GM for various members of the solar system. To determine G we must measure GMM'/d^ in some case in which both M and M' are known, whether they be a mountain and a plumb-bob, as in Maskelyne's experiment, the surface strata and a pendulum-bob, as in Airy's experiment, or two spheres of known mass and dimensions, as in all the various forms of Cavendish's experiment. Knowing the gravitation constant G, we may at once find the mean density of the earth A. For if 7 be the volume of the earth— regarded as a sphere of radius i?— the weight of any mass M', being the attraction of the earth on it, is GVAM'/R^ But if g is the acceleration of gravity the weight is also expressible as M'g. Equating these we get A = gR^jGV. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 45 Method of Using the Common Balance. In using the common balance to find the attraction between two masses, perhaps the most direct mode of proceeding would consist in suspending a mass from one arm of a balance by a long wire, and counterpoising it in the other pan. Then bringing under it a known mass, its weight would be slightly increased by the attraction of this mass. The increase would be the quantity sought if the attracting mass had no appreciable effect before its introduction beneath the hanging mass, and if, when beneath it, the effect on the balance could be neglected. This is very nearly the principle of the method used by von Jolly, and it is that of the method used in the prehminary experiments referred to above, in which a mass of 453 grms. of lead was hung from one arm of a chemical balance (about 40 centims. beam) by a wire 1-8 metres long, and was attracted by a mass of 154 kilogrms. of lead. But the attraction to be measured was exceedingly small, rather less than 0-01 milligrm., and it therefore appeared advisable to use a much larger balance with a larger hanging mass so that the attraction might be made comparable with the weight of exactly determined riders. Other anticipations as to proportionate increase of sensibility and diminution of effect of air-currents, have hardly been justified in the way I expected, though, by the ultimate form of the apparatus, they have, I think, been more than realised. With increase in the length of beam, a differential method became appHcable, by means of which the attraction of the mass on the beam w^as eliminated, and the necessity for prolonging the case to allow of a long suspending wire was removed. This will be seen from a consideration of Fig. 1. Let A, B represent equal masses suspended from the two arms of the balance, and let M be the attracting mass put first under A, the position of the beam being noted. If M is then placed under B its attraction is not only taken away from A but added to B, so that the tilting of the beam is that due to nearly double the attraction to be measured. Of course there are what we may term cross-attractions, in the first position, of M on B, and in the second position, of ilf on ^, but these may be allowed for in the calculations. We cannot give any mathematical expression for the attraction of M on the beam and suspending wires, owing to their irregularity of shape. But this attraction is eliminated if a second experiment is made in which A and B are raised equal known distances to A' and B' . For the difference between the two increments of weight on the right, is due solely to the alteration of the positions of A and B relative to M, the attraction on the beam remaining the same in each. From the observed effect of a known alteration of distance the attraction at any distance can be found. This is, shortly, the method adopted. The arrangement was ultimately complicated by the addition of a second mass m. Originally the mass M 46 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND was alone on a turn-table which revolved about a vertical axis immediately under the central knife-edge of the balance. And some experiments which I made led me to suppose that mere change of position of the mass did not affect the level of the balance. However, after a complete determination in 1888 of the mean density, when I supposed that the work was finished, an examination of the results showed some curious anomahes, which I could only ascribe to a tilting of the whole floor on the displacement of the mass. Making new tests as to the effect of removal of the mass, I found that the Fig. 1. Elevation of balanoe-room and observing-room. The front of the case is removed, and the front pillar is not shown. The pointer and mirrors are at the back. previous tests had been quite wrong in principle, and that there was a very appreciable effect quite visible in the telescope when the masses A and B were removed, and M was removed from one side to the other, the slope of the floor changing by an angle comparable with a third of a second. If this had been absolutely constant in amount, the differential method would have eUminated it ; but, probably, it varied sKghtly in successive motions of the turn-table, and the results showed that there was also a secular change, the amount of tilt gradually increasing. This secular change was probably due to increasing rigidity of the floor, so that it tilted over bodily, moving the supports of the balance with it, an increase partly due, perhaps, to the THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 47 pressure of the building, which had only been erected ten or twelve years, but chiefly, I think, to a gas-engine recently erected next door. When this was doing heavy work, the vibrations were very plainly felt, and no doubt they greatly aided the floor in 'settling down.' A second balancing mass m was therefore added, half as great as M, and on the opposite side of the turn-table, but twice as far from the axis. The resultant pressure was now always through the axis, and I could detect no tilting of the floor when the turn-table was moved. Of course the balancing mass acted somewhat to reduce the effect of the larger attracting mass, but in a calculable ratio. Finally, in order to eliminate or reduce the effect of any want of symmetry in the moving parts or in the masses, a second set of experiments was made with all the masses turned over and moved from left to right, and the mean of the first and second sets was taken. I now proceed to a detailed description of the various parts of the apparatus and the mode of experiment. The Balance-Room. The balance-room is in the basement of the Mason College, immediately under my room, and about 20 metres from the street. On one side were three windows looking on to a small courtyard, entirely surrounded by high buildings, but the windows have been bricked up. On the two adjacent sides are two other rooms, and on the opposite side a closely fitting door opening on a short corridor with doors at each end. There is no chimney in the room, and only an opening in the ceihng through which the balance was observed from the room above. The floor is of brick, resting on earth, and is very firmly laid. The temperature of the room was taken by means of a thermometer with a protected bulb at the end of a long wooden rod hanging down from the room above. The thermometer was about 6 feet from the floor, near one end of the case, and it could be rapidly pulled up into the room above and read by the observer before its temperature sensibly varied. The tempera- ture never appeared to vary so much as 0-1° C. in the course of two or three hours. The Balance-Case and its Supports. The case (Fig. 1) is a large cabinet of IJ inch wood, 1-94: metres high, 1-63 metres wide, -61 metre deep, with three large doors in front giving access to the hanging masses and riders, and a small door at the back near the mirror hereafter described. It is lined inside and out with tinfoil, and under each of the suspended masses is a double bottom with a layer of wool between, making a total thickness of about 1 J inches or 4 centims. At the top is a small window about 10 centims. square, through which the oscillations of the beam were observed. On each side within the case are placed three horizontal partitions, like shelves, to hinder circulation of the air. The larger attracting mass and the attracted masses are gilded, and it is 48 03Sr A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND possible that some advantage may arise from having the surface of the case of different metal. For if it, too, were gilded, it would readily absorb radiation from the large mass, and when the inside temperature changed, the suspended masses would readily absorb radiation from the inner surface of the case. But gold probably absorbs considerably less of tin radiation than it absorbs of gold radiation, and so temperature changes are probably lengthened out more than if the case were gilded. Plan of turn-table, girders, pillars, and balance-case. w. Window in case. c. Usual position of cathetometer. It was necessary to support the case so that the attracting masses could be moved about underneath it, and also to make it independent of the floor. Two brick pillars, 58 centims. x 36 centims. and 56 centims. high, were therefore built on thick beds of concrete under the floor, and about 3J metres apart. They rise up free from the bricked floor. Stretching between them are two parallel iron girders (g, g), about 30 centims. apart, and with their under side 56 centims. above the floor. The balance-case is placed across the middle of these girders (see plan, Fig. 2), with its under surface level with THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 49 that of the girders. The square base-plate of the balance is placed on the girders on three levelling screws. Two horizontal screws attached to the girders bear against each edge of the base-plate, so that it can be adjusted and fixed in any position. To lessen vibration one tier of bricks is removed from each pillar, and in its place are inserted eight cyhndrical blocks of indiarubber (^, i, Fig. 1), originally 7-5 centims. diameter and 7-5 centims. high. These crushed down almost 1 centim. at once, but have not shown any further measurable con- traction in the coursfe of several years. Their effect in deadening vibration has been surprisingly great. The Turn-table. On a bed of concrete, and quite free from the brickwork of the floor, is a circular rail of cast iron, 1-3 metres in diameter. On this, on conical brass wheels and pivoted at the centre, runs the turn-table, about 1-5 metres in diameter. This is made of wood and covered with tinfoil. It is like a wheel with a flat circular rim, and with four flat spokes arranged as a cross. It is as nearly symmetrical as possible, and at opposite ends of a diameter are placed two shallow cups, in either of which the large attracting mass may rest. The centres of these cups are a distance apart, equal to the length of the balance-beam. There are cut slots through the bottom of each cup, so that the bottom of the mass can be seen for the purpose of measuring the vertical diameter. Two beams, 2-74 metres long, run across the turn-table 26 centims. apart, with the cups between them, and across the ends are two boards, each with a circular hole 12 centims. in diameter, and in either of these the smaller, or balancing mass, may rest. These beams are braced by brass rods to brass uprights at their middle points to diminish bending. The turn-table is moved by an endless gut rope passing round it, and fixed at one point of the rim. The two sides of the rope pass over pulleys on to a drum in the room above. There are stops on the circular rail, against which come brass pieces on the turn-table when the masses are in position at either end of the motion. The drum can be turned easily by the observer at the telescope. Since the knife-edges and planes of the balance are of steel, all other moving parts of the apparatus were made free from iron. As an illus- tration of the necessity of this, I may mention that for some time I used what I supposed to be a brass wire rope to move the turn-table, but on looking out for the explanation of some irregularities, I found that the brass was wrapped round a core of steel wire, which acquired poles at the highest and lowest points in the position in which it always rested between different sets of weighings. These poles had quite an appreciable action on the balance-beam. The Balance. This is of the large bullion-balance type, with gun-metal beam and steel knife-edges and planes. It was made specially for the experiment by Mr. Oerthng, with extra rigidity of beam. Its performance p c. w. 4 50 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND has shown the great excellence of the design. The central knife-edge is supported on a steel plane by a framework rising 107 centims. above the base-plate, and the usual moveable frame can be raised or lowered from out- side the case, fixing the beam or setting it free to oscillate. The beam has often been left free to oscillate for months at a time, with the full load of 20 kilogrms. on each side, but I have no reason to suppose that the knife- edges have suffered at all. The length of the beam was measured by taking the length of each half separately by a beam-compass, and the mean of several measurements gave 123-329 centims. as the total length. The standard scale used throughout was that of a cathetometer made by the Cambridge Scientific Instrument Company. This scale has been verified at the Standards Office, and taking its coefficient of expansion as ewoo' ^^ ^^J ^® regarded for our purpose as perfectly correct at 18°, any errors being at that temperature much less than the errors of experiment. Comparing the beam-compass with this scale, it was found that -06 centim. must be subtracted, reducing the length to 123-269 centims. Now both beam and scale are of gun-metal and may, therefore, without serious error, be assumed to have the same coefficient of expansion, so that this is the length of the beam at 18°. At 0° it is 123-232 centims. Mirrors, Telescope, and Scale. At first a mirror was attached to the centre of the beam and the reflection of a scale in it was observed, either in the ordinary method or in the method described in the former paper (Roy. Soc. Proc, vol. 28, 1879)*, where a second fixed mirror is used to throw the ray of light a second, or even a third time back on to the moving mirror, each return increasing the deflection of the ray. But it was then necessary to make the time of vibration very long, and even when the time was three minutes, the tilt due to the attraction, i.e. the change of resting-point, did not amount to more than two or three scale-divisions. Now certain irregularities observed when the apparatus was first set up at Cambridge, led to experiments on the time taken by heat to get through the case in sufficient quantity to affect the balance, and I found that a coil of copper wire placed close under the case on one side (the bottom of the case being then solid, 1 inch thickness), heated by a current yielding 100 calories per minute, began to produce an appreciable disturbance on the balance in about 10 minutes, doubtless by the creation of air-currents from the heated floor of the case. It appeared advisable, therefore, to reduce the time of a complete experiment to less than this if possible, and, consequently, the time of a single swing very much below 3 minutes. This could only be done if at the same time the optical sensibility were very greatly increased. The employment of what may be termed the double-suspension mirror method due I beUeve to Sir WilHam Thomson, and used by Messrs. G. H. and * [Collected Papers, Art. 2.] THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 51 Horace Darwin in their experiments on the Lunar Disturbance of Gravity {Brit. Assoc. Rep., 1881), has very satisfactorily solved the problem, giving a greatly increased deflection on the scale, even when the time of oscillation is as short as twenty seconds. This method, which deserves to be more generally known and applied for the detection of small motions, consists in suspending a mirror by two threads, Microscope sijage Bnrad^xt 1 Tir^ «q Varus ivorkUn^ im dashpot Fig. 3. Double-Suspension Mirror (half size). one from a fixed point, the other from the point which moves. The angle through which the mirror turns for a given motion of the latter point is inversely as the distance between it and the fixed point, so that by diminishing this distance the sensibility of the arrangement may be almost indefinitely increased. To apply it to the balance, a small bracket (Fig. 3) is fixed to the ordinary pointer of the balance, about 60 centims. below the central knife-edge. This projects horizontally at right angles to the axis of the beam, and it is bevelled at the edge. Close to it is another bevelled edge attached to a microscope- 4—2 52 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND stage movement which is fixed on to the central pillar of the balance. A thread of silk (as supphed for the Kew magnetometer) is fastened to the stage, passes over the bevelled edge, through two eyes (e, e) on a Hght frame holding the mirror, up over the bevelled edge of the bracket, and is fastened to the bracket. The microscope-stage movement allows the distance between the threads to be adjusted, and also enables the azimuth of the mirror to be altered. Of course, if the mirror were weightless, it would not affect the sensibihty of the balance, and the threads might be brought very close together. But the weight of the mirror — it is silver on glass, 56 milhms. x 38 miUims. x 10 miUims. — has a considerable effect on the sensibility, diminishing it with de- crease of distance between the points of suspension. In practice it has been found convenient to work with the threads parallel, and from 3 to 4 miUims. apart, the time of swing one way being adjusted to about 20 seconds. A less time hardly suffices for a correct determination and record of the scale reading. Taking 4 milHms. as the distance, and supposing the bracket to be 600 millims. below the knife-edge of the balance, the mirror evidently turns through an angle 150 times as great as that through which the beam turns. The drawback to this method of magnification is that the mirror has its own time of swing and is easily disturbed. The swings of the mirror and the disturbances are, however, effectually damped by having four light copper vanes attached to the end of a thin wire, projecting down from the mirror and working in a dashpot with four radial partitions not quite meeting in the centre, one vane being in each compartment. I found that mineral lubri- cating oil is very suitable for the dashpot, as the surface keeps quite clean and there is little evaporation. The swings of the balance are also very greatly damped by this arrangement, but the effect of this will be discussed later. The telescope and scale are in the room over the balance-room (see Fig. 1), a hole being cut through the floor, and a small glass window being fixed in the top of the case. As the suspended mirror is in a vertical plane it is necessary to have an inclined mirror fixed in front of it to direct the light from the scale horizontally on to it and back again to the telescope. With the magnification used it was necessary, for good definition, to have an ex- ceedingly good inclined mirror, and several were rejected before a suitable one was obtained. That finally used is a silver on glass oval mirror, 60 milHms. x 40 milhms., by Browning. The glass window in the case is optically worked and carefully adjusted to be normal to the path of the light. The telescope has a 3-inch object-glass of about 4 feet focal length. It is fixed on a brick pillar, on one of the brick arches which form the ceiHng of the balance-room, and it rises free from the floor of the observing-room. To destroy vibration one course of bricks is replaced by blocks of india- rubber. The scale has 50 divisions to the inch (say J milhm.), ruled diagonally, THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 53 and divided to tenths by cross-lines. It is photographed on glass from a scale drawn on paper with very great care, 50 inches long (say 127 centims.), and with 500 divisions. The photograph is ^^^th of this length, and only the central part of the scale, about 60 divisions in length, has been used. The diagonal ruling enables a tenth of a division to be read with certainty, and the readings recorded in the Tables, pp. 98-128, are in tenths. Though the lines appear somewhat coarse, I have not been able to find another scale equal to it in distinctness and in ease of reading. As all the results depend on the ratio of measurements, taken almost simultaneously, of deflection due to attraction and rider respectively, in the same part of the scale, I have not thought it necessary to calibrate it. The scale is fixed horizontally on the end of the telescope close to the object-glass with a piece of ground glass over it. It was illuminated in general by an incandescent lamp placed above it, once by an Argand burner. The distance from the scale to the mirror and back is about 5 metres. It follows that 1 division of the scale corresponds to an angular motion of the mirror through -0001 radian. But this is at least 150 times the angle through which the beam turns for the same defiection. So that 1 scale division implies an angular motion of -0000006 radian, or -^" , in the beam. As the total length of swing in Table III is never more than 12 divisions, the angular vibrations of the beam are at the most about l"-6, and the hnear vibrations of the masses, since the half beam is about 60 centims., are at the most about -005 millim. This shows that it is quite unnecessary to consider any change of distance due to vibration. The greatest deviation from the mean in any of the series of weighings recorded is about 1 per cent, of the rider-value, corresponding to about -^-^th. of a division, or an angle of -^" in the beam, and a distance of -00004 miUim., say goo'ooo inch, in the motion of the masses. This seems to show that the method is accurate as well as sensitive. Determination of the Value of the Scale-Divisions by means of Riders. This was done by means of centigramme-riders (Fig. 4), these being the least weights which appeared capable of sufficiently accurate determination. Instead of transferring the same rider from point to point, it was much easier to use two equal riders, and to take one up while the other was being let down a given distance from it. The distance selected was about 2-5 centims., since the deflection due to the transfer of one centigramme so far along the beam was nearly equal to that due to the greatest attraction to be measured. At first the riders when on the beam rested in V-notches in a pair of parallel brass strips fixed on and parallel to the beam. But this plan was soon abandoned, as there was no certainty about the position of the rider in the notches. The riders were then supported in little wire frames, each hung 54 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND by two cocoon-fibres from the edges of a plate fixed to the beam, the edges being parallel to the central knife-edge. The only objection to this method was the very considerable time spent in replacing the fibres after the breakages which occurred on dusting or any readjustment of the balance. Fig. 4. Rider, actual size, and end of lifting-rod, r^ ^JX Fig. 5 a. Subsidiary rider-beam, 66, attached to centre Fig. 5 6. Wire frames depending like of balance-beam, BB, by plate jp just above central scale-pans from ends of 66, Fig. 5 a, knife-edge, k (half size). side and end views (half size). Ultimately a small subsidiary beam, about 2-5 centims. long, was attached to the centre of the balance-beam just above the knife-edge (Fig. 5a), the scale-pans being represented by small wire frames in which the riders could rest (Fig. 56). These frames depend from agate pieces resting on steel points at the extremities of the subsidiary beam in the way now usually adopted in delicate assay-balances. This mode of supporting the riders appears to be perfectly satisfactory. To raise or lower the riders two short horizontal lifting-rods parallel to the beam move up and down within the supporting wire frames with a nearly parallel motion, and on them are two metal pieces with their upper surfaces shaped so that the riders rest on them without swinging (Fig. 4, r). They are the extremities of L-shaped projections from a jointed parallelogram- framework (Fig. 6), supported on an upright in front of the subsidiary beam. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 56 The framework is moved by a tongue engaging with it, and projecting from a horizontal rod, which rotates about its axis in bearings, one within the case and the other outside. The rod is turned through an angle of about 30° between stops by an endless string passing upwards and round a wheel in the observing-room. The parallelogram-framework and the bearing of the rotating rod within the case are both supported, independently of the case, from the ceihng. At first they were supported respectively on the central pillar of the balance and on the case ; but when the increase of optical sensitiveness enabled me to detect small irregularities, I realised how essential it was for accurate weighing that all parts of the apparatus moved from the outside should be supported quite independently of the balance. Even the string moving the Fig. 6. Lifting-rods to raise or lower riders (half size). rod transmitted great and continual vibration. The rod and the framework with the lifting-levers were, therefore, supported by iron rods coming down from the ceiling through holes in the top of the case, large pieces of cardboard stretching from these rods over the holes to hinder the passage of dust into the case. Once or twice in the course of prehminary experiments irregularities were traced to accidental contact of outside bodies with the case. It appeared just possible that there might be electrification of the riders by friction with the lifting-rods, especially when they were supported by cocoon-silk. It was, therefore, advisable that the surface of the lifting-rods should be of the same kind as that of the riders. As the latter are silver wire gilded, the lifting-rods are also gilded. It may not be uninteresting to note here a curious phenomenon which occurred during some early preliminary experiments. The shaped pieces on the Hfting-rods were then of wood covered with gold leaf, put on with ordinary paste. After they had been on for some months, I obtained some very various results for the deflection due to the riders, and on examining the lifting-rods I found that a number of long 56 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND needle-growths projected from the wood pieces and interfered with the supporting wire frames. At first I thought these were organic, but my colleague, Professor Hillhouse, examined them and found that they were crystalhne. Doubtless, the hygroscopic paste set up electric action between the gold leaf and the brass to which the wood pieces were attached, and the crystals were probably zinc sulphate. The wood was then replaced by brass gilded, and no further difficulty of the kind was experienced. The length of the subsidiary beam was kindly determined for me by Mr. Glazebrook at the Cavendish Laboratory. The steel points are hardly sharp enough to determine the distance to 1 in 10,000, but the mean of the results is sufficiently exact. The following are Mr. Glazebrook's determina- tions ; the four points being denoted by a, h,c,d\ Date 1889 July 4 Julv 11 July 12 Temperature Number of readings , , Number of , , , «t°^ readings ^ ^ ^^ ^ 22^5 21-5 23 6 3 3 inches \ inches •9985 i 6 ^9979 •9990 i 3 j ^9982 •9988 3 ^9979 These are in terms of a gun-metal standard of which the error is only 3 in 100,000 at 0°, and, therefore, for my purpose negUgible. The beam is of brass, and we may assume with sufficient exactness that it has the same expansion as the standard. The temperature may, therefore, be left out of account. The mean value of \{ah + cd) is therefore -9983375 inch, or taking 2-539977 centims. to the inch we obtain Length of beam at 0°, 2-53575 centims. There is an advantage in fixing this beam at the centre, which should be noted here. Suppose the riders are not quite equal, but have values w and w + S. Let the two ends of the subsidiary beam be distant a and a -f I from the central knife-edge. Then the effect of picking up the rider w from the nearer, and letting down the rider -ii; + S on the further end, is equivalent to putting at unit distance {w + h)(a-\-l)-wa==wl-\-h{a-\-l) = wl (l + ^^ ^-\ , or the error hjw is multiplied by {a + 1)11, and, if the beam is not central, (a -f l)jl may be greater than 1, so that the error is magnified. If, however, the small beam is central, I is equal to — 2a, and the error is multipHed by + J. If the riders are interchanged and the weighings are then repeated, the mean result is the same as if riders with the mean value were used, for w[a + l)- {w-\-Z)a = wl~ Sa THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 57 and the mean of this and the above is (-1) The Attracting and Attracted Masses. These are all made of an alloy of lead and antimony, for the sake of hardness, the specific gravity in each case being about 10-4. They were made at various times and places, the large attracting mass M being made more than 12 years ago by Messrs. Storey, of Manchester. The smaller balancing mass m was made in 1889 by Messrs. Heenan, of Manchester and Birmingham. These were both cast with a 'head' on, and then turned. The attracted masses A and B were made by Messrs. Whitworth, and subjected to hydraulic pressure before turning. The dimensions have been measured from time to time, and there is no evidence of any sensible change of shape. The larger mass M and the attracted masses A and B were weighed at the Mint through the kindness of the Deputy Master and Professor Koberts- Austen. For the weight of the balancing mass m, I am indebted to Messrs. Avery, of Birmingham. The large mass M has suffered two accidents since it was weighed, once being slightly cut into by a saw during some alteration of the case, and once being scratched by coming into contact with a piece of metal fixed to the turn-table in taking it out of its place. The saw-cut was carefully filled in with lead, and the scratch removed only a fraction of a gramme, as was determined by taking a mould of the hollow. I should be glad to think that the determination of the attraction was sufficiently exact to make reweighing necessary, but I am afraid that the alteration in weight is very far beyond the important figures, and I therefore take the original weight as sufficiently near the truth. The masses A and B have been gilded since the original weighing, but I carefully determined their increase of weight by the balance used in the gravitation experiment. The values given below in the second column are the true masses. In the third column are the masses of M and m, less the air displaced by them, this being taken as 18-4:1 and 9-2 grms. respectively. It will be shown later that the true masses of A and B and the reduced masses of M and m may be used in the calculation of the result. True mass in grammes Mass less that of air displaced M m A B 153407-26 76497-4 21582-33 21566-21 153388-85 76488-2 58 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Suspension of the Attracted Masses. Each of the attracted masses is drilled through along a diameter, the hole being -6215 centim. in diameter, and a brass rod (Fig. 7) terminating in an eye e below, is passed through the hole. The mass is secured in position by a nut n working in a screw-thread cut for a short distance in the rod. An exactly similar rod terminating in a similar eye e', and with a similar nut n', is fastened end to end to this by a union u. The nuts and the inner sides of the enlargements for the eyes are hollowed out so as to fit exactly on to the spheres. From the ends of the balance beam hang down stout brass wires terminating in hooks. If these hooks are passed through the eyes e' the attracted masses are close to the floor of the balance-case, and their centres are adjusted to be about 32 centims. from the centre of the large attracting mass when under either of them. If the masses are turned over so that the hooks pass through the eyes e, they are about 30 centims. higher or at nearly double the distance, the length ee' being about 48 centims. The rods being perfectly symmetrical about the union u, the attraction on them is the same in either position. The weight of each is about 212 grms., or about j-J^ of the attracted mass, so that any small variation in their position would produce a negligible variation in the total attraction. By the differential method, the attraction on them entirely disappears from the results. The Mode of Support of the Attracting Masses M and m. This has already been described when describing the turn-table. The Riders. Four centigramme-riders, A, B, C, D, of silver wire gilt were made by Mr. Oerthng of the form shown in Fig. 4. These were weighed in 1886 at the Bureau International des Poids et Mesures, by M. Thiesen. The following is an extract from the certificate : ' Densite et volume. Comme densite on a accepte celle de 1' argent, et par consequent comme volume de chacun des cavahers, 0-0010 miUihtre. ' Deterynination des poids des cavaliers. L'etude des poids de ces quatre cavaliers a ete faite par M. le Dr. Thiesen, adjoint du Bureau International, Fig. 7. Suspender for Attracted Mass (one-fourth size). THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 59 charge de la section des pesees. M. Thiesen au moyen de la balance Stuchrath, destinee a des poids au dessous du gramme, a d'abord determine les differences entre les quatre cavaliers pris deux a deux dans les six combinaisons possibles, et ensuite la difference entre 1' ensemble des quatre cavaliers et le poids de 40 milligrms. de la serie du Bureau, serie en platine iridie recemment etalonnee par M. Thiesen. Les comparaisons ont ete faites du 19 au 29 Mars, 1886. ' Resultats. De I'ensemble de ces comparaisons resultent les poids : A = 10-1247 milligrms. B = 10-0615 0=10-1196 Z)= 10-1262 ' L'incertitude de ces determinations ne depasse pas 0-001 milhgrm.' A and D were selected for use as being the nearest to each other in value. B and C were kept untouched in boxes till 1890. In the various experiments made between 1886 and the final weighings, A and D had necessarily been handled to some extent, especially through the frequent breaking of the silk fibre suspension used before the subsidiary beam described above, and it appeared possible that their weights might be altered. It was also necessary to determine whether an appreciable amount of dust was deposited on them in the course of several weeks as it was inconvenient to dust them frequently. The riders B and C might be assumed to have the same weight as in 1886, and could be taken as standards. To make the weighings a 16-inch chemical balance was arranged with a double-suspension mirror on exactly the principle already described for the large balance. The apparatus was put together quickly with materials at hand, and might easily be greatly improved. It is only described here to show how accurate the method is, even with such rough apparatus, and that it is applicable to a small as well as a large balance. A cork sliding on the pointer with a horizontal needle stuck in it, served to support one thread of the mirror; a stand with a projecting arm — one made to hold platinum wires in a Bunsen flame — served to support the other thread. A wire with a small copper vane depended from the mirror and was immersed in an oil dashpot. The telescope and a miUimetre-scale were on a level with the mirror about 2 metres distant on one side of the balance. Two brass strips, parallel to each other and the beam, were fixed on the top of one arm of the beam, and in each of these were two V-notches in which centigramme-riders could rest. Two levers, worked by cams on a rod rotated by the observer, picked one rider up and let down the other, so that the effect was equivalent to the transfer of 1 centigrm. from one notch to the other. Their distance apart was such that this was equivalent to the addition of 60 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND •3284 mimgrm. to one pan of the balance. This was the arrangement described in my former paper. Attached to one pan was a pair of brass strips parallel to each other, and such that the riders A, B, C, or D, would just rest across them. Two lifting-rods worked up and down between these strips, so that of the two riders to be compared, one could be picked up at the instant the other was let down. The lifting-rods were worked by a rod rotated by the observer and supported quite independently of the balance, and of the slab on which it rested. By this plan the value of the scale-divisions and the shifting of the centre of swing on changing the weights to be compared, could all be determined without raising the beam of the balance between the successive weighings, an essential condition, I beheve, for exact work. The weighings were made in the large room of the Physical Laboratory, and no precaution was taken to protect the balance-case beyond placing a board in front of it. The room is draughty and subject to great variations of temperature, so that the weighings were made under very disadvantageous circumstances. One result of this was a rapid and sometimes very great change of resting-point in the course of a few hours, so that the scale passed out of the field of view. In order to bring it back without opening the case, two glass tubes passed through the top of the case, almost down to the scale- pans, and small bits of wire could be dropped through these on to either pan as needed. Caps fitted on to the tubes to prevent draughts. This plan appears worthy of mention, as it suggests a mode of determining the value of a scale-division when a balance is either too sensitive for riders or has no special arrangement for their accurate use. If a piece of wire weighing, say, 1 milligrm. is cut into say ten nearly equal parts, and if these are dropped on to the two pans alternately the shiftings of the centre of swing will be to and fro, about equal distances, due to about -1 milligrm., but the sum of the shiftings will be that due to 1 miUigrm., and the balance at the end will be nearly in the same position as at the beginning. The following is an abstract of the comparisons of the riders. They were made soon after the first determinations of attraction on February 4, when A and D had not been dusted for three months. In each case three extremities of swing were observed, and the centre of swing was determined from these by the graphic construction described later (p. 72). The centres of swing were combined in consecutive threes in the usual way to give the differences in scale-divisions. Thus, in the first series, the successive centres of swing with D and A alternately in the scale-pan were D A D A D !31 223 217 211-9 208 THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 61 whence (D - A)^ = ^ - 223 = + 1-0 division, fn A^ 017 223 + 211-9 ^ .^ .. . . (D — ^)2 = 217— = — 0-45 division, (D - A)^ = ^1+^ _ 211-9 = + 0-6 division. Mean D — A = -SS division. Successive values of the differences alone are given below. The time of swing one way was about 16 seconds. February 16, 1890. (1) Comparison of A and D, undusted. Deflection due to -328 milligrm., 83-45, 82-45, 84-45 divisions. Mean 83-45 divisions. D- A = 1-0, - -45, + -06 division. Mean -38 division; therefore D = .4 + -0015 milligrm. (2) Comparison of A undusted, D dusted. Value of scale-division taken as in the last. D- A = -'6, + -3, - -1, - -4, + -25. Mean - -09 division ; therefore D = A — -0004 milligrm. Februanj 17, 1890. (3) Comparison of A and D, both dusted. Value of scale-division taken as below (4). B- A = -'I, - -2, + -3, - -3, - -8. Mean - -22 division; therefore B -= A — -0008 milhgrm. (4) Comparison of C and B. Beflection due to -328 milligrm., 85-35, 85-4, 84-65. Mean 85-13 divisions. B-C = + -15, -00, + -05, - -15, + -05, + -3, + -05, - -05, - -35, -05, -35, -45, -50, -2. Mean -114 division; therefore D = C + -00044 milligrm. February 18, 1890. (5) Comparison of C and B repeated. Beflection due to -328 milligrm., 92-75, 92-3, 91-65. Mean 92-23 divisions. B-C= -35, - -05, - -8, - -95, - -1, + -05, 0, - -15, - -1, + -05. Mean — -17 division; therefore B = C — -0006 milligrm. 62 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Combining this with the last, and weighting them in the ratio of the numbers of determinations in each, D = C + (-00044 X 14 -- -0006 x 10) h- 24 = - -0000 milHgrm. (6) Comparison of A and D. Value of scale-division taken as above, -328 milHgrm. = 92-23 divisions. D-A = -45, -25, -1, - -2, - -1, -35, -25, -45, -60, -5, -5, -55, -5, -75, -55, •1, -05, -45, -10, -30, -8, -9, -35, -2, -30, -55, -50, -35, -45, -45. Mean -378 division; therefore D = A + -00134 milligrm. Examining the values obtained in (1), (2), and (3), it will be seen that no trustworthy evidence is given of a difference due to dusting. Any existing difference was probably under -002 milHgrm., and since the weighings on February 4, before dusting, were made with the attracted masses in the upper position, when the attraction was only one-fourth of that on which the final results depend, we may safely neglect the effect. After this the riders were dusted more frequently, so that we may probably assume their values more constant. The comparisons of C and D, and of A and D, in (4), (5), and (6), were made more carefully. That of A and D in (6) is much the best of the series, the air in the laboratory happening to be steadier while it was made. The range between the greatest and least values of the difference is one scale- division, or -0036 miUigrm., and the different results are grouped fairly closely about the mean. The centres of swing and the differences are plotted in Diagram VIII (p. 136). I do not claim that these results show any remarkable accuracy when compared with those obtained at the Bureau International des Poids et Mesures, but remembering how rough the apparatus was, and how little precaution was taken to ward off air-currents, I have not the slightest doubt that, with special design of apparatus and more suitable locality, the results could be very greatly improved, and the accuracy carried far beyond anything hitherto reached. As they stand, they seem to show the value of the com- bination of a short time of swing with optical magnification. The results of comparisons (4), (5), and (6), is, that if C has its Paris value, viz., C = 10-1196 miUigrms., then, A = 10-1183 milligrms., and D = 10-1196 milUgrms. ; whence 1{A ^ D) ^ 10-119 milhgrms. This value may be used in calculating the result, since the riders were interchanged before Set II was taken. The losses experienced since 1886 by A and D are respectively, by A -0064 miUigrm., and by D -0066 milligrm., i.e., they have diminished by THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 63 practically equal amounts. This was to be expected as they have probably received equal amounts of rough usage. The substitution of the subsidiary beam for the cocoon-fibre suspension of the riders having greatly diminished the handling to which they were subjected, I have not thought it necessary to weigh them again during the work. Linear Measurements. In the mathematical theory it will be shown that the lengths required are those marked in Fig. 14, viz., the horizontal distances, L and I, and the vertical distances, D^ D^ , d^d^, H^ H^ , h-^h^. The Horizontal Distances. Except when estimating the moment of the rider, the distance L is really that between the verticals through the centre of M and the centre of the more distant attracted mass. But the verticals through the centre of M in each position so nearly passed through the centre of the mass above it, and, therefore, through the knife-edge from which it hung, that L was taken as equal to the length of the beam (p. 50). The accuracy of this adjustment was secured as follows. A horizontal cross-piece was fixed on the top of each attracted mass, with two horizon- tal cards at its two ends, each with a portion of a circular arc on it, with radius equal to that of the large mass M, and with centre over that of the attracted mass (Fig. 8). A plumb-line was then hung just in front of the case, and the balance was moved by the horizontal screws bearing against the base-plate until the plumb- line always appeared to touch the circular arc above, when it appeared to touch the large mass below. The adjustment was not quite perfect, but the error in the worst case was probably not more than 1 millim., and certainly less than 2 millims. Such an error in the horizontal distance is negligible. The distance I had different values for the two positions occupied by m on the turn-table. Calling these values l-^ and l^ respectively, l-^ + l^ was found by measuring a, the inside distance between M and m, arranged as in Set II, and 6, the inside distance between them, when m was put on the same side of the turn-table as M, and adding to a + h the sum of the diameters of M and m in the radial direction of the turn-table as taken by square calipers. Fig. 8. Plumb-line Adjustment of Masses. 64 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND The following are the values in terms of the cathetometer-scale already referred to, the temperature being 15° C. : a = 157-01 h= 33-95 Diameter of M = 30-52 m= 24-23 therefore h + k- 245-71 The value of l^ - l^. was found by measuring the shortest distance of m from the wall when respectively in the first and the second positions on the turn-table. It was found that whence h = 122-91o l^ = 122-795. We may obtain from these measures an independent value of the radius of the circle in which the centre of M moves. With perfect adjustment this should be IL = 61-66 at 18°. It is equal to a + radius of M + radius of m - l^ , or, by the above measures, = 157-01 + 15-26 + 12-115 - 122-795 = 61-59, which is only -07 centim. less than \L. Inasmuch as the wood probably expanded less than the cathetometer- scale, while the metal expanded more, I have assumed as a rough approxi- mation that the total expansion equalled that of the scale, so that the values of /j and I2 are correct at 18° (see p. 50). No importance is, however, to be attached to this temperature-correction. The Vertical Distances. At the conclusion of each set of weighings with the attracted masses in a given position, the vertical distances between the top of the attracting masses and the bottom or top of the attracted masses (accordingly as they were in the upper or lower position) were measured by the cathetometer already referred to. This instrument is of the well-known design of the Cambridge Scientific Instrument Company, and is especially adapted for measuring differences of level at different distances in different vertical planes. It reads to -002 centim. The account of these measurements will be found in Table II (p. 89, et seq.). To find the distances D, f/, H, h (Fig. 14), it was necessary to add to the actual distances measured the sum or difference of the vertical radii of the attracting and attracted masses, and, therefore, the vertical diameters of all the masses were measured. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 65 For this purpose I used a cathetometer which has lately been constructed for me by Messrs. Bailey, of Bennett's Hill, Birmingham. I have to thank Mr. Potts, of that firm, for his care in its construction, and also for the trouble which he has taken in the construction and alteration of much of the apparatus used throughout the work recorded in this paper. As the cathetometer is, I beheve, new in design and satisfactory in its performance, it appears worthy of description. The Cathetometer used to measure Vertical Diameters (Fig. 9). There are two telescopes, one to sight the upper the other to sight the lower of the points between which the vertical height is required. There is no scale on the instrument, but after the telescopes are fixed to sight the two points the instru- ment is turned round a vertical axis, so that the telescopes sight a vertical scale at the same distance from them as the points. In general, of course, the cross- wire will appear to lie between two divisions, but by means of the fine adjustment, to be described below, the two nearest scale-divisions are brought in succession on to the cross-wire, and by interpolation the reading correspond- ing to the point first sighted by the telescope is determined. The telescopes are fixed on collars running up and down the main pillar, which has a section of the form shown in Fig. 10 (shaded). The guides consist of three knobs, k, k, on the inside of the collar, two sliding in a vertical V-groove and one on a plane, both groove and plane being at the back of the pillar. A screw, s, clamps the collar in any position. Gut strings running up over pulleys and supporting counterpoises, sliding on the thinner pillars (see Fig. 9), are attached to the collars so that these move easily. At first springs were used to keep the knobs always in contact, but I found it much better to remove these and trust merely to hand-pressure to keep the collars in the proper position before clamping with the screw s. p. c. w. 6 Fig. 9. Cathetometer used to measure Vertical Diameters. 66 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND The fine adj ustment is secured by the use of a piece of plate-glass {g, Fig. 10), placed in the front of each object-glass and capable of rotation about a Fig. 10. Section of pillar and collar of new Cathetometer. 5, clamping screw, k, k, guiding knobs, g, glass plate for fine adjustment, turning on axis hh, with pointer at %> perpendicular to plane of figure. horizontal axis, hh. A pointer is fixed on the end of this axis at f, and at its end is a small glass plate with a scratch on it moving close against a straight scale. If the plate is initially normal to the optic axis of the telescope, on turning it through an angle </>, the ray which now comes along the optic axis has been shifted, by transmission through the plate, parallel to itself, a distance t sin ((/> - j/f)/cos xjj, where t is the thickness of the plate and ifj is the angle of refraction within it (see Fig. 11). Fig. 11. Section of fine adjustment plate. For small angles this shifting happens to be nearly proportional to tan</), and, therefore, to the reading on the straight scale. To show how nearly this is the case the following table gives the shifting for angles of 5°, 10°, and 20°, with a thickness oit = I centim. and a refractive index a -= I: ngle Shifting* 5= itan 5°(1 + -00042) 10° itanlO°(l + -00160) 20° itan20°(l + -00526) The error in taking the shifting as proportional to tanc/) is, up to 20°, quite negUgible in ordinary telescope-cathetometer work. If it is desirable to have greater accuracy, it is probably best to use a table of corrections to the tangent ; but it is possible to get an exact scale thus : * [These expressions are given with the wrong sign in the original paper. A sHght correction has also been made in the values of the numerical coefiicients. Ed.] THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 67 Let OP, Fig. 12, represent the pointer of length /*, making an angle </> with a line MN. Let a pointer PM jointed to this at P be of length /ur, and let its extremity M move on the Hne MN. Drawing OD at right angles to MN, if s is the shifting, we have sin (<^ — ijj) or OD^OP _ rs s = ~OD. r cos l/j Probably the practical difficulties in the use of such an arrangement would render it troublesome and uncertain. The plate is used as follows: Adjust it normal to the optic axis of the telescope, and move the telescope till the required point is brought as near to the cross- wire as is possible by the hand. Clamp the telescope, and then turn the plate till the point is exactly on the cross-wire. Read on its scale the position of the pointer attached to the plate. Repeat these operations with the other telescope on the other point, then turn the instrument about its vertical axis till the telescopes sight the vertical scale placed at the same distance away as the two points. Looking through one of the telescopes the cross- wire is in general not exactly on a division. Turn the plate so that first the nearest division above, and next the nearest division below, is on the cross-wire. Reading the position of the pointer in each case, interpolation gives us the reading on the vertical scale corresponding to the position of the pointer when the cross- wire was between the two scale divisions. Doing this for each telescope the difference between the two points is found in terms of the vertical scale. The plates I have used are about 9 millims. thick, and the pointers about 9 centims. long. They move over scales such that 25 to 27 divisions corre- spond to a shifting of 1 milhm. The lower scale is graduated from to 50, the upper from 50 to 100, to prevent confusion. The 50 divisions occupy a distance of 66 millims. It will be observed that in this form of instrument the 'level error' is practically entirely obviated. It can only come in if the scale is not at the same distance as the height to be measured, and may then be made neghgible in practice by levelling the telescopes. Indeed, the uncertainty of measure- ment appears only to depend on the uncertainty with which the cross-wire 5—2 Fig. 12. 68 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND can be brought to the proper point, that is, it depends only on the magnifying power and definition of the telescopes used. To illustrate the use of the instrument, a full account of the determinations of the vertical diameters is given in Table II. Below are the results, and for the sake of showing that there has certainly been no great change in shape, I give results obtained with a cathetometer more than 10 years earlier at the Cavendish Laboratory at Cambridge. 1890 1880 centims. centims. Large attmcting mass J/ ... 30-526 30-5192 Small „ „ m ... 24-176 Attracted mass .4 15-8203 15-8166 B 15-7829 15-7842 The diameters of M and m in a horizontal direction parallel to a radius of the turn-table measured by square calipers were M = 30-52 centims. m = 24-23 „ Temferature-Correction. Though the expansion of the masses was to be expected of an unimportant amount, I thought it advisable to attempt to measure it, in case there might be anything anomalous. One of the attracted spheres, B, was for this purpose placed between two vertical levers, in a tank through which could be run a continuous stream of cold or warm water. These levers depended from horizontal rods which could rock or slightly rotate on fine point-suspensions. This was, in fact, a kind of double Lavoisier and Laplace apparatus. The motion of each lever was shown by another lever of about the same length, rising vertically up from each horizontal axis, and serving as the moving support for a double-suspension mirror in which was viewed the reflection of a millimetre-scale. Two telescopes and one scale were used for the two mirrors, though it would not have been difficult to arrange one telescope and two scales. The value of one scale-division was determined by inserting a piece of thin glass between the sphere and each lever in turn. The method is exceedingly sensitive, but I have not been able to make it exact, owing to the warping produced in the rods due to unequal temperatures. The measures of the expansion varied between -0000214 and -0000277, both vertical and horizontal diameters (in the position in the balance) being tested. The true value is probably nearly -000025 or 1/40000. It will, therefore, lead to no appreciable error if we take the expansion as equal to that of the scale of the cathetometer, say 1/60000 (see p. 92, Table II). THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 69 Determination of the Attraction by the Balance. When the balance is used to measure such small forces and weights as those with which we are here concerned, it must be left swinging on its knife- edge throughout any set of weighings in which the deflections are to be com- pared one with another. For there is not the slightest reason to suppose that if the beam is Hfted up and let down again, its new position of equilibrium will coincide with the old. And again, the beam, especially with such loads as the attracted masses, is put into a state of considerable strain, and continues to alter its shape sensibly for hours, and probably even days, after the masses are put on to it. I have, therefore, always left the beam free for at least two or three days before commencing work with the balance, and it has of course remained free during the course of each day's work. The balance- room was never entered just before any weighing, as it took many hours for the disturbance due to entrance and interference with the case to die away. When the turn-table supporting the attracting masses is moved half round, from one stop to the other, the bulk of the attraction is taken away from one attracted mass and put on to the other. The balance, being free, is slightly tilted over to the side on which is the larger attracting mass. But the deflection in the apparatus as arranged is so very small — at the most only 10 scale-divisions — that errors of reading can only be neutralised by making a great number of successive measures. Probably other errors are also largely eliminated, such as those due to the deposition of dust particles, shaking, change of ground level, and varying air-currents. Of such errors I have found those due to varying air-currents by far the worst. Sometimes — especially in autumn and winter — the balance will move quite irregularly through more than a scale-division, and continue to move to and fro in this way for days or weeks. When in such an unsteady condition it is useless for accurate work. In spring and summer, however, it is much more steady as a rule, and frequently the scale can hardly be seen to move. I have never worked when on looking into the telescope for some time the irregular movements appeared to be more than a fraction of a tenth, i.e., a fraction of one of the diagonal divisions, though, doubtless, irregularities comparable with a tenth of a whole division have often made their appearance in the work. It is perhaps not safe to ascribe these always to air-currents. I have always found the air steadiest in warm quiet weather, with a slowly rising temperature in the balance-room, and most unsteady after a sudden fall of temperature. As the alteration of temperature spreads downwards, this is fully in accord with Lord Rayleigh's observation that when the air is steady the ceiling is warmer than the floor, and that when it is unsteady the 70 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND floor is the warmer of the two. In the observing-room I had a gas stove often kept burning day and night, in the hope that the higher temperature it produced in the ceihng of the balance-room below might steady the air. But the vertical walls of the balance-room interfered with the action of the ceihng, and often produced unsteadiness. A door opening or shutting anywhere in the building had a visible though transient effect, doubtless through an air- wave. In a high wind the balance was always unsteady, partly, I suspect, through rushes of air into and out of the case with sudden pressure-changes, and partly through changes of grouDd-level, with variations of wind-pressure against the building. At all times there was a march, in one direction or the other, of the centre of swing. This was especially marked soon after the frame was lowered and the beam left free. As already remarked, readings were not taken till changes due to change in strain of the beam had subsided. But the march was very appreciable at other times, as will be seen from the diagrams. Perhaps the change was sometimes due to tilting of the ground, with barometric variation, since the balance was a very dehcate level, and sometimes due to the change in buoyancy of the air affecting the two sides unequally, though I have not been able to make out any direct connection between barometric height and position of centre of swing. I believe that the explanation is to be sought for the most part in unsymmetrical effect on the beam of sHght changes of temperature, for I have frequently noticed that a rising temperature produced an upward march, and a falling one a downward march. This explanation is supported by the following table (p. 71) of observations of the centre of swing, extending from May 9 to May 22, 1890, the balance being free, and the balance-room undisturbed meanwhile. The relation between temperature and centre of swing is represented in Diagram IX (p. 136). Of course, after a change in the position of the attracting masses or of the riders, the balance does not at once settle in a new position of equilibrium, but oscillates about it. Inasmuch as the balance never rests in this position, it is better to term it the centre of swing rather than the equihbrium position or resting-point. The dashpot used to damp the vibrations of the mirror reflecting the scale serves also to damp those of the balance-beam, and they die down rapidly. Instead of waiting, however, to observe directly the point on which they are closing in, it is much more exact, and also saves much time, to find the centre of swing, as with an undamped balance, from the extremities of the swmgs. I have always observed and recorded four extremities of three successive swings, occupying in all a httle more than a minute. Notwithstanding the very considerable damping, the successive lengths of swing are still in geometrical progression, but the rate of reduction is too great to allow the ordinary approximation, in which the geometrical is assumed THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 71 I I Temperature Date, 1890 Time Centre of Barometer swing Balance- room Observing- room May 9 11.5 a.ni. 1360 12°0 13-4 739-8 12.55 p.m. 1330 120 15-0 739-2 „ 12 11.15 a.m. 133-8 12-05 14-5 738-6 1.15 p.m. 131-9 12-05 15-8 738-3 2.40 p.m. 133-7 12-05 16-6 738-1 Stove left on all night of 12th-13th „ 13 11.0 a.m. 181-7 12-6 17-5 740-2 12.35 p.m. 181-5 12-6 18-4 740-3 3.15 p.m. 185-0 12-7 18-6 740-3 Stove turned off 5.25 p.m. 189-4 12-7 16-5 740-5 „ 14 11.20 a.m. 167-4 12-6 14-3 745-8 1.10 p.m. 165-5 12-6 14-4 746-0 „ 15 11.5 a.m. 156-8 12-4 13-8 749-5 2.45 p.m. 160-0 12-4 140 749-3 „ 16 1.25 p.m. 158-3 12-4 14-0 744-3 „ 17 8.50 p.m. 171-0 12-55 13-7 741-9 „ 19 10.30 a.m. 174-3 12-55 14-0 743-5 6.5 p.m. 181-8 12-6 14-0 741-0 „ 20 11.30 a.m. 175-2 12-75 14-0 739-8 1.5 p.m. 173-7 12-75 141 740-0 5.20 p.m. 172-3 12-7 13-8 741-7 ,, 21 11.10 a.m. 172-7 12-6 13-7 751-9 „ 22 11.30 a.m. 192 about 12-85 14-1 757-6 to be an arithmetical progression. The exact method of determining the centre of swing is as follows : Let a, h, c, d be four successive readings of extremities of swing, and let X be the reading of the required centre. Let the constant ratio of each swing length to the next be A. Then a-x^X(x- b), (1) X — 6 = A (c — x), (2) c-x = X{x-d) (3) Eliminating A from (1) and (2), we may readily obtain x in the form and from (2) and (3) x=b+-, — i'~y ., (5) (c — 6) + (c — d) 72 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND With no disturbances and no errors of reading, the values of x in (4) and (5) will coincide ; but usually there is some small difference, the result of error or disturbance, and it is better to find both and take the mean. A third value might be obtained from (1) and (3); but it appears unadvisable to combine directly observations so far separated in time. These formulae lend themselves to easy arithmetical treatment, especially with the aid of a shde-rule; but the following graphic method of finding the centre of swing is much less tiring and quite sufficiently exact. Let the line OA, Fig. 13, represent the scale; its zero, and A, B, C, D the points distant respectively a, b, c, d from 0. Let O'C be a parallel line, B\ C, D' being points opposite to B, C, D respectively. Let AB' and BC intersect in iiTi . Draw X^K^X^^' perpendicular to OA. Then Zj is the centre of swing given by equation (4). For AX^ AX, K,X, ^ X^^X.B X.B'X.'B'" K,X,' Z/C ZiC i.e., Zj is the point dividing AB and BC in the same ratio. Similarly if BC and CD' intersect in K2, and X2K2X2 be drawn perpendicular to OA, X2 is the point given by equation (5). The third point given by equations (1) and (3) is obtained fronj the intersection of AB' and CD', but evidently a small error in C or D' may considerably alter the position of this point, and it is better not to use it. The construction was carried out thus: a large opal glass plate, 10 in. x 11 in., was etched with cross- Fig. 13. lines 10 to the inch, so as to present the appearance of ordinary section-paper. The glaze was taken of! so that pencil-marks could be made. A diagonal line ran at 45° across the plate through the corners of the inch squares, and this was always taken as the line BC in the figure. Taking any convenient horizontal line, usually, of course, far below the plate, as zero, each inch represented a scale-division, each tenth a diagonal division. The values of b and c fixed the hues to be taken as OA, O'C, and on these were marked the points A, C, B', U. A long glass shp, with a straight scratch on it, was then laid across from A to B' so that the scratch passed through A and B' , and its intersection A\ with the diagonal BC w^as x-^ from the zero hue. The slip was then laid with the scratch passing through C and U , and its intersection K^ with BC gave x^. It will be observed that all the actual construction for a set of readings of the balance-swings consisted in marking four points on the plate. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 73 The following cases, the first of very regular, the second of very disturbed swing, will serve to compare the results by this exact method with those obtained from the ordinary arithmetic-mean method. At the same time they will show how nearly constant is the ratio of swing decrease. Date and Number, 1890 Scale-read- ings in diagonal divisions Length of swing Ratio of each to preceding Centre of swing, exact Centre of swing, approximate a 4- 26 -H c 4 May 4, No. 3 865 939 896 921 74 43 25 •581 •581 911-8 91b8 909-75 913-0 Mean 911^8 Mean 911-375 Sept. 17, No. 45 1118 1053 1093 1068 65 40 25 -615 -625 1077-8 1077-6* 1079-25 1076-75 Mean 1077-7 Mean 1078-0 In finding the attraction the observations were always made in the same order, the determination of the scale-value of rider and attraction being sandwiched so that each might be equally affected by any comparatively slow changes. Starting with the initial position, the attracting masses and riders were so arranged that, on moving either, the balance was deflected in the same direction and over the same part of the scale. The following was the order of proceeding always observed, the column headed ' Centre of swing ' being supposed to contain the values of the position in each case determined from four swing extremities as just explained : Centre of swing. (1) Initial position *i (2) Riders moved r^ ( 3 ) Riders moved back to initial position i^ (4) Masses moved round m^ (5) Masses moved back to initial position i^ (6) Riders moved ^2 (7) Riders moved back to initial position i^ (8) Masses moved round Mg and so on. To minimise the effect of progressive changes these observations were always combined in threes in the following way. Denoting the scale-value of rider by R, and of attraction by M : * [The original has 1078-6 which is evidently a slip ; hence this is not really a good example of a 'very disturbed' swing. Ed.] 74 ON A DETEEMINATION OF THE MEAN DENSITY OF THE EAETH AND From (1), (2), (3) Ri = ri-'^^, „ (3), (4), (5) M, = m,-'^\ „ (5), (6), (7) R2 = r^~'^\ and so on. These again were combined in threes, so that (the notation being continued) the successive values of attraction /rider are i?2 + jB2 ' 2i?2 -^2 "^ -^3 The successive centres of swing i-^, r-^, i^, m-^, etc., correspond to instants of time following each other at intervals of about 2 minutes, rather more than 1 minute being taken up in making and recording the four readings for each, and the rest in making the change of position in rider or mass and waiting for the next readings. It will be seen that each value of M or i^ is based on three successive centres of swing, the w^eighings extending over about 6 minutes, while each value of M/R is based on seven successive centres of swing determined in about 14 minutes. A series of readings was usually continued for about 2 or 3 hours. The temperature in both observing and balance rooms was read at the beginning and end of the series, and the barometric height was also observed. As soon as possible after the desired number of determinations was completed with the attracted masses in one of the two positions, the vertical distances between attracting and attracted masses were measured by the cathetometer in the manner explained in Table II, and the position of the attracted masses was then altered. A full account of all the weighings is given in Table III, and the results are represented in Diagrams I-VI (pp. 130-135). The three upper rows of points in each diagram represent the centres of swing, those in the initial position being marked • . After movement of the rider they are marked x , and after movement of the masses they are marked o. The base-lines for the different rows are altered to save space, as described on the diagrams, for on the scale adopted the rider series would always be about 10 inches above or below the initial series. In Diagram I the rider and mass series are also brought down and superposed on the initial series, so that each of the three has the same average height. It will be seen that all three are affected by the same disturbances. The advantage of the short time of swing and the mode of combining the results in threes will be realised more easily from this superposition. The base-hne may be regarded as a time-scale, as the instants corresponding to successive centres of swing were almost exactly equidistant. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 75 In each case, under the representation of the centres of swing, are plotted the resulting values of MjR, and at the side will be found a representation of the distribution of results about the mean. Assuming that each day's mean value is correct, and that the differences for different days are to be set down to variation of distance, etc., we can find the distribution of all the values about the mean by simply superposing the marginal curves at the side of the figures. The result fairly shows the accuracy as far as the weighing alone is concerned. It is represented in Diagram VII, where A is the mean value of the attraction in the lower, and a that in the upper position. A and a are brought near together to save space, but really they should be 40 inches apart. It will be seen that the range is about 2 per cent, of ^ — a on each side of the mean, or taking the value of ^ — a in milHgrammes weight as about \ milHgrm., and the load on each side as 20 kilogrms., the range is about 1/3 x 10^ of this load on each side of the mean. A comparison of the values of MjR in Diagrams I and II, shows a very curious similarity in the fluctuations, and at first I was inchned to think there was some common external disturbance producing these fluctuations. But an analysis of the two sets of values appeared to show that the resemblance is merely accidental. When the values of M and R are set out separately, it is seen that the fluctuations depend chiefly on M, of which the fluctuations are shghtly hke each other for the two series, while those of R are quite different, but such that they make the fluctuations in MjR resemble each other more closely than those in M alone. Further, it is not easy to see how fluctuations due to some external source would aflect the values of M equally in the upper and lower positions and not have any effect on R. Some periodic change of level might be suspected, but this ought certainly to be traced in R. I have examined all the other diagrams and plotted out the component values of M and R, but have found no trace of resemblance, so that I think the curious likeness in I and II must be set down to accident. There is a curious step by step descent of the centre of swing in the initial position on September 23, Diagram VI, which I cannot explain. It may be due to some error in the method of finding the centre of swing which comes in with a rapid march of that centre. The effect on the result is probably only small, for the value of MjR obtained with a march in the reverse direction on September 25 is very nearly the same, the two values being September 23 -2112753. 25 -2112533. The following is a hst of the weighings recorded, with the distances measured and the mean values of the attraction: 76 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Date 1890 Position of attracted Feb. 4 Upper April 30 and May 4 Lower May 25 Upper Date 1890 Position of attracted masses July 28 Lower Sept. 17 Lower Sept. 23 and 25 ... Upper Set I. No. of values oiM/R 50 100 50 Set IL No. of values oiM/E 25 25 52 Mean value of M/R •2142212 1-0109685 •2157379 Mean value otM/R •9973168 •9984148 •2112647 Dord in centims. 62-318 31-783 62-308 D or d in centims. 32-106 32-116 62-708 Horh in centims. 61-416 30-824 61-373 H ork in centims. 30-965 30-954 61-566 On the completion of Set I the four masses were inverted, and changed over from right to left or left to right, and the initial position was after this always arranged so that movement of rider or mass decreased the reading. This was done in order to lessen errors due to want of symmetry. If reversal had no effect, Set II should, with the increased distance recorded above, give a value of M/R in the lower position of about -990, instead of -998. The larger value actually found is no doubt chiefly due to a want of symmetry in the large attracting mass M, The effect of this want of symmetry will be discussed after the investigation of the mathematical formula, and an account will be given of an independent method of detecting it. I think there is still outstanding a small difference, due, perhaps, to want of symmetry in the turn-table or in the attracted masses. The result of the reversal shows how necessary it was to make it. I should have Hked to have in Set II as many determinations as in Set I, so that the mean should be based on values of equal weight. During June and July, 1890, a complete set of 100 in each position, upper and lower, was made; but, owing to the pressure of other work, I was unable to calculate the results till the completion of the set. I then found that the value of M/R was still more than in Set II, and, on plotting out the results, it appeared that occasionally the rider-value fell very considerably, and in an irregular way. On examination, there was Httle doubt that the rider came in contact occasionally with the suspending frame, when it was raised and should have been clear from it. Very likely tempera- ture-changes had brought about a displacement of the lever-apparatus. Comparison with Set I seemed to show that during that set no such contact had taken place, for there was no comparable irregularity. As it appeared dangerous to attempt to disentangle the good from the bad, the set of June and July was rejected, and Set II was taken as recorded. When I had made the weighings giving 50 and 52 values in the two positions respectively, the balance became so irregular, through the cooler weather, that it was useless THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 77 to continue work. Rather than carry over the experiment into another season, when it might be necessary to repeat the whole of the work, I have preferred to take Set II as it stands, and give it the same weight as Set I. The final results are calculated from the means of Sets I and II, as explained hereafter. I may here state the results obtained : ^ r ■ ^ 6-6984 Constant of attraction "^ = — tt^— . Mean density of the Earth ... A = 5-4934. General Remarks on the Method. Comparing the common balance with the torsion-balance, there is no doubt that the former labours under the great disadvantage that the dis- turbances due to air-currents are greatest in the vertical direction, that of the displacement to be measured. But even with this disadvantage the common balance may, I beheve, be made to do much more than has hitherto been supposed possible. As an instrument in itself, apart from the external disturbances of air-currents, dust, etc., I believe its accuracy would be far beyond anything approached when these external disturbances are, as they always are, present to interfere with its action. I have always found that every precaution to ward off air-currents and external disturbance has been accompanied by a corresponding increase in steadiness ; and I have seen no sign of a limit of accuracy depending on the instrument itself. Besides the protection from air-currents, there are two conditions essential above all others for accurate work : 1st. That during any set of weighings in which the deflections are to be compared with each other, the beam should be supported on its knife-edge, and should be under constant strain. 2nd. That all moving parts, such as apparatus for changing riders or weights, should be supported quite independently of the balance or its case. With regard to the first condition, it seems impossible to make the supporting frame move so truly and with so little disturbance that the knife- edge shall return exactly to the same line. Even were it possible, the beam after raising and lowering would be practically a different beam, for, as my observations show, the condition of strain changes considerably after the load is first put on, and it would be merely a chance coincidence if the mean state of strain were the same during successive weighings. I have, in my former paper {Proceedings of the Royal Society, vol. 28, 1879)*, described one method of comparing weights of nearly equal value with the beam throughout on its knife-edge and equally strained f, and I should now only modify that * [Collected Papers, Art. 2. J t I am glad that Dr. Thiesen urges the importance of this condition {Travaux et M^moires du Bureau International des Poida et Mesures, vol. 5, 'ilfitudes sur la Balance'). 78 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND method in having regard to the second condition, of which I have since reahsed the importance when working with the large balance and with increased optical sensitiveness. It is surprising to find how much disturbance is produced by having the moving parts of the apparatus connected with the balance or its case. As to air-currents there is no doubt that, as Professor Boys has shown, the greater the apparatus the greater the errors produced by them. At the time my apparatus was designed I did not know this, and there seemed to be a great advantage in making it large, as riders could be used of weight large enough to be measured accurately. Were I about to start with a new design I should certainly go towards the other extreme and make the apparatus small, attempting to get over the rider-difficulty by some such method as that explained on p. 60. For not only is a smaller apparatus kept more easily at a uniform temperature, and, therefore, freer from the source of air-currents, but it is much more handy to adjust, and even if the adjustments are not more accurate they will at least take much less time to make. At the same time it is only fair to say, on behalf of the large apparatus, that some errors have been magnified on a like scale till they have become observable, and so could be investigated and eliminated. Starting with a small apparatus they would probably never have been detected, and would, therefore, have appeared in the final result. II. Mathematical Investigation. The Value of the Attraction Expressed in Terms of the Masses and Distances, and the Investigation of the Effect of Want of Syynmetry in the Masses. Let us suppose that initially the attracting masses are in the positions Ml, »?i, Fig. U, the larger on the left, the smaller on the right, and that the attracted masses are in the lower positions A, B. When the turn-table is moved round so that the positions of the masses are M^, m^, the greater attraction is taken from the left and put on to the right. Let the centre of swing of the balance alter by an amount corresponding to a total change of vertical pull of n dynes. Assuming that a spherical mass M attracts another spherical mass M' when their centres are D centimetres apart with a force of GMM'/D" dynes, we can express the change of vertical pull due to the change of position of the masses as 6^ x a function F of the masses and distances. There is also a change of pull on the suspending-rods and the balance-beam which we may denote by E. Then n = GF + E. In order to eliminate E let the attracted masses be moved into their upper positions A', B\ and let the change on moving round the attracting THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 79 masses be n' dynes. If / is the function of the masses and new distances corresponding to F, Subtracting whence n — n' G = F-f and knowing G, the mean density of the earth may be at once found in the manner shown later. Fig. 14. We have then to find the form of the functions F, f, and as a prehminary step it is necessary to find the effect of the holes bored through the attracted masses A, B. This may be made to take the form of a correcting factor to the attraction which would be exercised on them if they were spheres. The piece bored out in each case has radius -31 centim. This we denote by c. It may be taken as practically a cylinder with plane ends and length equal to 15-8 centims., the diameter 2r of the spheres. The intensity due to such a cylinder of mass /x at Z) from its centre is (Todhunter's An. Stat., Ed. 5, p. 292), 2r - V{{D + r)2 + c2} + ^{(D - r)2 + c^} GfJL c^r which equals, to a sufficient approximation, GfJL 80 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND If the mass remaining after fx is removed is A, and if the centre of the mass Mis D below that of A, the attraction of M on A is GMJJ^ji) GMji GMA , rM (^ ^ \ = ^{l -^(J + ^ig^er powers of JJl. 1 and the greatest value of N- f = -^J^^r = I % ^'^^'^ = - ^^-'^ = ■'^''''^ ' _ / 79 \2 _ .Afil Then the higher powers may be neglected, and the attraction may be written GMA /, 3 c2 \ GMA IMAf. 3 C^\ brMA When A and B are in the lower position, D = 32, and 1 - 6 ^ -99986. When they are in the upper position, D = 62 and 1 - ^ = -99996, a value so near 1 that we shall in this position omit the correction, since it is only applied to one-fourth of the final result. In the cross-attractions we shall also omit the correction. Referring to Fig. 14 let the vertical differences of level between the centres of the various spheres be denoted as follows, the suffixes to M and m denoting their first and second positions respectively : A-M^ = D^, B-M^ = D^, B B M^=D^, A-M2 = D^, nil - H^, A - mi = i?/. m^ = H2, B - m^^ H^. When the masses A, B are placed in their upper positions, let the corre- sponding distances be denoted by small letters. Let the horizontal distance between the centres of A and B be L, being within sensible limits equal to that between the centres of M in its two positions, and to the length of the beam, and let the radius of the circle in which m moves be I. Then we have the following horizontal distances : A-M^^B-M^=^L, A — m-^ = B — m^ = I + \L, A — 7)12 = B — m^ ^ I — \L. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 81 We may now write the change in vertical pull on the left by the motion of M from left to right, and of m from right to left, as follows — the first four terms representing the vertical attractions on A and Bhj M and m in their first position, the next four their attractions when moved round, and the last term E representing the change in attraction on the beam and suspending rods : ( MA (1 - 6) _ MBD,' mBH^ , mAH^' \ Di' {D^^ + L^f MB(\-d) MAD^ mAH^ mBH: H{' '-r «."+('+ 1) LV)^ + D, {D,'^ + L^)^ H,+ il-^f + 2)] H''^ '-!) LV)^ + E. We may arrange all but the last term in nearly equal pairs. Thus the first and fifth go together, and if we put B^^- 3^ = 2D and Di + S = D2 — S = D, their sum is A B (?M(l-^)(^^+^^,) GM(l-d)\'^,{l = GM(l- d) 382 2 Z)2 D^V^ D^ D^^ '"J^ D^V D^ D (^1 + ^ + higher powers of -^^j ^ A + B d[ ^ D^ Now (S/Z))2 is negligible, as will be seen by reference to the table of distances, p. 92, and {A — B)I(A + B) is less than jjji-Qjj, or less than S/D.* To a sufficiently close approximation then the sum of the two terms is GM (A -\-B)(l- d) 2)2 The second and sixth terms may also be taken together, and putting D^ + D^ = 2B' and Z)/ + 8' = D/ - 8' = Z)', we may show that to a sufficient approximation BD^ , .4Z)./ I _ GM {A + B)U^ (Z)'2+ 2,2)1 GM The two pairs with m give similar results with H = i{H^ + H2) and H' = i {H,' H,'). * [A slight correction of the original, obviously required, has been made here. Ed.] P. c w. 6 82 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Now 2D=D^ + D, = A-M^-hB-M^ = B-M, + A-M,-=D,'-\-D,' = 2D', and similarly 2H = 2H' , so that we may put the expression in the form (M(A-^B\i\-e) _M{A + B)D m(A + B)H m(A + B)H \ -{- E=GF + E say. It is evident that we may combine experiments at different distances on different occasions in the same way by taking D and H to represent the mean values of these distances, so long as there is only a small variation from the mean. If the attracted masses are now moved into their upper positions the expression for the change in attraction may be at once deduced from that in the lower position by replacing D and H hj d and h, and omitting the factor 1 - e. Let it be denoted by Gf + E. Subtracting one expression from the other E is ehminated, and we have 0{F-f) [M^ + B){l-e) M{A + B )D m{A + B)H ^ m(A + B)H M(A + B) , M(A + B)d , m{A + B)h m{A + B)h ^ d^ "^ {d^ + L^)'^ M-Wf M-W This is to be equated to the difference in the values of the change in attrac- tion in the two positions, as determined by the rider. Let b = the length of the small rider-beam, ir = the mass of each rider, A = mass deflection -^ rider deflection in lower position*, « = » ^, „ „ upper „ *, gjj = acceleration of gravity, or dynes weight per unit mass at Birmingham. Then . G(F j^ _ i^ - f bwgj,^ Whence we may find the gravitation-constant ^ 2hwgj^ {A -a) where all the quantities on the right-hand are given in the tables at the end. * [It will be noted that this symbol is used with two distinct significations. Ed.] THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 83 The value of gjg may be found sujSiciently nearly from the formula (Everett's Units, p. 21): g = 980-6056 - 2-5028 cos 2A - -000003/i, where A is the latitude = 52° 28' at Birmingham, and h is the height above sea-level, which may be taken as 450 feet, or 13,725 centims. Whence g^ = 981-21. Since all the operations are conducted in air, the effective masses should throughout be less by the mass of air each displaces. But since they all have nearly the same densities, and w and A -^ B appear respectively in numerator and denominator, it is sufficient to take their true masses, and to correct for air displaced in the case of M and m only. To obtain the mean density of the earth A, we must express the acceleration of gravity in terms of G and the mass and dimensions of the earth. The ordinary formula (Pratt, Figure of the Earth, 4th ed., p. 119) is based on the assumption that the earth is a spheroid. It is sufficiently correct for our purpose, the departure of the assumed spheroid from the actual shape being very small. Adding a term — 3 x 10~^h, or approximately, — 41 x 10~®, since the balance-room is taken as 13,725 centims. above sea-level (see above), the value of gravity at Birmingham may be written g^ = ^^|l +^-^^+(1^-^) sin^ 52° 28' - 41 X 10-« where F = volume of the earth = 1-0832 x 10^7 (Everett's Units, p. 57), a = mean radius of the earth = 6-3709 x 10^ {loc. oil.), A = mean density of the earth, m = equatorial ' centrifugal force ' -^ gravity = ^^ , € = ellipticity of the earth = ^b 2 • The value of the ellipticity is taken to make the formula agree with that quoted above from Everett's Units. The uncertainty in the value is quite unimportant, for were e as low as 2^5, the error in A, introduced by taking it as 2^2 ' would be less than 1 in 50,000. Substituting for G, the value of the mean density of the earth is a^L{F-f) 2bwV |l + I - I m + f| m - e") sin^ 52° 28' - 41 x 10-4 (A - a) Here, as in the value of G, w and A + B may have their true values, M and m their values less the mass of air displaced. In the foregoing investigation we have supposed that all the masses are homogeneous and spherical, with the exception of the borings through A and B. We have supposed, also, that the turn-table is exactly symmetrical about a vertical plane through its axis, so that its motion through two right angles is without effect. Doubtless, these suppositions and the formula based on 6—2 84 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND them are not quite true. But, if we invert all the masses and change their sides, or pervert the whole arrangement of them, on taking the mean of the results obtained in the original and inverted and perverted positions we ought to greatly reduce the errors. Indeed, those due to want of symmetry in the turn-table should evidently be quite ehminated, and those due to want of homogeneity in the masses should certainly be lessened. To show this, we shall calculate the effect of a spherical ' blow-hole,' or gas-cavity in M, in the first and most important term of F. This we shall take as being GM (A + B) on the supposition that M is homogeneous and spherical. If the mass of metal which would fill the blow-hole is A, supposing it to be placed there, the sphere is completed and its attraction is G(M + X){A + B) ^ 2)2 but the vertical attraction is less than this in reahty by the vertical component of the attraction of A. Let B be the centre of the cavity, P the centre of the attracted mass, the centre of the attracting mass, 8 the distance of B from the centre of M, e the angle BOP. The vertical component of the attraction of A is GX (A +B) cos BPO PB^ but 5P2 = Z)2 + §2 _ 22)3 cos d, 1 7? on D-Scosd and . cos BFU = „„ — , Br whence the attraction of A may be put GX(A + B){D-Scosd) GX{A + B) "^ ^ ( D2 + §2 _ 22)8 cos d)^ ' ' dD VZ)2 + 8^ _ 2Z)8 cos 6 GX(A + B)f^ , ,„ 8 , ,„ 8^ A + B)( 6 6^ 2)2 ^^ + -^12) + ^^2 2)2 where P^, P^, ... are zonal harmonics. The attraction of the sphere with the cavity is therefore GM(A + B)L A / „ 8 , ,„ 8^ 2)2 |i MV^^D^^^' D^^ 'D^ THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 85 If the mass is inverted, the vertical component is obtained by changing the sign of 8, and the mean of the two values is 6MiA + B)i^_X_j^pJ^^^^p^8^ 2)2 M\ ^D^ " D* ■)i- the first power of hjD being ehminated. If ^ = 0, Pg ^^^ ^11 ^h^ other harmonics = 1. If ^ = 90°, Pg = - 1, P4 =- I, etc. Now, with the actual dimensions of the apparatus, (S/Z))^ cannot be so great as (J)^ or J, and may, of course, be much smaller. The first term of those involving A, therefore, is the most important, and it lies between + f (A/M) (8VI>2) and - 3 (XjM) {h^jB^), changing sign for the value of 6 given by Pg == 0. The second set of experiments recorded in this paper was taken after inversion and change of side of all the masses, and the final result obtained from this set differs by a little more than 1 per cent, from that obtained from the first set, the observed attraction being slightly greater at the same distance. The difierence may be due to irregularities in any or all of the masses and in the turn-table, and to other undetected effects, such as change of level on rotating the turn-table. It would be a very long task to disentangle these, and I have contented myself with trying to find how much must be set down to irregularity in the large mass M, by taking a set of weighings with it alone inverted. After the weighings on July 28, and the subsequent measures of distances, M was inverted only, and the other masses remained as in Set II. Some weeks later, on September 14, 25 values of MjR -^ A were obtained, the mean being -9926. The distances were B = 32-118, H = 30-978. The mass M was then put in its original position, as in Set II, and on September 17, as will be seen on referring to the tables, the value of MjR obtained was •9984, the distances being B = 32-117 and H = 30-955, practically the same as on September 14. Assuming that the difference in attraction is due to cavities in various places, and that, for each, the term ^tP^^h'^jB^ is negligible, we have, approxi- mately, 2AP,8 " MP _ 9926 ^ .SAPi8~9984' Whence, approximately, since B = 32, ,>r— = -0464 centim. M 86 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND This result may be tested by independent experiment. For, let the centre of gravity be x below the horizontal plane through the point bisecting the vertical diameter (i.e., the centre of figure), in the position of Septem- ber 14. The distance of any missing particle A from the horizontal plane is 8 cos ^ = Pi§. Completing the sphere by the addition of all such particles, the centre of gravity is brought to the centre of figure, so that we have Mx = SAPiS, SAP^S and X = — ^-^ . We have, therefore, to determine the vertical distance of the centre of gravity from the centre of figure. In order to do this, a large flat-bottomed scale-pan (one belonging to the balance used in the gravitation-experiment) was suspended by two parallel wires about 8 centims. apart and 3 metres long. In the middle of the pan was a shallow cup about 7-5 centims. internal diameter, arranged so that it could turn freely but truly about a vertical axis. The mass, M, was placed on this cup with the diameter, which had been vertical, arranged horizontal, and perpendicular to the plane of the suspending wires. A vertical flat plate, worked by a horizontal micrometer-screw, could be brought just in contact with the end of the diameter, and the reading of the micrometer gave the position of the point of contact. The position of the scale-pan was deter- mined by a plumb-line hanging over one edge in front of a horizontal scale. On turning the cup and mass through 180°, and repeating the readings, knowing the weight of the scale-pan, and the position of its centre of gravity, X could at once be found. Two separate experiments gave X = -0536 centim., and X - -0516 centim., not very different from the value -0464 obtained from the attraction-experi- ments. The agreement is, I think, very close when it is noted that a difference of 1 in 1000 in the attraction in one of the sets of weighings would make x either -038 or -054. This result appears to justify the rejection of all terms in the expansion above the first, and so supports the belief that the reversal largely ehminates errors due to irregularity of shape. For it is in the case of M that there is the greatest danger of a large value for S/D, and the above experiments seem to indicate that even in this case it is small. It is, perhaps, noteworthy that the largest term rejected in the attraction of M, viz., ^XP^S^/MD^ is, if we give P^ its maximum value 1, 3AS S^^3x S MB' D~ D'D' THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 87 which is not greater than since the radius of the mass is 15. This is in a term about 5/4 of the final result, so that the greatest error which can be introduced by neglecting this term is -0025, or 1 in 400. In calculating the results of the experiments the means of Sets I and II have been taken. Equal weights have been given to each set. It would have been more satisfactory if the number of experiments had been the same in each set ; but I should have had to wait for another season to obtain more, and then it would, probably, have been necessary to repeat the whole series in both arrangements, as it is not safe to assume that the various disturbing causes remain the same over a wide interval of time. The second set, though fewer in number, are, in some respects, I beheve, better ; partly owing to the additional experience gained when they were taken. In order that the various terms in F —f may be compared, I give below their numerical values, as determined from the values of the masses and distances given in the tables. The meaning of each term in the first column will be seen on referring to Fig. 14. The second column contains the actual values ; the third column the values in terms of the lowest term, the fourth. Value oi F -f. M{A-{-B){l-e) 2)2 M(A + B)D (Z)2 + L^)^ miA + B)H m(A + B)H M(A + B) M{A + B)d m(A + B)h + 6483938-8 416 - 102416-3 6-6 - 316243-3 20 + 15579-9 1 1693687-2 109 156728-0 10 310695-0 20 ^iA + B)h _ 27597.7 ].7 {'-('-ST Whence F -f= 4826997-2. 88 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND The mean value oi A - a (see Table III) is ^_a = -791295; substituting these values of J -/and ^ - a in the formula for G (p. 82), we obtain 6-6984^ substituting them in the formula for A we obtain A = 5-4934. The values given by Sets I and II, treated separately, are to two figures of decimals : Set I. A = 5-52 Set 11. A = 5-46. i III. Tables. Table I. Constants of the Apparatus and Dimensions of the Earth. Masses Attracting mass M, in vacuo Less air displaced, say . . . Attracting mass m, in vacuo Less air displaced, say ... Attracted mass A, in vacuo 55 55 -^5 ?5 ,, ,, A + B, in vacuo Riders each, in vacuo grms. 153407-26 153388-85 76497-4 76488-2 21582-33 21566-21 43148-54 0-010119 Vertical Diameters of Masses in terms of Catheto meter -Scale correct at 18°. The masses are taken as having the same coefficient of expansion as the scale. centims. M = 30-526 m = 24-176 A = 15-8203 B - 15-7829 The diameters of the masses A and B are taken between the nuts securing them on the suspending wires. centims. Balance beam at 0°, L _ =123-232 Rider beam at 0°, 6 ^ 2-53575 L/b (as occurring exphcitly in G and A, independent of tempera- ture, assuming them to have the same coefficient of expansion) = 48-59775 THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 89 Latitude of Birmingham Height of balance-room above sea-level Gravity at Birmingham, g^ ... Mean radius of earth ... Volume of earth Equatorial ' centrifugal force '/gravity Ellipticity of earth f]sin2 52° 28' -41x10-6 3 , /5 = 52° 28' = 13725 centims. = 981-21 centims./sec.2 = 6-3709 X 108 centims. = 1-0832 X 1027 cub. centims, = -999161. Table II. Vertical and Horizontal Distances. Vertical Diameters of Masses taken by the Cathetometer, described p. 65. In the tables below p.s. signifies divisions on the scale over which moves the pointer, which is attached to the small adjustment-plate, v.s. signifies divisions on the vertical millimetre-scale. Diameter of Large Attracting Mass M. Reading on pointer-scale Upper telescope sighting top of mass ... 73-2, 73-4, 73-2 Lower „ „ bottom „ ... 23-0, 23-0, 23-2 Mean 73-27 P.s. 23-07 p.s. Turning round to the Vertical Scale. Reading on pointer-scale Upper telescope sighting 459 millims. v.s. ... 94-6, 94-9, 94-4, 95-0, 94-0 458 „ ... 68-8, 68-8, 69-7, 69-0, 68-7, 70-0, 69-6, 69-4 Therefore 25-33 p.s. divisions = 1 millim. v.s., Mean 94-58 P.s. 69-25 p.s. and scale-reading for top of mass = 458 = 458-158 millims. v.s. 73-27 - 69-25 25^33 Lower telescope sighting 153 millims. v.s. „ 152 Reading on pointer-scale Mean 27-3, 27-4, 27-7, 270 27-35 r.s. 0-0, 0-3, - 5, 0-0 - 0-05 p.s. Therefore 27-40 p.s. divisions = 1 milhm. v.s., 23-07 -f 0-05 and scale-reading for bottom of mass = 152 + = 152-844 milhms. v.s. The difference = 30-5314 centims. 27-40 90 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND This IS rather greater than the diameter of the mass, as the cross- wire was made to touch the image of the mass in each case. A series of measures of 1 milUm. on the scale, in which the cross-wire was on the centre of each division, and of 1 miUim. between the jaws of a wire-gauge, in which the wire touched the images of the jaws, showed that at the distance at which the scale was, -005 centim. must be subtracted, leaving Diameter of if = 30-526 centims. Vertical Diameter of Small Attracting Mass m. Reading on pointer-scale Mean Upper telescope sighting top of mass ... 75-6, 75-6, 75-0 75-40 p.s. Lower „ „ bottom „ ... 26-5, 26-3, 26-8 26-53 p.s. Turning round to the Vertical Scale. Reading on pointer-scale Mean Upper telescope sighting 388 millions, v.s. ... 100, 99-9, 99-7 99-87 p.s. 387 „ ... 73-9,73-4,74-0 73-77 p.s. Therefore 26-10 p.s. divisions = 1 millim. v.s., 75-40 — 73-77 and scale-reading for top of mass = 387 + — ^r^^r. = 387-062 millims. v.s. Reading on pointer-scale Mean Lower telescope sighting 146 millims. v.s. ... 45-9, 45-9, 45-0 45-60 p.s. 145 „ ... 20-4,19-4,20-0 19-93 p.s. Therefore 25-67 p.s. divisions = 1 millim. v.s., and scale-reading for bottom of mass = 145 + "^'^^ ~ ^^'^^ 25-67 = 145-257 milhms. v.s. The difference = 24-1805 centims. Subtracting the same correction as in the last case for the cross-wire, Diameter of m = 24-176 centims. Vertical Diameters of Attracted Masses A and B taken between the Junctions of the Securing Nuts ivith the Sphere. A. Reading on pointer-scale Mean Upper telescope sighting top of mass ... 82-7, 83-0, 82-9 82-9 p.s. ^'''''^'' •' » bottom „ ... 31-0,' 31-5,! 31-3 31-3 p.s. I THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 91 • Turning round to the Vertical Scale. Reading on pointer-scale Mean Upper telescope sighting 429 millims, v.s. ... 95-0, 95-2, 95-3 95-2 p.s. 428 „ ... 69-0, 69-0, 68-9 69-0 p.s. Therefore 26-2 p.s. divisions = 1 millim. v.s., and scale-reading for top of mass = 428 H ^r^-^ = 428-531 millims. v.s. Reading on pointer-scale Mean Lower telescope sighting 271 millims. v.s. ... 48-0, 47-8, 48-4 48-1 p.s. „ 270 „ ... 23-4, 230, 22-8 23-1 p.s. Therefore 25-0 p.s. divisions == 1 millim. v.s., 31*3 — 23*1 and scale-reading for bottom of mass = 270 H ^r^^r ^ 25-0 = 270-328 millims. v.s. The difference gives the diameter since the middle of the cross-wire was used, so that Diameter of ^ = 15-8203 centims. B. Reading on pointer-scale Mean Upper telescope sighting top of mass ... 72-0, 71-0, 71-0 71-3 p.s. Lower „ „ bottom „ ... 24-6, 25-0, 25-2 24-9 p.s. Turning round to the Vertical Scale. Reading on pointer-scale Mean Upper telescope sighting 430 millims. v.s. ... 94-0, 94-6, 940 94-2 p.s. 429 „ ... 68-0, 68-1, 68-1 68-1 p.s. Therefore 26-1 p.s. divisions == 1 milHm. v.s., and scale-reading for top of mass = 429 H -^^ ,, = 429-123 minims, v.s. Reading on pointer- scale Mean Lower telescope sighting 272 millims. v.s. ... 43-3, 430, 430 43-1 p.s. „ 271 „ ... 171, 17-3, 17-4 17-3 p.s. Therefore 25-8 p.s. divisions = 1 milhm. v.s., 24-9 — 17-3 and scale-reading for bottom of mass = 271 -f ^„ „ = 271-294 milhms. v.s. And diameter oi B = 15-7829 centims. 92 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Vertical Distances between the Levels of the Centres of the Attracting and Attracted Masses Measured by Cathetometer. The measurements were made as soon as possible after the completion of a set of weighings, usually on the following day. It was necessary to fix the attracted masses in the position occupied during the weighings, and with the beam of the balance in the same strained condition. This was done in some cases by gripping the left suspending wire by a pair of jaws ; in others, by adding a small weight to one side, and placing a block of the right thickness under the mass on that side. The cathetometer was placed in front of the left side of the balance-case, from which position all the masses could be viewed by turning the telescope round the central pillar (Fig. 2). It was read when sighting the top of each attracting mass and the top of each attracted mass when in the lower position, the bottom of each attracted mass when in the upper position, the top and bottom being taken at the junctions of the securing nuts with the masses. It is therefore necessary to add to the distances measured by the cathetometer the difference of the radii of attracting and attracted masses in the lower position, and their sum in the upper position (see p. 88). The work is shown in full for February 5 and May 5. Tests were made at various times, showing that there was no change in the distances (at least within errors of reading), either through moving the turn-table or in the course of a few days (see February 5 and May 5 for examples). Temverature-Corrtction. The cathetometer-scale is taken as correct at 18°, and its coefficient of expansion is assumed to be 1/60000. That of the masses is probably about 1/40000, but, for simphcity, is taken as equal to that of the scale, the difference, 1/120000, never amounting to as much as the errors of reading, since the greatest length concerned is 23 centims. The temperature was estimated to be about 1° above that observed during the immediately preceding weighings, the presence of the observer and the lights used tending to raise it. The cathetometer rested always on the brick floor of the room. Its vernier reads to -002 centim. Set I. Attracted masses A on the left, B on the right. Attracting mass M moving round from left to right in front of the balance-case. February 5, 1890. Attracted masses in upper position. Assumed temperature 11°. Half-way through the measurements the cathetometer was accidentally moved, and could not be exactly replaced. Repeating the reading of A it THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 93 was found that -197 must be added to the previous readings to compare with the following ones. This addition is made where the numbers have an asterisk. 23-448 A 64-999* 65-001* ^1 25-895* 25-889* B 65-284 65-282 26-070 26-064 mi 23-947* Differences: ^ - m, = 41-552 A- Mt^ = 39-108 B - M^ = 39-216 B - m^ = 41-336 Table I P- 88, the sums of the radii of the masses are ' Rm + Ra = 23-173, Rm+ ^B = 23-154, Rm-^RA = 19-998, Rm+RB = 19-979, d = 1 {39-108 + 23-173 + 39-216 + 23-154} = 62-326, h = 1 {41-336 + 19-979 + 41-552 + 19-998} = 61-433. and These are in terms of a scale correct at 18°, so that the value is too great by about 7/60000. We take as true values Corrected d = 62-318, h = 61-425. Test Experiment. At the conclusion, the distance A — M^ was measured again and found to be 39-110. May 28, 1890. Attracted masses in upper position. Assumed temperature 14°. A 64-674 64-674 B 65-286 65-288 mo 23-422 23-424 25-726 25-724 25-920 25-920 m^ 23-766 23-756 Differences: J. - ma = 41-251 A- M^ = 38-949 J5 - -¥2 = 39-367 5 - m.i = 41-526 whence d = 62-312 5 h = 61-377 94 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Subtracting temperature-correction -004, Corrected d = 62-308, /i- 61-373. Mean values in Set I, d = 62-313, h = 61-399. May 5, 1890. Attracted masses in lower position. Assumed temperature 13°. A B 50-324 50-622 50-324 50-634 50-328 50-630 7m, M, M, Ml 23-672 25-972 26-138 23-998 23-674 25-970 26-138 24-008 25-972 26-132 Differences : A - m.^ = 26-652 A - Mi = 24-354 B- M^ = 24-493 B - m^ = 26-626 From Table I, p. 88, ^31 -Ra = 7-353, Rjyj -Rb= 7-372, R,, -i?^== 4-178, R,, -i?5^ 4-197, whence and D l_ {24-354 + 7-353 + 24-493 + 7-372} = 31-786, H = \ {26-626 + 4-197 + 26-652 + 4-178} - 30-827. Subtracting temperature-correction -0025, Corrected values for Set I, D = 31-783, H = 30-824. Test Experiment. The balance was set free at the end of these measures, and two days later, on May 7, it was again fixed, and the distance D was determined by the cathetometer described on p. 65. The value obtained was D = 31-786. Note. If the apparatus were perfectly rigid and constant in its dimensions we should expect D - H = d~h = constant. The values actually given by the above experiments are February 5 ... ... -892, May 5 .959^ May 28 .935. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 95 There is apparently a slight increase during the course of the spring, probably due to the warping of the wood supporting the mass m. But there was some uncertainty in sighting the top of the mass m, especially when in the distant position on the right. Set II. Attracted masses A on the right, B on the left. Attracting mass M moving round from left to right behind the balance-case. All the masses inverted. July 29, 1890. Attracted masses in lower position. Assumed temperature 16°. mi 22-434 22-436 B 49-014 49-014 M^ 24-584 24-586 A 49-846 49-844 M, 24-788 24-782 ^2 22-868 22-864 Differences : B - m^ = 26-579 B- M^ = 24-429 A- M^ = 25-060 A - m^^ 26-979 whence D - 32-107, H = 30-967. Subtracting temperature-correction -001 , Corrected D - 32-106, H = 30-966. September 18, 1890. Attracted masses in lower position. Assumed temperature 16°. B A 49-076 49-768 49-074 49-766 mi J/2 Ml ^2 22-467 24-576 24-756 22-840 24-576 24-758 Differences : B - m^ = 26-608 B- M^ = 24-499 A - M^ = 25-010 A - m^ = 26-927 whence D - 32-117 H = 30-955 Subtracting temperature-correction -001, Corrected D = 32-116 H = 30-954 Mean values in Set II, Z) = 32-111 H = 30-960 96 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND September 27, 1890. Attracted masses in upper position. Assumed temperature 16°. ) B A 63-880 63-876 64-540 64-544 Ml 22-450 22-448 24-570 24-572 24-756 24-758 Mo 22-8^10 22-816 Differences: B - - Wj = 41-429 B- - M^ = 39-307 A - - M, = 39-785 A - mo = 41-729 whence d - 62-710, h= 61-568. Subtracting temperature-correction -002, Corrected values for Set II, d = 62-708, h = 61-566. Note. The values oi D — H and d — h, which should be constant, are from the above, and from another set of measures (not here recorded, see p. 85) on September 15, as follows. (We have no reason to expect the same value as in Set I, as the masses M, m have changed sides.) July 29 1-140, September 15 1-110, September 18 1-162, September 27 1-142. From July 29 to September 15 inclusive, the balance was swinging freely without alteration. The values of H should, therefore, be the same on those dates. They were July 29 30-967, September 15 30-978, equal almost within errors of reading for the top of m. Means of Sets I and II : Z) = 1(31-783 + 32-111) = 31-947. H = \ (30-824 + 30-960) - 30-892. d=\ (62-313 + 62-708) - 62-511. h = \ (61-399 -t 61-566) = 61-483. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 97 At 18° and whence Horizontal Distances, Set I. L = 123-269 centims. Zi = 122-915 ^= 61-635 h + %= 184-550 h-^= 61-280 „ Taking the mean temperature of the Set as 12°, and assuming 1/60000 as the coefficient of expansion, on correcting to 12°, l^ + ^ = 184-532 centims. L At 18' h ±J ~ 2^ 61-274 Set XL 55 \ = 122-795 centims. L 2 61-635 55 h *\' 184-430 " h L 2 ~ 61-160 55 Whence Taking the mean temperature of the Set as 15°, and correcting to 15°, ^2 + - = 184-421 centims. ''2 9 61-157 Mean values for the two Sets L = 123-260 l-\-%= 184-477 l-~= 61-216 p. c. w. 98 ON A DETERMmATION OF THE MEAN DENSITY OF THE EARTH AND Table III. Determination of Attraction by the Balance. Determinations of the Attraction in terms of the Riders by the Balance. In each case four turning-points of three successive swings are recorded in tenths of a division, i.e., in divisions on the diagonal lines. In the columns headed i the masses and riders are in the initial position, in those headed r the riders are moved, and in those headed m the masses are moved. Under each set of four readings is the calculated centre of swing (see p. 71). In the next hne are the deflections due to movements of riders and masses, each placed under the middle one of the three centres of swing from which it is calculated. In the next Hne are the values of deflection due to mass -^ deflection due to rider, or M/R (see p. 74). Set I. Attracted Masses in Upper Position. Feb. 4, 1890, 7.59 p.m. to 10.49 p.m. Temperature: in Observing Koom, 15°-7-16°-5; in Balance Eoom, 10°-05. Barometer, 752-2-752-0 miUims. Weather mild and still, after shght frost on the two previous nights. Time between successive passages of centre about 20 seconds. I. i r i 771 r m Scale-readings . . . (1) (2) (3) (4) (5) (6) (7) (8) 725 912 725 804 764 913 726 804 798 838 799 787 779 838 800 787 759 878 759 796 771 879 759 797 780 856 781 791 776 857 781 791 Centre of swing 772-55 863-85 773-00 792-80 773-90 864-60 773-40 793-20 Deflection due to rider or mass... ... 91-075 ... 19-350 90-950 19-700 Mass deflections rider deflection ... ... -212608 -214688 ... -217110 / r i m i r i m (9) (10) (11) (12) (13) (14) (15) (16) Scale-readings... 763 913 724 804 764 914 725 805 779 837 801 787 779 838 801 789 771 879 759 796 771 880 760 796 Centre of swing Deflection due to 775 857 782 792 776 857 783 792 773-60 864-25 773-85 79305 773-90 865-05 774-50 793-65 rider or mass... Mass deflections 90-525 19-175 90-850 ... 18-950 rider deflection -214720 1 -211440 ... •209824 •208758 THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 99 Table III (continued). i r i m i r i m (17) (18) (19) (20) (21) (22) (23) (24) Scale-readings... 765 916 726 803 765 914 725 805 780 839 803 788 779 838 800 787 772 881 762 797 771 879 759 796 111 859 784 792 776 857 782 791 Centre of swing 774-90 866-30 776-30 793-65 774-00 864-50 773-60 792-85 Deflection due to rider or mass... ... 90-700 ... 18-500 90-700 ... 19-225 Mass deflection -^ rider deflection ... •206174 •203966 ... -207966 ... •211438 i r i m i r i m (25) (26) (27) (28) (29) (30) (31) (32) Scale-readings... 763 914 121 805 764 913 725 804 780 838 800 789 779 838 800 786 770 880 760 797 771 879 760 796 776 857 782 792 115 857 781 791 Centre of swing 773-65 865-05 774 15 793-90 773-65 864-65 773-80 792-50 Deflection due to rider or mass... ... 91-150 ... 20-000 ... 90-925 19-275 i Mass deflections rider deflection ... •215167 ... -219690 ... -215975 •211494 i r i m ^• r • i m (33) (34) (35) (36) (37) (38) (39) (40) Scale-readings... 763 913 724 803 763 912 726 802 778 838 801 789 778 837 799 787 770 879 759 796 769 879 759 797 774 857 782 792 774 857 780 792 Centre of swing 772^65 864-60 773-85 793-50 772-30 864-20 772-95 793-30 Deflection due to rider or mass... ... 91-350 ... 20-425 ... 91-575 ... 19-875 Mass deflections rider deflection -217296 ... •223316 ... -220038 ... •216503 7—2 100 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III (continued). i r i m i r i m (41) (42) (43) (44) 803 (45) (46) (47) (48) Scale-readings... 764 915 725 763 913 726 803 779 839 800 786 778 838 799 787 771 880 759 795 770 879 759 796 776 858 780 791 774 857 781 792 Centre of swing 773-90 86-555 773-15 792-25 772-70 864-60 773-15 792-95 Deflection due to rider or mass... 92-025 ... 19-325 91-675 ... 19-200 Mass deflection ^ rider deflection •212986 ... -210397 ... •210117 ... -209693 * r i m i r i m 1 (49) (50) (51) (52) (53) (54) (55) (56) Scale-readings... 764 914 724 803 762 912 721 801 779 839 802 787 778 836 798 785 772 880 759 795 769 877 755 794 776 858 781 790 774 855 779 788 Centre of swing 774-35 865-55 773-85 792-10 772-20 862-55 770-35 790-55 Deflection due to rider or mass... 91-450 19-075 91-275 20-275 Mass deflections rider deflection -209267 -208784 -215557 ... -222161 (57) (58) (59) m (60) Scale-readings, 760 911 724 776 836 799 767 1 877 758 772 855 779 803 785 795 789 (61) 762 777 769 773 (62) (63)* 911 836 877 855 Centre of swing Deflection due to rider or mass... Mass deflection s rider deflection 70-20 722 800 758 780 862-50 772-20 791-35 , 771-70 862-50 ; 772-50 91-250 ... 19-425 | ... 90-400 •217534 ... -213873 •217506 (63 a) 725 799 759 780 m (64) 803 786 796 790 772-90 I 792-30 19-900 217873 * After 63 the riders were moved by mistake instead of the masses, therefore it was necessary to return to the initial position, and take the readings in (63 a). THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 101 Table III (continued). i r i m i r i m (65) (66) (67) (68) (69) (70) (71) (72) Scale-readings... 762 913 725 802 759 909 722 802 in 838 800 785 776 834 797 784 769 879 758 794 767 875 757 794 774 857 780 789 772 854 779 789 Centre of swing 771-90 864-60 112'lb 790-80 770-15 860-80 771-05 790-55 Deflection due to rider or mass... 92-275 19-350 90-200 19-450 Mass deflection -:- , rider deflection •213170 ... -212084 ... -215078 -214947 * r i m i r i m (73) (74) (75) (76) {11) (78) (79) (80) Scale-readings... 760 911 724 803 762 911 721 800 777 835 798 785 111 835 797 784 768 877 151 795 769 877 756 793 773 854 779 790 773 854 778 788 Centre of swing 771-15 862-05 771-40 791-50 771-70 862-05 770-30 789-75 Deflection due to rider or mass... 90-775 19-950 ... 91-050 ... 20150 Mass deflection -^ rider deflection ... •217020 ... -219442 -220209 ... -221064 i r . m i r i m (81) (82) (83) (84) (85) (86) (87) (88) Scale-readings . . . 759 910 722 801 759 910 723 802 774 833 796 783 115 834 797 783 766 876 757 793 767 876 757 793 771 854 778 787 771 854 778 787 Centre of swing 768-90 860-95 770-50 789-30 769-60 861-25 770-90 789-35 Deflection due to rider or mass... 91-250 19-250 ... 91-000 19-300 Mass deflection -^ rider deflection ... •215990 ... -211248 ... •211813 -212995 102 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III i r i m i r i m (89) (90) (91) (92) (93) (94) (95) (96) Scale-readings . . . 759 908 719 800 760 910 723 800 775 831 795 783 775 835 798 785 766 874 754 792 767 876 756 793 771 852 776 788 772 854 779 789 Centre of swing 769-20 859-00 768-35 789-05 769-90 861-60 770-90 790-30 Deflection due to , rider or mass... ... 90-225 ... 19-925 ... 91-200 19-350 Mass deflection -^ rider deflection -217373 •219650 •215323 ... •212462 i r i m i r * m i Scale-readings . . . (97) (98) (99) (100) (101) (102) (103) (104) (105) 761 910 721 798 759 909 721 800 759 111 835 796 783 775 833 796 783 115 768 876 756 791 765 874 756 793 767 772 853 111 787 770 852 777 787 771 i Centre of swing 771-00 861-35 769-80 788-35 768-60 859-65 769-80 789-35 769-60 Deflection due to rider or mass... 90-950 19-150 90-450 19-650 Mass deflections rider deflection -211655 -211136 ... -214483 Feb. 4, 1890. Mean of 50 determinations of M/R = a Attracted masses in upper position •21422122. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 103 Table III (continued). 11. Attracted Masses in Lower Position. April 30, 1890, 7.45 p.m. to 10.32p.m. Temperature: in Observing Room, 17°-16°-1 ; in Balance Room, 11°-1. Barometer, 748-6-749-2 millims. Weather clear; S.E. wind; sunny during day. Time between successive passages of centre not quite 20 seconds. i r i m i r i m (1) (2) (3) (4) (5) (6) (7) (8) Scale-readings . . . 1046 1133 951 1127 955 1134 952 1123 969 1055 1024 1062 1025 1059 1028 1069 1012 1098 984 1099 986 1102 985 1099 988 1075 1007 1078 1007 1077 1009 1082 Centre of swing 996-60 1082-85 998-35 1085-50 999-80 1086-25 1000-40 1088-25 Deflection due to rider or mass... ... 85-375 86-425 86-150 87-350 Mass deflections rider deflection ... ... 1-00772 ... 1-00856 ... 1-01437 i r i m i r i m (9) (10) (11) (12) (13) (14) (15) (16) Scale-readings . . . 962 1136 955 1129 961 1136 956 1133 1023 1060 1029 1067 1027 1064 1031 1069 989 1104 987 1102 989 1105 989 1103 1009 1079 1012 1083 1012 1081 1013 1085 Centre of swing 1001-40 1088-00 1002-45 1089-55 100315 1090-00 1004-15 1091-25 Deflection due to rider or mass... 86-075 86-750 86-350 86-350 Mass deflection-^ rider deflection 1-01133 ... 100623 ... 1-00232 1-00101 i r i m i r i m (17) (18) (19) (20) (21) (22) (23) (24) 1134 Scale-readings... 967 1141 957 1135 965 1143 958 1027 1064 1034 1070 1031 1066 1036 1073 994 1108 990 1106 994 1110 993 1108 1012 1083 1015 1086 1015 1085 1017 1089 Centre of swing 1005-65 1092-00 1006-00 1093-20 1007-40 1094-00 1008-35 1095-40 Deflection due to rider or mass... ... 86-175 ... 86-500 86-125 86-825 Mass deflection -H rider deflection ... 1-00290 ... 100406 1-00624 ... 1-01106 104 ON^ A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table HI (continued). i r * m i r i m (25) (26) (27) (28) (29) (30) (31) (32) Scale-readings... 976 1143 963 1141 966 1145 960 1141 1027 1069 1037 1073 1037 1070 1040 1075 998 1110 996 1112 996 1112 995 1113 1016 1087 1019 1090 1019 1089 1020 1092 Centre of swing 1008-80 1095-35 1010-65 1097-90 1010-85 1097-05 1011-15 1099-30 Deflection due to rider or mass... ... 85-625 87-150 86-050 87-575 Mass deflections rider deflection ... 1-01591 1-01529 1-01264 101713 i r i m i r i 1 m 1 (33) (34) (35) (36) (37) (38) (39) (40) Scale-readings... 973 1147 962 1137 975 1146 965 1138 1034 1071 1041 1079 1034 1075 1042 1080 1000 1114 996 1112 1002 1114 999 1113 1020 1090 1022 1094 1020 1091 1021 1094 Centre of swing 1012-30 1098-55 1012-50 1100-20 1013-40 1099-80 101400 1101-00 Deflection due to rider or mass... 86-150 87-250 86-100 ... 86-650 Mass deflections rider deflection 1-01465 ... 1-01306 1-00987 ... 1-00858 i r i m i r i m ' (41) (42) (43) (44) (45) (46) (47) (48) Scale-readings... , 977 1150 964 1089 976 1153 968 1 1148 1 i 1035 1073 1043 1 1110 1038 1074 1044 1078 1003 1116 1000 1098 1004 1118 1001 1118 1022 1093 1025 1104 1023 1094 1025 1095 Centre of swing 1014-70 1100-80 1015-451 1102-20 1016-15 1102-35 1016-45 1103-40 Deflection due to rider or mass... ... 85-725 86-400 86-050 86-475 Mass deflections rider deflection ... 1-00933 j 100597 1-00450 1-00625 THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 105 Table III (continued). i r i m i r i m (49) (50) (51) (52) (53) (54) (55) (56) Scale-readings... 978 1153 969 1149 976 1153 968 1145 1039 1076 1045 1080 1043 1079 1047 1085 1005 1119 1002 1118 1005 1121 1004 1118 1025 1094 1027 1097 1026 1096 1028 1099 Centre of swing 1017-40 1103-35 1017-65 1104-45 1018-60 1105-55 1019-25 1106-15 Deflection due to rider or mass... ... 85-825 ... 86-325 86-625 86-875 Mass deflection -^- rider deflection ... 1-00670 1-00116 ... -99971 ... 1-00973 i r i m i . i m (57) (58) (59) (60) (61) (62) (63) (64) Scale-readings . . . 984 1155 971 1151 977 1157 972 1152 1039 1078 1048 1083 1046 1081 1051 1087 1008 1120 1004 1122 1007 1123 1007 1123 1026 1097 1029 1100 1030 1100 1031 1102 Centre of swing 1019-30 1105-10 1020-00 1107-85 1021-25 1108-10 1022-60 1109-95 Deflection due to rider or mass... ... 85-450 87-225 ... 86-175 86-850 Mass deflection -^ rider deflection 1-01872 ... 1-01646 ... 1-01011 1-00798 i r I m i r i m (65) (66) (67) (68) (69) (70) (71) (72) Scale-readings... 983 1159 976 1153 983 1161 978 1158 1046 1082 1051 1088 1049 1083 1053 1088 1011 1125 1008 1126 1012 1127 1011 1127 1031 1102 1033 1104 1032 1102 1034 1106 Centre of swing 1023-60 1109-80 1023-70 111205 1025-15 1111-05 1025-95 111315 Deflection due to rider or mass... ... 86-150 87-625 85-500 ... 86-700 Mass deflections- rider deflection ... 101262 ... 1-02097 ... 1-01944 ... 1-01226 106 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III (continued). i r i m * r * m (73) (74) (75) (76) (77) (78) (79) (80) Scale-readings... 985 1163 980 1156 990 1163 982 1161 1051 1086 1056 1092 1051 1087 1057 1093 1014 1129 1013 1129 1017 1131 1015 1132 1033 1104 1036 1108 1036 1107 1039 1109 Centre of swing 1026-95 1113-40 1028-25 1115-50 1029-20 1115-20 1030-15 1117-65 Deflection due to rider or mass... ... 85-880 ... 86-775 85-525 ... 87-100 Mass deflection -^ rider deflection 1-01093 ... 1-01299 ... 1-01652 ... 1-01782 ^ r i m i r i m (81) (82) (83) (84) (85) (86) (87) (88) Scale-readings... 991 1167 984 1158 992 1169 984 1161 1054 1090 1059 1098 1056 1092 1063 1100 1018 1133 1018 1132 1021 1136 1018 1135 1038 1108 1041 1112 1041 1111 1042 1115 Centre of swing 1030-95 1117-40 1032-60 1119-55 1033-55 1120-00 1034-00 1122-20 Deflection due to rider or mass... 85-625 86-475 86-225 87-600 Mass deflection -^ rider deflection 1-01358 1-00640 1-00942 1-01890 i r i m i r i m (89) (90) (91) (92) (93) (94) (95) (96) , Scale-readings... 996 1171 987 1165 995 1172 989 1169 1058 1094 1064 1100 1061 1097 1066 1099 1022 1137 1022 1137 1024 1137 1023 1140 1043 1114 1045 1117 1045 ■ 1116 1046 1117 Centre of swing 1035-20 1121-75 1036-85 1123-75 1037-35 1123-15 1038-20 1125-05 Deflection due to rider or mass... 85-725 86-650 85-375 86-575 Mass deflection ^ rider deflection 101633 ... 1-01286 ... 101450 ... 1-01065 THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 107 Table III (continued). i r i m i r i m i (97) (98) (99) (100) (101) (102) (103) (104) (105) Scale-readings... 998 1175 992 1174 995 1176 995 1169 1001 1062 1097 1066 1102 1064 1098 1067 1105 1066 1026 1141 1025 1141 1027 1143 1026 1143 1029 1045 1116 1047 , 1119 1048 1118 1049 1121 1049 Centre of swing 1038-75 1125-05 1039-45 1127-15 1040-15 1126-70 1040-75 1128-90 1042-20 Deflection due to rider or mass... 85-950 87-350 86-250 87-425 Mass deflection -^ rider deflection ... 1-01178 ... 1-01452 101319 April 30. Mean of 50 determinations of M/R = A Attracted masses in lower position 1-010905. May 4, 1890, 11.11 to 11.50 a.m. Temperature: in Observing Koom, 13°-5 to 13°-8; in Balance Room, ll°-7. Barometer, 742-0 to 741-7 millims. Weather inclined to rain; a little cooler than previous day; wind S. to S.W. * r i m ^• r * m (1) (2) (3) (4) (5) (6) (7) (8) Scale-readings . . . 875 1045 865 1044 865 1045 861 1041 936 969 939 970 938 967 940 971 900 1013 896 1014 897 1013 894 1012 920 988 921 989 920 986 921 989 Centre of swing 91310 996-95 911-80 997-75 911-60 996-00 910-95 997-10 Deflection due to rider or mass... 84-500 86-050 ... 84-725 ... 86-275 Mass deflection -h rider deflection ... 1-01699 ... 1-01697 ... 101950 108 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III (continued). * r i m i r i m (9) (10) (11) (12) (13) (14) (15) (16) Scale-readings... 869 1046 862 1036 867 1045 860 1040 936 966 939 972 936 964 937 968 896 1012 894 1009 896 1011 893 1009 919 985 920 988 t 918 984 919 986 Centre of swing 910-700 995-10 910-45 995-55 910-40 993-80 909-20 994-20 Deflection due to rider or mass... ... 84-525 ... 85-125 ... 84-000 ... 85-450 Mass deflections rider deflection 1-01390 ... 1-01024 1-01533 1-01454 i r i m i r . m i (17) (18) (19) (20) (21) (22) (23) (24) (25) Scale-readings .-. . 863 1043 859 1033 863 1040 857 1035 862 934 964 936 971 932 963 936 966 930 894 1009 892 1006 892 1007 889 1006 891 916 983 917 985 915 982 915 983 914 Centre of swing 908-30 992-60 908-00 993-15 906-60 991-00 906-10 991-40 905-25 Deflection due to rider or mass... 84-450 85-850 84-650 85-725 Mass deflections rider deflection 1-01421 ... 1-01538 1-01344 May 4, morning. Mean of 10 determinations of MjR = A Attracted masses in lower position 1-015050. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 109 Table III (continued). Same Day. 2.40 to 4.54 p.m. Temperature: in Observing Koom, 13°-9- 14°-1 ; in Balance Room, ll°-7-ll°-75. Barometer, 740-3-739-7 millims. i r i m i r i m (1) (2) (3) (4) (5) (6) (7) (8) Scale-readings... 847 1035 853 1031 864 1035 853 1026 933 957 930 961 925 960 931 965 883 1003 886 1002 890 1003 885 1001 912 977 911 979 909 977 912 980 Centre of swing 901-40 986-20 90200 987-05 902-55 987-10 90200 987-65 Deflection due to rider or mass... ... 84-500 ... 84-775 ... 84-825 85-300 Mass deflection -^ rider deflection ... ... ... 1-00133 1-00251 1-00783 1 ^• r i m i r . m (9) (10) (11) (12) (13) (14) (15) (16) Scale-readings... 866 1035 853 1023 864 1036 854 1024 924 959 932 968 925 960 931 968 891 1004 886 1000 889 1004 886 1000 1 909 977 912 982 910 978 912 982 Centre of swing 902-70 987-20 902-80 988-35 902-30 987-75 902-50 988-45 Deflection due to rider or mass... ... 84-450 ... 85-800 ... 85-350 ... 85-925 Mass deflection -^ rider deflection 1-01303 1-01060 1-00601 ... 1-00940 i r i m i r i m (17) (18) (19) (20) (21) (22) (23) (24) Scale-readings . . . 864 1039 855 1025 866 1040 855 1011 925 958 931 969 925 958 932 977 890 1005 887 1001 891 1005 888 996 909 978 913 982 911 978 913 985 Centre of swing 902-55 987-80 903-25 989-20 903-50 987-90 904-00 989-10 Deflection due to rider or mass... ... 84-900 ... 85-825 ... 84-150 ... 85-225 Mass deflection -f- rider deflection ... 1-01148 ... 1-01538 1-01634 ... 101232 110 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III (continued). i r i m i r i m (25) (26) (27) (28) (29) (30) (31) (32) Scale-readings... 864 927 1036 961 854 933 1024 970 865 926 1037 962 854 934 1031 967 890 1004 887 1001 892 1005 888 1004 912 979 914 984 912 980 915 983 Centre of swing 903-75 988-10 904-00 989-85 904-35 989-25 904-90 990-55 Deflection due to rider or mass... 84-225 ... 85-675 84-625 85-625 Mass deflections rider deflection 101454 ... 1-01481 ... 1-01211 ... 1-01182 * r m i r ^ m , (33) (34) (35) (36) (37) (38) (39) (40) Scale-readings... 864 1039 855 1024 864 1041 856 1025 928 961 934 972 929 961 934 971 892 1006 888 1002 891 1006 888 1002 912 980 914 985 913 980 915 984 Centre of swing 904-95 989-50 904-80 99M0 905-00 989-65 905-00 990-65 Deflection due to rider or mass... 84-625 86-200 84-650 ... 85-500 Mass deflections rider deflection ... 1-01521 ... 1-01846 ... 1-01418 ... 1-00796 i r i m . r . m (41) (42) (43) (44) (45) (46) (47) (48) Scale-readings... 866 1038 857 1030 865 1043 858 1035 927 962 934 960 930 962 934 966 893 1007 889 1004 893 1008 891 1006 913 981 915 984 915 981 915 984 Centre of swing Deflection due to 905-30 990-40 905-50 991-30 906-60 991-15 906-55 991-55 rider or mass... Mass deflections ... 85-000 ... 85-250 84-575 ... 84-95 rider deflection ... 1-00441 1-00546 ... 1-00621 1-00741 THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 111 Table III (continued). i r i m i r i m (49) (50) (51) (52) (53) (54) (55) (56) Scale-readings... 869 1041 858 1037 867 1041 856 1029 i 927 961 935 965 929 963 936 972 895 1008 891 1007 894 1007 890 1004 914 982 916 984 914 982 916 986 Centre of swing 906-65 990-90 907-00 991-80 906-60 991-05 906-70 992-50 Deflection due to rider or mass... 84-075 ... 85-000 84-440 85-750 Mass deflections rider deflection 1-01070 ... 1-00905 ... 1-01155 ... 1-0] 509 i r i m ^ r i m (57) (58) (59) (60) (61) (62) (63) (64) Scale-readings... 871 1041 859 1035 869 1044 860 1030 928 963 936 969 931 963 937 973 895" 1008 890 1008 895 1010 892 1005 913 982 917 985 916 983 917 986 Centre of swing 906-80 991-50 907-10 993-50 908-20 992-85 908-25 993-30 Deflection due to rider or mass... 84-550 85-850 84-625 ... 84-775 Mass deflections rider deflection 1-01478 1-01493 1-00812 ... 1-00162 i r i m i r i m (65) (66) (67) (68) (69) (70) (71) (72) Scale-readings . . . 863 1042 863 1039 840 1042 861 1037 935 965 935 969 949 965 937 971 894 1010 894 1008 886 1009 893 1009 917 984 917 985 923 985 918 987 Centre of swing 908-80 993-45 908-80 993-75 909-15 993-25 909-05 995-05 Deflection due to rider or mass... 84-650 ... 84-775 84-150 85-750 Mass deflection s rider deflection ... 1-00148 100444 ... 1-01322 101449 112 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III (continued). i r i m i r i m (73) (74) (75) (76) .(77) (78) (79) (80) Scale-readings... 865 1045 860 1038 868 1045 863 1041 935 965 938 969 934 965 937 969 895 1011 893 1010 895 1012 894 1011 918 985 919 987 918 985 919 988 Centre of swing 909-55 994-30 909-25 995-00 909-50 994-75 909-80 995-80 Deflection due to rider or mass... 84-900 ... 85-625 ... 85-100 85-625 Mass deflection-^ rider deflection ... 1-00928 ... 100735 ... 1-00617 1-00765 i r i m i (81) (82) (83) (84) (85) Scale-readings... 864 1044 860 1036 867 938 967 940 974 936 895 1012 894 1010 896 919 986 920 989 919 Centre of swing 910-55 995-45 910-65 996-80 910-60 Deflection due to rider or mass... 84-850 86-175 Mass deflections rider deflection ... 1-01238 May -1, afternoon. Mean of 40 determinations of MIR = A] r 1 '01 on'?7ft Attracted masses in lower position J ^ * April 30 and May 4. Mean of 100 determinations of MIR = A] \4.. . 1 . . . r 1-0109685. Attracted masses m upper position j THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 113 Table III (continued). III. Attracted Masses in Upper Position. May 25, 1890, 11.20 to 12.53 noon. Temperature : in Observing Room, 15°-4-16° ; in Balance Room, 13°-3. Barometer, 748-5-748-1 millims. Weather, E. wind, warm, very- bright. Time of swing not recorded. i r i m i r i m (1) (2) (3) (4) (5) (6) (7) (8) Scale-readings... 1071 1175 960 1049 1003 1173 956 1049 986 1085 1047 1028 1021 1085 1047 1028 1033 1134 998 1041 1010 1134 996 1040 1005 1108 1025 1034 1017 1107 1024 1034 Centre of swing 1015-90 1116-90 1015-50 1036-20 1014-25 1116-55 1014-25 1035-80 Deflection due to rider or mass... 101-200 ... 21-325 ... 102-300 21-500 Mass deflection ^ rider deflection ... ... ... -209582 -209311 •210320 i r i m i r i m (9) (10) (11) (12) (13) (14) (15) (16) Scale-readings... 1001 1173 958 1049 1002 1173 957 1049 1021 1085 1046 1028 1020 1084 1046 1028 1011 1134 996 1038 1010 1134 995 1040 1016 1106 1024 1033 1015 1105 1023 1032 Centre of swing 1014-35 1116-35 1014-05 1034-70 1013-50 1115-80 1013-25 1035-90 Deflection due to rider or mass... 102-150 20-925 102-425 ... 22-850 Mass deflection ^ rider deflection ... -207660 ... -204571 ... -213688 ... -223297 i r i m i r i m (17) (18) (19) (20) (21) (22) (23) (24) Scale-readings... 1003 1173 956 1048 1001 1172 958 1048 1019 1082 1043 1025 1019 1081 1042 1026 1009 1133 994 1038 1008 1131 994 1038 1015 1104 1023 1032 1014 1103 1021 1030 Centre of swing 1012-85 1114-60 1011-90 1033-60 1012-05 1113-10 1011-35 1033-50 Deflection due to rider or mass... 102-225 21-625 ... 101-400 ... 22-375 Mass deflections rider deflection ... -217535 -212400 -216962 ... •220172 p. c. w. 114 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III (continued). i r i m i r I m (25) (26) (27) (28) (29) (30) (31) (32) Scale-readings... 1000 1171 953 1047 1000 1171 955 1046 1017 1081 1044 1025 1016 1080 1041 1025 1007 1130 992 1037 1007 1130 992 1036 1014 1103 1021 1031 1013 1103 1021 1030 Centre of swing 1010-90 1112-70 1010-80 1032-90 1010-45 1112-40 1010-00 1032-15 Deflection due to rider or mass... 101-850 ... 22-275 ... 102-175 22050 Mass deflection -^ rider deflection -219195 •218356 ... •216907 ... •215885 i r ; m i r i m (33) (34) (35) (36) (37) (38) (39) (40) Scale-readings... 999 1168 952 1046 999 1170 955 1046 1017 1080 1043 1024 1018 1082 1043 1026 1006 1131 992 1037 1007 1130 993 1039 1013 1102 1021 1030 1012 1102 1021 1030 Centre of swing 1010-20 1112-40 1010-40 1032-35 1010-70 1112-65 1011-05 1033-75 Deflection due to rider or mass... ... 102-100 21-800 101-775 22-100 Mass deflection -^ j rider deflection ... •214740 •213857 ... •215672 -216858 i r i m i r i m (41) (42) (43) (44) (45) (46) (47) (48) Scale-readings... 998 1171 955 1046 1000 1173 956 1048 1019 1082 1043 1027 1019 1082 1046 1028 1009 1132 994 1038 1009 1131 995 1039 1014 1104 1022 1031 1014 1104 1023 1032 Centre of swing Deflection due to 1012-25 1114-00 1011-65 1033-85 1012-35 1113-80 1013-20 1034-85 rider or mass... Mass deflections 102050 ... 21-850 ... 101-025 21-900 rider deflection ... •215388 ... -215197 ... •216531 -216350 THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 115 Table III (continued). i r i m i r i (49) (50) (51) (52) (53) (54) (55) Scale-readings . . . 1001 1172 956 1048 1000 1173 956 1019 1083 1046 1027 1020 1082 1044 1009 1132 996 1039 1009 1133 995 1015 1105 1023 1032 1016 1105 1023 Centre of swing 1012-70 1114-60 1013-65 1034-60 1013-15 1114-70 1012-65 Deflection due to rider or mass... ... 101-425 ... 21-200 101-800 Mass deflection -^- rider deflection ... •212472 ... -208636 May 25, morning. Mean of 25 determinations of MjR = a Attracted masses in upper position •21446168. Same Day. 3.15 to 4.50 p.m. Temperature: in Observing Room, 16°-0 to 16°-25; in Balance Room, 13°-3 to 13°-35. Barometer, 747-7-747-4millims. i r i m i r i m (1) (2) (3) (4) (5) (6) (7) (8) Scale-readings . . . 1001 1205 990 1081 1034 1207 991 1083 1069 1116 1077 1061 1055 1120 1080 1062 1031 1165 1029 1073 1044 1168 1030 1076 1052 1138 1057 1066 1049 1139 1059 1068 Centre of swing 1044-55 1147-60 1046-30 1068-55 1047-60 1150-50 1048-35 1070-65 Deflection due to rider or mass... ... 102-175 ... 21-600 102-525 21-675 Mass deflection -^ rider deflection ... -211041 ... •211046 -212162 i r i m i r i m (9) (10) (11) (12) (13) (14) (15) (16) Scale-readings . . . 1037 1209 995 1086 1039 1213 994 1088 1056 1119 1082 1066 1058 1121 1086 1069 1046 1169 1030 1078 1048 1172 1034 1078 1052 1141 1059 1071 1054 1145 1064 1073 Centre of swing 1049-60 1151-10 1049^00 1073-50 1051-60 1154-10 1052^90 1074-90 Deflection due to rider or mass... ... 101-800 ... 23-200 ... 101-850 ... 21-675 Mass deflection ^ rider deflection ... •220408 ... -227842 ... •220299 ... •216738 8—2 116 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III {continued). i r * m i r i m (17) (18) (19) (20) (21) (22) (23) (24) Scale-readings... 1045 1212 996 1088 1043 1215 1002 1094 1059 1126 1088 1071 1063 1126 1088 1072 1050 1175 1037 1080 1053 1177 1039 1083 1056 1147 1066 1076 1058 1148 1066 1077 Centre of swing 1053-55 1157-20 1055-30 1077-10 1056-30 1158-50 1056-55 1079-25 Deflection due to rider or mass... ... 102-775 21-300 ... 102075 ... 22-350 Mass deflection ^ rider deflection ... •209073 ... •207957 ... •213813 ... •219575 i r ^ m i r i m (25) (26) (27) (28) (29) (30) (31) (32) Scale-readings . . . 1044 1217 999 1095 1048 1221 1002 1096 1064 1127 1093 1073 1067 1131 1094 1076 1053 1178 1041 1086 1056 1180 1042 1088 1061 1150 1069 1079 1061 1152 1071 1081 Centre of swing 1057-25 1159-80 1059-35 1081-35 1059^70 1162-50 1060^70 1083-55 Deflection due to rider or mass... 101-500 21-825 102-300 22-800 Mass deflection ^ rider deflection ... •217611 •214181 •218112 •223147 (33) r (34) (35) m (36) (37) (38) (39) m (40) Scale-readings... 1049 1067 1057 1064 1221 1131 1182 1154 1004 1095 1045 1072 1097 1079 1089 1082 1053 1069 1059 1066 1224 1134 1183 1156 1007 1096 1047 1075 1101 1079 1091 1085 Centre of swing 1060-80 Deflection due to rider or mass... Mass deflections rider deflection 1163-75 102-050 -221583 1062-60 1085-20 22-425 -219907 1062-95 1165-70 101-900 -217983 1064-65 1086-90 22-000 -215898 THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 117 Table III {continued). * r i m i r i m (41) (42) (43) (44) (45) (46) (47) (48) Scale-readings . . . 1054 1226 1009 1101 1054 1225 1008 1102 1072 1135 1096 1081 1073 1135 1099 1081 1061 1185 1048 1093 1064 1187 1048 1094 1068 1157 1076 1086 1068 1159 1080 1089 Centre of swing 1065-15 1167-15 1065-35 1088-55 1066-85 1168-40 1067-00 1089-70 Deflection due to rider or mass... ... 101-90 22-450 101-475 21-625 Mass deflection -^ rider deflection •218106 ... -220774 -217172 ... •213607 i r i m i r i (49) (50) (51) (52) (53) (54) (55) Scale-readings . . . 1058 1228 1014 1104 1059 1229 1019 1075 1138 1102 1086 1076 1141 1103 1066 1189 1053 1097 1068 1192 1055 1072 1161 1080 1091 1074 1163 1081 Centre of swing 1069-15 1170-80 1070^45 1093-00 1070-95 1173-40 107215 Deflection due to rider or mass... ... 101-000 22-300 ... 101-850 Mass deflection -f rider deflection •217458 ... •219867 May 25, afternoon. Mean of 25 determinations of MjR = a Attracted masses in upper position Mean of 50 determinations, morning and afternoon, '2157379 21701412. Summary of Set I. February 4 ... a= -2142212 May 25 Mean value of April 30 May 4 Mean value of a = -2157379 a = -2149791 A = 1-010905 A = 1-011032 A = 1-0109685 therefore a = -7959894. 118 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III (continued). Set II. All Attracting and Attracted Masses inverted and changed over, each to the other side. The Suspending Rods also reversed and Riders interchanged. The initial position always the higher reading on the scale. I. Attracted Masses in Lower Position. July 28, 1890, 8.10 to 9.43 p.m. Temperature: in Observing Room, 17°-16°-9; in Balance Room, 15°4. Barometer, 747-6-748 millims. Weather fine and calm; wind W. r i m i r i m (1) (2) (3) (4) (5) (6) (7) (8) Scale-readings... 1099 912 1130 917 1126 914 1131 922 1051 1007 1034 1005 1036 1008 1035 1005 1081 951 1093 952 1091 951 1093 954 1063 985 1057 985 1058 986 1057 984 Centre of swing 1069-65 971-95 1070-55 97210 1070-20 972-60 1070-95 973-15 Deflection due to rider or mass... ... 98-150 98-275 97-975 97-575 Mass deflections rider deflection ... 1-00217 -99949 -99541 i r i m i r i m (9) (10) (11) (12) (13) (14) (15) (16) Scale-readings... 1128 913 1134 924 1130 915 1137 919 1035 1010 1035 1006 1038 1013 1034 1012 1092 951 1095 956 1094 953 1098 955 1058 987 1058 987 1061 989 1060 989 Centre of swing 1070-50 973-30 1072-25 975-00 1073-05 975-65 1073-80 976-40 Deflection due to rider or 7nass... 98-075 97-650 91-775 97-700 Mass deflections rider deflection ... •99528 ... -99719 ... -99898 ... -99719 THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 119 Table III (continued). i r i m i r i m (17) (18) (19) (20) (21) (22) (23) (24) Scale-readings... 1132 917 1140 924 1134 916 1136 960 1040 1014 1036 1009 1042 1016 1041 991 1095 954 1099 958 1098 955 1098 972 1062 989 1060 990 1064 993 1064 983 Centre of swing 1074-40 976-55 1075-05 977-40 1076-90 978-20 1076-70 979-05 Deflection due to rider or mass... ... 98-175 98-575 ... 98-600 ... 98-100 Mass deflections- rider deflection ... •99962 ... 100191 ... -99734 ... -99506 i r i m i r i m (25) (26) (27) (28) (29) (30) (31) (32) Scale-readings... 1133 916 1142 925 1134 918 1143 925 1044 1018 1039 1013 1045 1019 1042 1018 1098 956 1103 960 1101 957 1103 961 1065 994 1064 994 1068 997 1066 996 Centre of swing 1077-60 979-55 1078-65 980-30 1079-80 981-00 1080-00 982-65 Deflection due to rider or mass... ... 98-575 ... 98-925 98-900 97-075 Mass deflection -^ rider deflection ... -99937 1-00190 ... •99090 -99031 i r i m i r i m (33) (34) (35) (36) (37) (38) (39) (40) Scale-readings... 1136 924 1145 930 1140 918 1143 928 1046 1019 1042 1016 1046 1022 1045 1018 1099 961 1104 964 1104 959 1104 962 1068 996 1067 995 1069 997 1068 996 Centre of swing 1079-45 982-95 1080-75 983-50 1082-00 982-80 1081-80 983-30 Deflection due to rider or mass... 97-150 97-875 ... 99-100 ... 98-875 Mass deflections rider deflection ... 1-00335 ... -99745 ... •99268 ... 100051 120 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III (continued). i r i m i r * m (41) (42) (43) (44) (45) (46) (47) (48) Scale-readings... 1138 926 1146 928 1142 927 1144 937 1047 1020 1045 1022 1047 1021 1047 1015 1104 963 1107 964 1106 963 1107 967 1071 997 1069 998 1071 999 1070 996 Centre of swing 1082-55 984-45 1083-45 985-75 1083-70 985-10 1084-05 985-10 Deflection due to rider or mass... ... 98-550 ... 97-825 ... 98-775 98-900 Mass deflections rider deflection ... •99797 -99151 -99582 1-00139 i r i m i r i (49) (50) (51) (52) (53) (54) (55) Scale-readings... 1140 923 1148 932 1141 924 1144 1049 1024 1045 1021 1050 1024 1048 1105 962 1108 966 1106 963 1107 1072 999 1071 998 1072 1001 1072 Centre of swing 1083-95 985-40 1084-35 986-60 1084-80 986-25 1084-75 Deflection due to rider or mass... ... 98-750 97-975 98-525 Mass deflection s rider deflection ... -99684 ... -99328 July 28, 1890. Mean of 25 determinations of MjR Attracted masses in lower position •9973168. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 121 Table III (continued). September 17, 1890, 8.0 to 9.31 p.m. Temperature: in Observing Room, 17°-17°-5; in Balance Room, 15°-8. Barometer, 746-2-746-4 millims. Weather warm, cloudy. i r i m i r i m (1) (2) (3) (4) (5) (6) (7) (8) Scale-readings . . . 1085 908 1118 921 1109 905 1126 921 1051 1004 1029 995 1036 1006 1026 996 1073 945 1085 949 1081 944 1087 951 1058 981 1050 978 1053 981 1050 978 Centre of swing 1064-20 967-35 1063-40 966-70 1063-75 967-35 1063-95 967-90 Deflection due to rider or mass... 96-450 96-875 96-500 96-450 Mass deflection -^ rider deflection ... ... ... 1-00415 ... 1-00168 •99613 i r i m i r i m (9) (10) (11) (12) (13) (14) (15) (16) Scale-readings . . . 1113 907 1126 929 1110 910 1131 934 1034 1006 1027 993 1038 1007 1027 993 1084 944 1088 953 1083 947 1092 956 1053 982 1052 978 1056 984 1052 979 Centre of swing 1064-75 967-75 1065-05 968-40 1065-90 969-90 1067-10 970-20 Deflection due to rider or mass... 97-150 97-075 96-600 96-850 Mass deflection -;- rider deflection ... -99601 ... 1-00206 ... 100375 ... 1-00026 i r i m i r i m (17) (18) (19) (20) (21) (22) (23) (24) Scale-readings . . . 1104 910 1129 924 1116 909 1121 927 1044 1008 1030 1001 1040 1009 1036 1000 1081 947 1091 953 1086 947 1088 956 1059 985 1054 983 1057 986 1056 984 Centre of swing 1067-00 970-44 1067-90 971-45 1066-70 970-85 1067-05 972-80 Deflection due to rider or mass... 97-050 958-50 96-025 ... 95-550 Mass deflections rider deflection ... •99279 -99288 -99662 ... •99105 122 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III (continued). i r * m i r * m (25) (26) (27) (28) (29) (30) (31) (32) Scale-readings... 1112 914 1131 929 1114 916 1132 934 1043 1009 1033 1002 1044 1011 1035 1002 1086 951 1093 957 1087 952 1094 960 1060 987 1056 984 1061 988 1058 986 Centre of swing 1069-65 973-10 1070-15 974-00 1070-70 974-50 1071-70 976-00 Deflection due to rider or mass... 96-800 ... 96-425 96-700 96-550 Mass deflection -^ rider deflection -99161 ... •99664 -99780 -99690 [ - i r i m ^• r . m (33) (34) (35) (36) (37) (38) (39) (40) Scale-readings... 1097 916 1135 942 1119 919 1136 935 1058 1015 1037 999 1048 1017 1039 1006 1083 954 1098 965 1093 957 1099 963 1067 991 1061 986 1066 993 1063 989 Centre of swing 1073-40 977-10 1074-80 977-85 1075-80 979-70 1076-25 979-10 Deflection due to rider or mass... 97-000 97-450 96-325 97-025 Mass deflections rider deflection 1-00000 ... 1-00815 1-00947 1-00362 (41) (42) (43) m (44) i (45) (46) (47) m (48) Scale-readings . . . 1122 1048 1093 1065 917 1018 956 994 1141 1038 1101 1062 929 1011 962 993 1118 1053 1093 1068 921 1019 958 996 1141 1041 1103 1065 941 1009 966 993 Centre of swing Deflection due to rider or mass... M;iss deflections rider deflection 1076-00 979-45 97-025 -99948 1076-95 980-65 96-925 •99704 1078-20 981-40 97-400 -99538 1079-40 982-65 96-975 •99628 THE GBAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 123 Table III {continued). i r i m i r * (49) (50) (51) (52) (53) (54) (55) Scale-readings . . . 1134 920 1143 932 1134 925 1140 1047 1022 1041 1016 1048 1021 1045 1100 958 1104 964 1102 962 1104 1067 998 1065 996 1068 997 1068 Centre of swing 1079-85 982-65 108000 983-85 1081-15 984-20 1081-55 Deflection due to rider or mass... ... 97-275 96-725 97150 Mass deflection -h rider deflection ... •99563 ... -99499 September 17, 1890. Mean of 25 determinations of M/R = A] Attracted masses in lower position J July 28 and September 17. Mean of 50 determinations of M/R = A, -9978658. II. Attracted Masses in Upper Position. September 23, 1890, 7.52 to 9.30 p.m. Temperature: in Observing Eoom, 15°-3-15°-4; in Balance Room, 15°-05. Barometer, 749-8-750-2 millims. Weather, light S.W. wind and clear after heavy showers. Scale-readings between about 1100 and 1300; 1000 omitted. i r i m ^• r ^• ?/i (1) (2) (3) (4) (5) (6) (7) (8) Scale-readings . . . 307 113 329 235 281 112 326 233 248 210 235 257 261 208 232 256 285 151 293 243 273 149 290 241 263 186 257 251 265 185 256 249 Centre of swing 271-00 173-10 270-95 248-25 268-35 171-45 268-25 246-60 Deflection due to rider or mass... 97-875 21-400 96-850 21-175 Mass deflection h- rider deflection ... ... ... -219797 -219799 ... •218581 124 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III {continued). i r i m i r i m (9) (10) (11) (12) (13) (14) (15) (16) Scale-readings . . . 279 110 331 232 277 110 324 230 260 207 228 255 258 205 229 254 272 148 290 239 271 147 288 239 264 183 253 248 262 182 252 247 Centre of swing 267-30 170-15 266-80 245-15 265-80 168-90 265-55 244-50 Deflection due to rider or mass... ... 96-900 21-150 ... 96-775 ... 20-225 Mass deflection -^ rider deflection •218395 ... -218407 ... -213769 -209179 i r ^ m ^• r ^■ m (17) (18) (19) (20) (21) (22) (23) (24) Scale-readings... 275 108 323 228 276 107 328 226 256 204 228 253 255 203 224 252 269 145 286 237 268 145 287 236 261 181 251 247 260 179 249 245 Centre of swing 263-90 167-40 264-10 243-15 263-00 166-65 263-^5 241-85 Deflection due to rider or mass... 96-600 20-400 96-475 20-225 Mass deflections rider deflection ... •210274 -211317 •210547 ... -209652 i r i m i r i m (25) (26) (27) (28) (29) (30) (31) (32) Scale-readings... 274 106 320 232 271 100 317 222 254 199 224 245 252 197 221 247 265 143 283 237 262 138 281 232 258 176 246 241 255 175 243 241 Centre of swing 260-90 163-90 260-40 239-85 258-25 160-50 257-90 237-60 Deflection due to rider or mass... 96-750 19-475 97-575 19-100 Mass deflection -^ rider deflection -205323 -200437 -197668 -196730 THE GBAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 125 Table III (continued). i r i m i r i m (33) (34) (35) (36) (37) (38) (39) (40) Scale-readings . . . 266 98 317 221 265 97 311 218 249 197 219 244 246 193 218 242 259 136 279 228 257 135 275 226 254 173 241 237 251 171 240 235 Centre of swing 255-50 159-10 255-90 234-20 253-05 157-00 253-30 232-10 Deflection due to rider or mass... ... 96-600 ... 20-275 96-175 20-150 Mass deflection -^ rider deflection ... -203804 -210351 ... -210164 ... -209271 i r i m i r i m (41) (42) (43) (44) (45) (46) (47) (48) Scale-readings... 264 95 310 215 261 91 311 212 243 191 216 239 241 188 210 236 256 133 273 223 253 128 272 220 249 168 238 232 246 166 234 229 Centre of swing 251-20 154-90 251-40 229-10 248-55 151-10 248-40 226-10 Deflection due to rider or mass... 96-400 ... 20-875 ... 97-375 20-675 Mass deflection -^ rider deflection ... -212785 ... -215456 ... -213350 ... -213585 i r i m i r ^• m (49) (50) (51) (52) (53) (54) (55) (56) Scale-readings . . . 257 90 306 211 256 88 303 208 237 186 208 234 236 184 209 232 250 127 269 218 248 125 265 216 243 162 232 227 242 160 231 225 Centre of swing 245-15 149-25 245-80 224-10 244-25 147-20 243-95 222-10 Deflection due to rider or mass... 96-225 20-925 96-900 20-350 Mass deflections rider deflection -216160 -216699 •212977 ... •209956 126 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III (continued). (57) (58) (59) m (60) (61) Scale-readings... 253 233 246 238 86 180 122 157 301 205 263 227 203 228 213 222 293 204 280* 245 Centre of swing Deflection due to rider or mass... 240-95 144-00 96-950 240-95 September 23, 1890. Mean of 27 determinations of M/R = a) .2112753 Attracted masses in upper position J September 25, 1890, 7.10-8.43 p.m. Temperature: in Observing Koom, 15°-15°-2; in Balance Room, 15°. Barometer, 760-8 millims., steady. Weather cloudy, with westerly airs. Time of swing 21 seconds. 1000 omitted in scale-readings. I r i m i r *■ m (1) (2) (3) (4) (5) (6) (7) (8) Scale-readings... 246 84 301 206 248 82 297 202 238 179 205 229 233 178 204 228 243 121 263 215 243 119 260 212 239 156 228 224 236 156 226 222 Centre of swing 240-90 142-90 240-95 220-40 238-95 141-60 238-95 218-10 Deflection due to rider or mass... ... 98-025 19-550 97-350 20-850 Mass deflections rider deflection •200128 •207499 •213163 * This is a considerable rise, showing either a sudden disturbance or a displacement of the apparatus; possibly the telescope was touched. The rise was maintained and therefore the observations were discontinued. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 127 Table III (continued). i r i m * r i m (9) (10) (11) (12) (13) (14) (15) (16) Scale-readings... 248 83 300 204 252 84 303 207 233 176 203 228 233 182 206 232 243 119 261 214 245 122 265 217 236 155 226 224 239 158 228 226 Centre of swing 238-95 140-80 239-20 •219-50 240-70 144-60 242-45 222-60 Deflection due to rider or mass... ... 98-275 20-450 ... 96-975 20-825 Mass deflection -^ rider deflection ... -210125 ... -209475 ... -212813 ... •214718 i r . m *■ r i m (17) (18) (19) (20) (21) (22) (23) (24) Scale-readings... 255 87 307 210 257 90 271 215 237 184 207 234 238 184 233 233 249 125 267 219 251 127 255 222 242 162 232 228 244 163 241 229 Centre of swing 244-40 147-55 244-70 224-65 24605 148-75 246-70 226-15 Deflection due to rider or mass... 97-000 20-725 97-625 ... 20-725 Mass deflection-:- rider deflection ... -214175 -212974 ... -212292 ... -212564 i r • m i r ^• m (25) (26) (27) (28) (29) (30) (31) (32) Scale-readings... 258 90 307 213 262 93 307 215 241 186 213 237 242 189 215 239 251 129 270 223 253 131 272 225 244 164 236 232 246 167 237 233 Centre of swing 247-05 150-45 248-60 228-30 248-85 153-00 250-25 23005 Deflection due to rider or mass... 97-375 20-425 ... 96-550 ... 19-750 Mass deflection-^ rider deflection -211297 •210649 ... •208053 -204425 128 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND Table III (continued). i r i m i r * m (33) (34) (35) (36) (37) (38) (39) (40) Scale-readings... 261 243 93 189 312 214 215 241 263 245 96 192 312 217 217 243 253 132 273 225 257 135 275 227 247 167 237 235 250 168 240 237 Centre of swing 249-35 153-40 250-80 231-10 252-40 156-05 253-05 23305 Deflection due to rider or mass... ... 96-675 20-500 96-675 ... 20-525 Mass deflection -f rider deflection ... •208172 •212051 ... •212180 ... •212254 i r . m i r i m (41) (42) (43) (44) (45) (46) (47) (48) Scale-readings... 264 98 314 220 267 100 321 224 247 194 219 243 249 197 223 246 259 136 277 231 260 138 281 233 251 171 242 238 255 174 246 242 Centre of swing 254-10 157-85 25505 235-20 256-25 160-35 259-30 238-05 Deflection due to rider or mass... ... 96-725 20-450 ... 97-425 ... 21-250 Mass deflection -^ rider deflection •211812 ... -210662 ... •214011 •218650 i r i m i r i (49) (50) (51) (52) (53) (54) (55) Scale-readings... 271 102 321 224 271 102 314 252 200 221 247 251 198 226 264 139 282 233 264 140 280 1 256 176 245 242 256 174 247 ' Centre of swing 259-30 162-20 259-00 238-40 258^95 161-60 259-50 Deflection due to rider or mass... 96-950 20-575 ... 97-625 Mass deflections rider deflection ... -215704 ... •211487 September 25, 1890. Mean of 25 determinations of MIR = a\ .01125332 Attracted masses in upper position J ' ^September 25^ } ^^^^ ^^ ^^ determinations of MjR = a, -2112647. THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 129 Table III (continued). Summa/ry of Set II. July 28 September 17 Mean value of September 23 25 Mean value of A = -9973168 A = -9984148 A = -9978658 a - -2112753 a = -2112533 a =-2112647 Therefore A-a= -7866011. Mean value, giving equal weights to Sets I and II, A-a = -791295. [The experimental data in these tables have been verified from the original MS. Certain slips in calculations from the data occur in Table III, but it was not considered advisable to correct these. Ed.1 p. c. w. 310 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND [|!|||l||||||ITI1IMlLllllll'i'l!IIWilllllittttt^^ ^--zrtiti±-S--xii""- X !xzzE--r:fzz-z:+:-=t:::j::::::::::i-zz§ __^j — ^^-oJ — m ^ ^ 'c ^---!---^ — \ U U uJ 1 Kr-h \ ^:^ M -s ^ \ U J . -1 l--'-r--i M h 7*- 1 1---* ' 8::X^::i::^?::4:g:gEE^ EE^EEE±EEE eIeEEEeSi EE^:|E::g::r^i-T— T-rt-^^-^-— ^-^-X--m-^ ^ S^i _ - - - -.-+- -. -- - zitztzz I zm MmN |i i|. 1 1 i|l <t 1 1 1 i 1 _L_-^_^ 1 1(- , ^s--l ^ \ ' ^+4-- ZfZ ^_^ \ |4-X / , -i pi-^- §+_z__X:i:x:::iX:::::::::-:::-::::^!::=:::-::.±-::i:::::-::±4XZg XX-±±-±T---E--zzzzzz^zzzi-zzzzfziz-zzzx:";::t:rzztz-zzzzJ-- _Z '' ' \ 1 . ■ \ . . ^X XX ■ X 1 ^^ - 1^ -|--+- j U . 1 1 ' ' n s ■ ; ' ■ ■ ■ .1 ! 1 1 '«_ _L_ 4 LJ ^4 ' '^ ' 1 . , ,. : 4 1 ,, ,, f5 ^^ - H_ ».-- -^ - -- ^ - -j-ph- - - - o __^^ . 1_< L-^ -^ -J ^A 1-ij h -T — ^ 1 — ^ hn ^-^--i ^ ^^ K-n H- "^ ^ XX ! ^ :|: ^ zLx^ nzt X ' 1 ^ i . I i ,* ■? * . ' . . ^ ' F . 4 /,, -. °' t ^1 .--I-4-- "" . > 1 -f ,,.-..- 3 I 1** . '"^ A .'- J X -^ ^•-— 4-X_^-J r-^TT— xP---H r--' ^--r ^- = J — ^ 1" ! ' 1 -X— ^^^--xl j^'^Tv ^x-Hx----J 1 . n . -, / M 1 ' i M ■'+ '■ ~^„ , H / 1 1 1 L^-! s-:::gi:gX:,:::,:,--,,,-|-l:::^|g:^|#.x:::g, q:-::-i±:::±^-::::?::;:X:i:::tt:iE"-"EJ:-X-"l4:--*----ZS: X::5±ii:|£:=ii'Ti:EX:||EEi|EEEE|EEEEEEEEE^ |||n|g:;|=|i|::|||||;;||. iP^^iiEiiiiliEiiiiiiEEE H ^ ^ hH — X-^ 1 h 1 ^ ■? r>- h-1 — 1 UXX — XH ^::::::=::::::i|pEEEXEE|E5z^===^:=::^^|4^^^^ THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 131 "H" i ^) "ji " '1 ~_ "' It: - _ -X -- i: : __:: 8 , -iL.-If-i"- " -a-^ 1 p^ :----- -"__X-!! _ .. . '^ . _ . . — : ± ' -s :: ± i\ .. ± :: _. ^ :: _ .^. :::-.;? s . . ^X ^I-'- . _ _ _ . ^- . .. 1 _ JL _SJe J.. i'. -.^!se ^ i, t \l ^ . 1 -"5: ": : : : ::: ::: : : ::"" i ----^ -.- ----__ ji^r. : _ _ ^ _ :~^3t : _ - _ ^ ± : : . _ . • •i .._ _a_S__i__i ._z:__u.- i : : i — ± tr it — J - _ ^ - - - 2 1- u _L it it • ^ , >i 1 1 • :i" -l+ -f- -t:* 4^-- ': ' i' ; ' * ' ~ - : :: :ix± ii:^-*^: _' :_ a_Lxit: _-it-- -dt-- -_: :_. ot : X " a i - L. - : : 1 1 ^ • ; 1 1 i • i K " ' • jt • * - < L • *" . O ^ . ' \ :: ^"5"! - t'-~ ~ " ^ ^ _^ ■: it "^ ■ 3 L »- • . - |- r- ' > '; ::_ ___ a i:___ ___ _:_ "t: --± :__:_: s 1 « ^- ---t- _ .. it k'^H^X X^i ii» " " - x5 n • 1 5 i - ' A '• * * ^ — ^ ^ _j_- _ _^. _» _ • ; ■ 3 ; ' "• ■* Q L ' ' ' — -_ ^_jj ^.f - . ^ ^ L ^ ? " "■ .*f " " ■ (5 "^ ^" 5" ~ - 5 -• 7 x_ it * ' , _ J. « , _ _ ^ - " Q :x ^" "13" " ^ _ _ -- - pj: ^ r- " i" " ' ' * ' S '1 ■" ;[ ^ :: :x -- ± - "~ \ ^ ~ ' . " " - >c" " ..:.u !:___ X ..± , .^ =.._ ---^ _ 1 § • v'/'^r § 6^^^y^s 5 J° «-«?»D i ^ ^ ^ J^ ^ ^M - O »v»»« ff 5 o o i § 1 t 1 1 1 9—2 132 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND 3 O OBSTyfX-i^pTH-^&W^yo a-fTLWjg THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 133 In 1 ° .* - ._.._.. ^ - - (. :::::::__;__:_:::- ;:;_::_:__:_:;:__:::_:_::_:_:_: _» X- X :::::i:i::4:::::::::=:::::i:::::::::==::::=::::=*:=: 1 1 1 j »i '- ]-- 1 p' -• — _ 1__) . ». — . _ _ _| — It ^ M J^ U « . 1, 1 y. _j l_l --■ 1 '■ 1 1 1 1 » . |_j <--r 1 1 I M \ — r- 1 1_| 1 I I I I ■ ►- 1 — I » pj 1 ! ^ 1 I I -t|- -J- mill 1""!!!!!" 1 »-- 1 1 > :::::E=E=::::EE::=:=|e=::;=::=±:::±^ , — p , , 4( 1 1_ -« ^1 h- : 4 1 ^^ : 1 ( i: ii: II II :!:::ii::::i~f":::;:r::::::::i:::::i: :•:-::: n:^ — ^ir ►-Tr4^ 6vy^ JO 9J}!U3^^ 8 • T°7r?^'^ Rl" ' It ' -___ _.-. sL"" * " ' 2C ' ":. ":" ::::j :: :: :_::_:&:!: " *" 1 ^ ' ► - -J- • : : it : :::::::; : \ '_ >> J- • 1 ± : -. ± : : : : it : ___ _ , X.o . . ^i » i :x ::::::::: ni ' ( _ir _ _ _^ : i.-Ia ^^ x_ * — ^ ■ . ■ - I_ ^ - - - - - - -}- - i c - i -■■, ■> _. ._._ _ ■ ! c, ? it _ ± I :: '^:::-:---± ;^ ■§ X « - i- - e T " it T " ^fe « :: ::: -4: ± : s i" ""■ . " "t \ _. _ ..-'".-^- u : - :: ±:: : a- : T *" " T ■' " 1 I J i ! : 8 . s 1 § ' ' f 134 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND 8 o ^ I ^ ^1 THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 135 _. .._j.._^_ TBI" « ' ±: :±:i:t:::_: iiiiisfii iL. ._ z . -XII 11^ iiiaSi -.11 i> 1 IKU i . _ 41 tw -«---+ _r - - _ _ "KS ' <----- -1 1 ' t- - :: h: : i.: _±: : '^ ._:--. _ . 41-. . - (L . . . __ _. ±1. : j^ < _ _.-:__. . - : : .s ^L. _. S •* ^^ _ : — '5 <^) — _ '\ ^ - J ;:_- : :: ::: * : ::: :: ^ - - - • ?^ " z X " It ---,-.. - .5 :::: :::::::::! : §- ~^'T ■''T''" 1 -----_ - _, vS "^1 . _^ _ ^ x 'i ' 1 i ' 1 ' i ' 1 ' i" x±" >- ' is' ■ -1 1 i 1 i:±iti::±t;:::i . _ ^ T — Js? I -"i — ■ — 1 1 — g .. . ^ ^^ ^ .^.. .- u. 1 ji :_ T . — 4^ ' -^ 1 ■ :: I": ii^r. ~z :_:::: *& " "^: s s u ■ 1" ■* P^ " ■2-x± ------: T i — 1 - 1 i 1 -1 1 1 1 1 i ■ i'::"x: § 1 § o '^^ ■Ll. 1 , H M 1 1 1 1 j 1 1 1 1 M M M 1 M 1 M 1 M^lo § P 1 i 1 1 ni^nn. .g | Sn tv^o-i? | k -wpitf | 6via»s Jo | «->noa § 1 136 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH ni yA% wm zti.^ v^ m ii vmi 222 ^2222Z2S2 7, 1 1 Z A 152Tt£ulja ISOrtsults Diagram VII. Distribution of results about the means assumed cor- rect for each set of distances. The numbers along the base are in percentages of the distance a A. The numbers on the side-line show the numbers in each interval 0-25 per cent, from the mean. The distance aA should be 40 inches to show both on the same diagram. J; 60 TT ' 1 rjT 1 _,,... - — r T - j 1 T7"^ r T- - — TT .^ . |. I" 1 Ml ! 1 t 1 j 1 "T"*"^ 1 J 1 ! 1 1 1 " - 1 '• 1 ? ■* 1 ' 1 ' 1 1 t 1 ♦ 1 i i 1 •5 1 ( 1 1 1 5 ! ' ' ' 1 i ; -T" : ' : i *■! 1 i ' 1 1 ■«• ' ! : , : ' 1 ■ 1 , i 1 t t 1 ! , 4 1 1 I ■ fi 1 • • V ■Ml t 1 5 ^ I ' ' ' ! i ^ \^\ t 1 t ^ \ ', \ \ ♦ xl ! 1 a; , 1 Mm 1 1 -i ? 1 1 ' i ; Hx X Xl^: ; r 1 ! ; 1 ^ ^ -r-j-^ ♦x , ^ •* •• 1 1 i 1 4 .- ^^ ^ ! 1 ! ! ■ !_;_[_ ! I'M 1 i ' ' c^i"^ i ' 1 ' 1 ( 1 w MM 1 ^ M 1 _L ' J N 1 ! ' to io 50 - _>. Norrx2yers of 3ux:c««sii^e ^^e^hjings vUh, A ojrA D - ; ; . . . -J Mil ' 1 ■ i 1 i j 1 i M 1 t : ■^-r -rr ; : , , , 1 I M ^ i ' 1 ! i ' ! 1 1 M ' ' 1 : 1 : i !-m ^_ _ ! 1 i h 1 - M- 1 1 --U- i 1 J'l j!' Ti- 1 1 1 . j — - i 1 1 1 j ; t r T,, ^TXSV^ ^-t ! T Idjv X: 1 6 T^A \k ^ J M ! ! 1 i 1 1 ■ 1 ' ' 1 r 1 - i ,1 . _ • ' ' 1 ' ' -i — U- 1 ■ U-- M i i -J"p ^-^^ - -- 1 """ ^ =91 t 5 X- a 53 — ■ i : - - I i -H \ \ \ \ #M 5 1 ^. \ W- fe Diagram VIII. Comparison of Riders A and i). Successive values of i) - ^ in terms of 1 scale-division = -0035 mgm. 1 1 J T ■ 1 1 '' ' \ ]" TM^" \ ' 1 ~1 r ! 1 1 i » 1 1 1 I ■ 1 ' : ... MM ; j 1 1 ' i i i MM: 1 ! i 1 ' Mm 1 i Mi M i 11 \ ' 1 ■ , 1 ' 1 : ; ' i 4 1 , 1 ' I i M '■ i 1 i i 1 ' j 1 1 1 ! M M ! ' ' ' \ ' ' 1 1 i 1 i i' < : : 1 1 ! _i i ^ : : ' 1 Ij M " ' 1 1 « 1 1 T ; I j 1 j 1 Mm ' t ' ' ! ' -^ M ' i i 1 ' ' 1 1 1 ' ! M 1 ' ^" ' 1 ! 1 t M M M 1 1 L ^_ -X 1 M ■ M ' ~'^ 1 ' 1 ■ 1 1 1 1 1 H : ! i 1 1 ■ : i 1 1 1 1 i 1 : I ' 1 1 J I "" * „ 1 700 1 : 1 ! ■ ■ MM , 1 "^ MM 1 1 ! 1 1 X ! 1 t ~[ 4I 1 ' M ' 1 Tin 1 1 ISO 1 ^ i 1 ' , ' ' U ' 1 1 ! 1 ~ "~ 1 ! I ~" ~i " "^ J ■ ' 130- l_ i_ _ _ - -. -- Diagram IX. Relation between temperature and scale-ieading. May 9 — 22. 4. AN EXPERIMENT IN SEARCH OF A DIRECTIVE ACTION OF ONE QUARTZ CRYSTAL ON ANOTHER. By J. H. PoYNTiNG and P. L. Gray, B.Sc. [Phil Trans. A, 192, 1899, pp. 245-256.] [Received September 27. Read November 17, 1898.] Since so many of the physical properties of crystals differ along the different axes, our ignorance of the nature and origin of gravitation allows us to imagine that the gravitative field of crystals may also differ along those axes. Dr. A. S. Mackenzie {Phys. Rev., vol. 2, 1895, p. 321) has described an experiment in which he failed to find any such difierence. Using Boys's form of the Cavendish apparatus, he showed that the attraction of calc-spar crystals on lead and on other calc-spar crystals was independent of the orientation of the crystalline axes, within the limits of experimental error — about one-half per cent, of the total attraction. He further showed that the inverse-square law holds in the neighbourhood of a crystal, the attractions at distances 3-714 centims., 5-565 centims., and 7-421 centims. agreeing with. the law" to one-fifth per cent. One of the authors of this paper had already pointed out (The Mean Density of the Earth, 1894, p. 7) that if the attraction between two crystal spheres were different for a given distance, according as their like axes were parallel or crossed, such difference should show itself by a directive action on one sphere in the field of the other. This directive action is suggested by the growth of a crystal from solution, where the successive parts are laid down in parallel arrangement — a fact which we might perhaps interpret on the molecular hypothesis as showing that, within molecular range at least, there is directive action. The experiment now to be described is a modification of one indicated in the work above referred to, carried out for two quartz spheres, and we may say at once that we have certainly not succeeded in proving the existence of a directive action of the kind sought for. To bring out the principle of the method, let us suppose that the law of the attraction between two spheres w^ith their hke axes parallel, as in Fig. 1 (a), 138 AN EXPERIMENT IN SEARCH OF A DIRECTIVE ACTION is GMM'jr^, where M, M' are the masses, r the distance between the centres, and G a constant for this arrangement. Let us further suppose that the law of attraction when the axes are crossed, as in Fig. 1 (6), is G'MM'/r^, where G' is a constant for this arrangement, and different from G. Let us start with the spheres r apart, as in Fig. 1 {a). The work done in removing M' to an infinite distance, in a Hne perpen- dicular to the parallel axes, is GMM'jr. Now turn M' through 90° to cross the axes, and bring it back to the original position, but with the axes crossed. The force will do work G'MM'jr. Then turn M' through 90° into its original orienta- ^^S- ^■ tion. Assuming that the forces are conserva- tive, the total work vanishes, so that there must be a couple acting during the last rotation, which does work equal to the difference between the works done on withdrawal and approach. If we take the average value of the couple as L, then Our suppositions as to the law of force are doubtless arbitrary, but they serve to show the probability of the existence of a directive couple accom- panying any axial difference in the gravitative field. In the absence of any distinction between the ends of an axis we may assume that the couple is 'quadrantal,' that is, that it goes through its range of values with the rotation of the sphere through 180° and vanishes in every quadrant, and we shall suppose that it is zero when the crystals are in the positions shown in Fig. 1 (a), and Fig. 1 (h). Taking the couple as a sine-function of amplitude F, we have F sin 26 cW whence F=^{G-G' MM' But it is conceivable that the two ends of an axis are different, having polarity of the magnetic type. The couple would then be 'semicircular,' going through its range of values once and vanishing twice in the revolution. We shall suppose that the couple is zero when the axes are parallel. We should now have G and G' constants for the axes parallel, the one when Hke OF ONE QUARTZ CRYSTAL ON ANOTHER 139 ends are in the same direction, the other when they are in opposite directions, and we have MM' r But if F is the amphtude of the couple ttL = [" J sin QdB = 2F, Jo MM' and 2F=(G-G')^^. r To seek for the directive action we have made use of the principle of forced oscillations, thereby obtaining to some extent a cumulative effect, and at the same time largely eliminating the errors due to accidental dis- turbances. Briefly the method was as follows : A small quartz sphere, about 0-9 centim. in diameter, was carried in a frame to which a light mirror was attached, and suspended by a quartz fibre inside a brass case, the position being determined by the reflection of a scale in the usual way. The complete time of torsional vibration was about 120 seconds. Outside the case was a larger quartz sphere, about 6-6 centims. in diameter, its centre being level with that of the suspended sphere, and 5-9 centims. from it. The larger sphere could be rotated about a vertical axis through its centre at any desired rate. The crystalline axes of both were horizontal, that of the smaller sphere being perpendicular to the hne joining the centres. To test for the quadrantal couple, the larger sphere was rotated once in 230 seconds — a period nearly double that of the smaller sphere. To test for the semicircular couple, the larger sphere was rotated once in 115 seconds, or nearly the period of the smaller sphere. Assuming that a couple exists, a continuous rotation of the larger sphere would set up a forced oscillation in the smaller sphere of the same period as the couple, and since the damping was very considerable, this forced oscillation would soon rise to approximately its full value. Meanwhile, any natural vibrations of the suspended system would be rapidly damped out. Though continually renewed by disturbances due to convection-currents and tremors, they would be irregularly distributed, and there was no reason to suspect that their maximum amplitude would recur at any particular phase of the period of the apphed couple. To secure the distribution of successive maxima of natural vibrations of the smaller sphere over all phases of the forced period, the latter was made sensibly different from the natural period in the ratio 23 : 24 ; and though the cumulative effect of the forced oscillations was reduced by the largeness of this difference, we did not think it advisable to make the periods more nearly coincident, lest the distribution of the disturbances, which were sometimes large, should not be sufficient. This 140 AN EXPERIMENT IN SEARCH OE A DIRECTIVE ACTION conclusion was arrived at from the results of preliminary experiments with more nearly equal periods. During each complete period of the supposed appHed couple, the position of the smaller sphere was read ten times at equi-distant intervals of time, and the scale-readings were entered in ten parallel columns, one horizontal line for each period. The observations were continued usually for 70 or 80 periods. Adding up the columns and dividing by the number of periods, any forced oscillation would be indicated by a periodicity in the quotients. The periodicities found were too irregular to be taken as evidence of the existence of a couple. Bescriftion of the Afj)aratus. The quartz spheres were placed in a cellar at Mason College, Birmingham, below the room in which the observing telescope and rotating-apparatus were fixed. The smaller sphere, 0-9 centim. diameter and weighing 1-004 grams, was held in an aluminium wire cage, and was suspended by a long, fine quartz fibre in a brass case from a torsion-head at the top of the case. A light plane mirror was fixed to the cage, and opposite this mirror was a glass window in the case; in front of the window was a plane mirror at 45°, by means of which the light from the scale was reflected into the case and back again to the telescope, as shown in Fig. 2. The case was surroimded by a double-sided wooden box, lined within and without with tin-foil, and with cotton-wool between its inner and outer walls. The box was supported on indiarubber blocks to lessen tremors. The larger sphere, 6-6 centims. diameter and weighing 399-9 grams, was held at the lower end of a vertical brass tube which terminated in a very carefully turned shallow brass bell, in which the sphere was held by tapes. The tube passed upwards through the top of the wooden casing without contact, a kind of air stuffing-box indicated in the figure serving to prevent currents through the hole. The tube came into the room above, and was there connected with a train of wheels, driven by an electromotor, the rotation of the motor being geared down from 1000 to 1. The observing telescope was fixed to a heavy stone slab resting on indiarubber blocks, standing on a brick-pillar, which was built on the brick arches forming the cellar-roof. A diagonal scale (of half-milUmetre graduations, divided into tenths by the diagonal ruhng) was clamped to the telescope-tube and illuminated by an incandescent lamp, aided by a concave mirror. A tenth of a division could be read with certainty, and as the distance from scale to mirror was 358 centims., the position of the suspended sphere could be determined within a Httle more than one second of arc. The steady rotation of the larger sphere was maintained by a regulator, OF ONE QUARTZ CRYSTAL ON ANOTHER 141 for which we are indebted to Mr. R. H. Housman. It consisted of two parts : (1) the governor proper, which automatically maintained approximate steadiness, and (2) a fine hand-adjustment, by which the motion could be accelerated or retarded when it got 'out of time.' One lead to the motor went through two mercury-cups, and the circuit was completed by a fork of platinum- wire dipping into the cups. This wire 7b Accumf ReguUa.U>f Inc Lined Mi •9 [ ni rm it,n"inr xr Fig. 2. Diagrammatic sketch of the apparatus. was fastened to one end of a wooden lever, the other end of which was attached to a sliding collar on the axle of the motor. To this collar were fastened the upper ends of the loaded springs of the governor, as shown in the figure. If the speed increased, the loads flying out pulled the collar down and so raised the wire out of the mercury-cups, and broke the circuit. As the speed diminished, the wire again dipped into the mercury and re-estabhshed the current. To diminish sparking the mercury was covered with alcohol, and the two cups were permanently connected by a high resistance shunt. 142 AN EXPERIMENT IN SEARCH OF A DIRECTIVE ACTION The fine hand-adjustment consisted of a small wooden plunger working in a tube connected with one of the mercury-cups ; by means of a screw the plunger could be raised or lowered, and the level of the mercury in the cup varied accordingly. If the revolving sphere was found to be gaining or losing, it was quite easy to bring it 'up to time' again by working the screw of the plunger. The last of the train of driving-wheels was fixed on the tube supporting the larger sphere ; its rim was divided into equal parts by numbered marks, the use of which will be explained directly. There were 20 numbered marks, at intervals of 18° ; of these only 10 alternate ones were used for the quicker rotation, while the whole 20 were used for the slower speed. The Observations. Two observers were required, one at the telescope to note the position of the smaller sphere, the other to regulate the speed of rotation of the larger sphere, and to notify when readings were to be taken by the first observer. The motion having been started, and brought to about the right speed, a time-table was rapidly prepared, showing the times, on the chronometer used, at which each of the numbered marks above mentioned should pass a fixed mark throughout the whole set of observations for one occasion. A signal was given at each passage of a mark past the fixed point, the observer at the telescope putting down the simultaneous scale-reading in a manner which will be understood from Table I, which may serve as a typical record. It does not appear to be necessary to give the full details in other cases. If the motion did not keep to the time-table, it was easily corrected by the hand-adjustment already described. Every reading in the same column is taken at the same phase in the rotation of the larger sphere, and therefore the mean readings of the columns should preserve any periodicity in the motion of the smaller sphere equal to that of the larger sphere, and more or less ehminate all others. These mean readings are given at the foot of Table I, and appear to indicate a sKght periodic vibration, but this might be due to a want of symmetry in the larger sphere and its attachments about its axis of rotation, since the system supporting the smaller sphere and mirror was necessarily not symmetrical. The observations for each couple were on this account divided into two sets : for the semicircular couple the larger sphere was, in the second set, turned through 180° about a vertical axis from its position in the first set ; for the quadrantal couple the rotation was 90°. For the final results the means of the results of the two sets were taken, in each case after the second set had been advanced by an amount corresponding to the change of position of the sphere. OF ONE QUARTZ CRYSTAL ON ANOTHER 143 Table II contains all the mean results obtained in the same way as the figures at the foot of Table I, the greatest range being given in the last column as an indication of the magnitude of the disturbances. In Table III are given the means for each azimuth of the larger sphere in its support, the B and D series being advanced as mentioned above. In combining the results it appeared useless to attempt to weight them according to the number of periods taken, since no accurate conclusion could be expected. It will be seen that in each case there is an outstanding periodicity, but the amplitude is less when the disturbances (as indicated by the greatest range during a period) are less, and it diminishes when the results are combined so as to lessen the effect of want of symmetry. In the 'quadrantal' observations (Series C, D), where the effect of want of symmetry of the apparatus should almost be eliminated, since it is approximately semicircular, the mean range is much smaller than in Series A and B. For these reasons we do not think that our observations can be taken as indicating the existence of a couple of the kind sought, but only as giving a superior limit to its value, should it exist. We now proceed to the Calculation of the Superior Limit of the Couple. Equation of Motion of the Smaller Sphere. Let / be the moment of inertia of sphere and cage. „ jjL „ torsion-couple per radian. ,, A ,, damping couple per unit angular velocity. ,, F cos 'pt be the supposed couple due to the larger sphere, having period Stt/^. Then Id + XO + yi^ = F cos ft. Putting K^XjI; n^ = ^II', E = F/I we have . S + k6 + n^d = E cos ft (1) The solution of this is d = — ^^ cos (pt -€) + Ae-^'^ cos {Vin^ - ^k^) t - a}, (2) where tan e = -ij^ — ^ and A, a are constants. n^ — f^ The first term in the value of 6 in (2) gives the forced, and the second term the natural vibrations, the period of the latter being = T, say. Vin^ - \k^) 144 AN EXPERIMENT IN SEARCH OF A DIRECTIVE ACTION The value of T was always very near to 120 sees., and the mean of various determinations during the observations gave r=/v-^-i^, = 120-8 sees (3) Value of K. When there are only natural vibrations any complete swing ^ i^ ly next complete swing The value of this ratio was usually near 14. The mean of a number of determinations taken at various times was 1-3953. Putting g3o-2. _ 1-3953, ,veget K' = 0-011033. Value of n. Substituting for k in the value of T in (3) we get ^2 - 0-0027359, and n = 0-052306. Value of e. The forced period 27r/p was always 115 sees., whence tan 6 =-,-^-^ = 2-420, and e ^ 67° 33', sin e = 0-9242. From equation (1) it will be seen that the steady deflection due to F is n while from (2) the amphtude of the forced oscillations is E sin c n^ sin e E or . -^ . Using the values found for uk and e we have or the forced oscillations give a cumulative effect, about four times the steady deflection due to the couple at its maximum value. Value of Moment of Inertia, I. This was found by vibrating the cage hung by a short quartz fibre, (1) when empty, (2) when containing the sphere, the times of vibration being respectively 8-38 sees, and 11-22 sees. The sphere weighs 1-004 grams, and its radius is 0-45 centim., so that its moment of inertia jMr^ - -08132. From this, and the times of vibration, we get 7 = 0-1821. Value of F. The vibrations were observed in scale-divisions, each 0-05 centim., the distance between mirror and scale being 358 centims. If OF ONE QUARTZ CRYSTAL ON ANOTHER 145 N is the number of scale -divisions in the ampHtude of vibration, i.e., in half the range, we have from (2) Eainc _ 6N pK " 2 X 35800' whence F = EI = 0-8293i\^ x lO-^, using the values already found for e, k, I. Taking the Hmiting values of the amplitudes as half the mean ranges given in Table III, the vibration due to the quadrantal couple has amplitude not greater than 0-033 div., and that due to the semicircular couple, amplitude not greater than 0-095 div. Whence F (quadrantal) is not greater than 2-737 x 10-^°, and F (semicircular) is not greater than 7-878 x 10"^°. Perhaps some idea of these values may be obtained by noticing that the times of vibration of the small sphere under couple F per radian would be respectively 32 hours and 25 hours. But it is probably best to interpret the value in terms of the assumptions we made as to the force in the introduction. We found for the quadrantal couple F = (G- G') MM'jf, G-G' GMM' '^ G ' r ' where M, M' are the masses of the spheres, r the distance between their centres, G, G' the parallel and crossed gravitation-constants. Now M, the mass of the larger sphere, is 399-9, say 400 grams, M' „ ,, smaller „ 1-004 grams, r is 5-9 centims., G and G' are exceedingly near 6-66 x 10"^, G-G' Ft whence -^ = GMM'^^^^' On the assumed law of force this imphes that the attractions between the two spheres, with distance 5-9 centims. between their centres, do not differ in the parallel and crossed positions by as much as yeioo ^^ ^^^ whole attraction. We may compare this result with Rudberg's values of the refractive indices of quartz for the mean D line ^,-^„ 1-55328 - 1-54418 _ , ^/x— r54418 -TT^^bout. For the semicircular couple ^^ G-G' GMM' 2F = —^ , G r whence — __- = 3^j^^. p. c. w. ^^ 146 AN EXPERIMENT IN SEARCH OF A DIRECTIVE ACTION On the assumed law of force, this imphes that the attractions between the two spheres, with distance 5-9 centims. between their centres, with their axes parallel and respectively in Hke and unhke directions, do not differ by as much as -^-^-^-^ of the whole attraction. This hmit is large, undoubtedly owing to the want of axial symmetry in the apparatus which produced a semicircular couple as already pointed out. This couple was large, and though we attempted to ehminate it by the two sets of observations with the different azimuths of the larger sphere, in all probability w^e failed. Table I. Showing Scale- Readings in Tenths of a Division at Phases at Heads of Columns. Time of Revolution of Larger Sfhere 115 sees. 1 2 3 4 5 6 7 8 9 55 61 61 64 61 42 25 26 31 40 50 55 60 53 52 45 44 40 45 50 54 51 49 49 48 49 52 57 52 54 57 52 40 33 28 25 30 44 57 66 70 64 52 40 38 36 39 46 55 60 63 61 52 44 44 45 44 43 49 50 52 48 42 30 32 37 45 50 62 71 69 62 52 42 39 38 44 55 58 65 65 66 61 51 45 45 41 38 40 49 56 61 62 60 56 50 48 42 39 37 40 42 58 69 69 68 58 48 41 38 38 42 48 54 60 57 55 50 43 41 41 42 47 49 55 57 63 60 58 49 46 47 46 44 51 52 50 54 48 45 44 40 36 40 50 60 67 67 62 54 44 33 35 35 38 50 57 62 68 62 52 45 36 36 39 44 51 56 59 53 47 48 53 51 50 49 50 49 52 50 50 51 53 52 54 55 48 47 44 41 44 52 55 58 60 56 49 41 41 42 43 47 50 55 60 60 60 56 58 47 43 47 49 50 50 50 54 54 54 48 50 51 49 52 52 45 42 43 48 49 bQ 56 52 52 52 57 56 51 46 42 42 43 49 51 55 55 55 52 49 57 50 50 50 44 43 50 50 50 43 43 46 50 58 54 55 50 49 49 49 48 50 51 54 53 5.6 56 57 58 56 56 51 43 40 38 41 51 60 60 60 58 52 48 48 48 52 57 58 60 57 47 41 41 51 62 63 59 53 46 40 40 40 43 49 51 61 60 60 56 51 48 42 42 43 51 59 63 62 61 55 52 51 50 51 51 52 56 58 58 53 45 40 41 49 60 70 70 60 52 48 48 50 50 54 55 53 51 50 50 1 47 50 50 52 53 53 50 48 48 46 48 50 51 51 50 50 52 52 49 46 44 44 49 50 55 59 57 58 51 49 46 43 44 51 59 68 64 56 50 42 40 49 57 68 71 70 59 50 OF ONE QUARTZ CRYSTAL ON ANOTHER 147 Table I (continued). Mean of 80 in divisions . . . 1 2 3 4 5 6 7 8 9 47 41 43 56 65 66 56 45 39 35 33 40 48 59 68 70 64 51 43 42 48 51 60 64 70 67 56 42 39 38 40 47 52 65 60 61 59 51 51 50 48 47 50 54 53 60 52 50 41 39 40 44 51 58 66 70 71 63 50 38 35 38 41 43 50 56 70 75 70 59 50 46 45 51 61 70 71 70 62 52 41 40 40 40 47 54 60 71 71 68 60 50 48 45 39 39 42 49 51 60 64 61 50 46 47 49 52 60 72 75 70 62 57 32 23 21 30 42 57 77 84 79 63 51 42 33 34 38 49 60 66 64 57 51 49 44 47 49 52 52 55 55 52 58 59 56 57 51 42 40 43 47 55 61 66 64 60 52 50 45 41 45 49 59 67 67 56 50 49 43 38 45 48 53 55 56 57 55 54 56 53 49 42 42 51 61 69 70 65 54 45 41 40 47 51 56 61 59 55 49 48 52 60 60 60 58 50 46 44 43 45 51 53 60 63 67 62 60 55 48 44 46 49 50 52 54 53 50 50 52 60 62 63 61 51 41 39 38 42 50 55 59 54 51 48 47 42 47 48 55 58 61 62 60 59 54 52 52 50 50 58 54 55 55 58 56 56 50 51 51 56 58 51 52 48 48 54 55 50 51 52 51 51 50 45 44 42 46 51 55 56 53 56 59 59 60 58 59 59 54 49 46 46 49 50 52 58 56 57 53 51 50 50 46 49 51 58 66 67 69 65 62 51 46 39 39 39 45 51 56 62 61 53 48 40 38 47 62 67 63 55 52 57 56 56 53 49 42 38 41 51 60 65 71 73 72 60 52 50 40 42 49 52 62 71 73 73 65 59 44 39 38 40 43 51 51 59 61 61 50 49 41 49 51 52 58 58 52 52 50 53 56 57 51 50 49 49 49 51 52 49 52 53 52 ... ... ... ... 5175 5163 5143 min. 5-186 5-246 5-294 5-355 max. 5-284 5-300 5-216 Mean range 5-355 - 5-143 - 0-212 division. Greatest range in one period 7-5 — 3-5 =^ 4-0 divisions. 10—2 148 DIRECTIVE ACTION OF ONE QUARTZ CRYSTAL ON ANOTHER Table II. 1 Series Azimuth of large sphere Mean readings at phases (whole numbers omitted) Greatest range in a period in Scale-divisions 1 2 3 \ 4 5 6 7 8 9 A 1 o sees. 115 80 •175 •163 •143 •186 ^246 •294 •355 •284 •300 •216 •212 40 A2 115 80 •653 •558 •566 •653 I-813 •950 1030 1-008 •929 •769 •472 31 Bl 180 115 80 •485 •590 •624 •648 -556 •464 •379 -328 •284 •364 •364 30 B2 180 115 70 •423 ■503 •650 •836 -941 •961 •843 •714 •540 •464 •538 75 B3 180 115 54 •650 •632 •619 •648 ^656 •680 •717 •739 785 •739 •166 2^7 CI °0 sees. 230 72 •708 •731 •708 •739 -717 •711 •676 •678 -642 •688 -097 1^3 (72 230 80 •370 •400 •358 •326 •271 •214 •173 •158 •253 •310 -242 3-4 C3 230 80 •616 •654 •686 •673 -663 •627 •571 •560 •566 -584 -126 2^0 D\ 90 230 50 b024 1-042 1-031 b004 -988 •920 •926 •954 -994 1-010 •122 ' 2-2 D2 90 230 70 •031 •090 -150 •210 -220 •230 •223 •176 •126 •096 •199 31 Table III. Series Mean readings at phases Mean 1 2 3 4 ' 5 6 7 8 9 •493 range •338 A •414 -361 -355 -420 -530 -622 ' -693 -646 •615 5 (advanced 180°) •702 •646 ^594 -536 -522 -519 •575 ^631 •711 •663 •718 •199 Means of A and B •558 -503 -474 -478 -526 -570 •634 ^638 •605 ^189 C •565 1 ' •595 -584 -579 -550 -517 •473 ^465 •487 -527 •130 D (advanced 90°) •575 -575 -565 -560 -553 -528 •566 ^592 •607 •604 -079 Means of C and D -570 1 i -585 -575 -570 -552 -523 •520 ^529 •547 •566 -065 i 1 AN EXPERIMENT WITH THE BALANCE TO FIND IF CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT. By J. H. PoYNTiNG and Percy Phillips, M.Sc. [Roy. Soc. Proc. A, 76, 1905, pp. 445-457.] [Received July 12, 1905.] In all the experiments hitherto made to determine the gravitative attrac- tion between two masses, the temperature has not varied more than a few degrees, and there are no results which would enable us to detect with certainty any dependence of attraction upon temperature even if such dependence exists. It is true, as Professor Hicks has pointed out*, that Baily's results for the Mean Density of the Earth, if arranged in the order of the temperature of the apparatus when they were obtained, show a fall in value as the temperature rises. But this is almost certainly some secondary effect, due to errors in the measurements of the apparatus, or to the seasons at which different attracted masses were usedf . The ideal experiment to find if temperature has an effect on gravitation would consist in one determination of the gravitative attraction between two masses at, say 15° C, and another determination at, say, the temperature of boiling liquid air. But the difficulties of exact determination at ordinary temperatures are not yet overcome, and at any very high or very low tempera- tures, they would be so much increased that the research seems at present hopeless. The question can, however, be attacked in a somewhat less direct method by examining whether the weight of a body— the gravitative attraction of the earth upon it — varies when the temperature of the body varies. The various parts of one of the attracting masses — the Earth — remain, each part, at the same temperature throughout, and this is, no doubt, a weak- ness of the method. For it is perhaps conceivable that in the expression for the attraction a temperature factor might exist of some such form as 1 + AC {Mt + mt')l(M + m), where M and m are the two masses, and t and t' are their temperatures. If m/M is negligible, this reduces to 1 + kI, and * Proc. Camb. Phil. Soc, vol. 5, p. 156. t Poynting, Mean Density of the Earth, p. 56. 150 AN EXPERIMENT WITH THE BALANCE TO FIND IF is independent of the temperature t' of the smaller mass. But it seems more hkely that each mass would have a separate temperature factor. If such a factor exists, and if its variation is appreciable, then we ought to be able to detect a change of weight with change of temperature. Observations on pendulums suffice to show that at the most any such effect must be small. The nearly constant period of vibration with the nearly constant length of a compensated or an 'invar' pendulum shows constancy of weight of the bob to a considerable degree of exactness. Again, the agreement of weight-methods and volume-methods of measuring the expansion of liquids with rise of temperature shows, though less conclusively, that there is no great variation. It appeared to us that it would be possible to go much further in testing constancy of weight by a direct weighing experiment, in which the weight on one side of a balance should be subjected to great changes of temperature while the counterpoise should remain at a uniform temperature. We give an account in this paper of a series of experiments carried out on the following principle. A brass cylinder weighing 266 grammes was hung by a wire from one arm of a balance so as to be near the bottom of a tube depending from the floor of the balance-case, the tube being closed at the bottom and opening at the top into the case, the wire passing down through the opening. The brass cylinder was counterpoised by an equal cylinder hung by a short wire from the other arm inside the case. To this short wire was attached a finely divided scale on which the swings of the balance could be read by a microscope looking through a window in the case. The balance was released and left free to swing. Then the case was exhausted till the pressure was not more than a small fraction of a milHmetre of mercury. Steam was passed round the lower part of the tube where the weight hung, and after a time the weight was allowed to cool again. In other experiments the lower part of the tube was cooled by liquid air and again brought up to the temperature of the room. While the changes of temperature were in progress there were considerable apparent variations in weight. But ultimately, when the temperature became steady, the weight, too, became steady. At 100° C. it was slightly less than at the temperature of the room. This difference was partly due to a rise in the temperature of the case, such a rise being always accompanied by an apparent diminution of the weight in the tube, whether steam was apphed or the balance was merely left to follow the temperature of the room. Probably this effect was due to some change in the balance-beam. But the difference was partly due to convection-currents, or at any rate to the residual air in the case, for it varied with the disposition of diaphragms in the tube. There were no doubt convection-currents, as there was always a tendency for the case to rise in temperature when steam was apphed, and this could hardly be accounted for by conduction or radiation, under the conditions of the CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT 151 apparatus. As effects due to residual air should depend upon surface and not upon volume, similar experiments were made with hollow weights, each about 58 grammes, and of the same size and form as the solid weights. There was again an apparent diminution in weight when steam was applied. Any- true diminution due to change of temperature should be shown by a difference in the diminution with the solid and with the hollow weights, the surface effects being eliminated, and this diminution should be that of 266 — 58 = 208 grammes. The net result of all the experiments was that there was not a greater change in 208 grammes between 15° C. and 100° C, than 0-003 milligramme. But an inspection of the detailed account given later shows that this result is probably accidentally small — within the limits of experimental error. It would imply that there is not a change greater than 1 in 6 x 10^ per 1° C. But the experiments hardly justify us in saying more than that there is not a change greater than 1 in 10^ per 1°. When liquid air was used, air-currents were absent, and the temperature variations of the case were much less. The net result of these experiments was that there is not a change of weight in 208 grammes between 16° C. and — 186° C. greater than 0-002 milHgramme. This would imply that there is not a change greater than 1 in 1-3 x 10^^ per 1° change of temperature. We may probably assert that the change is not greater than 1 in 10^^ per 1° C. We now proceed to a detailed account of the apparatus and of the mode of using it. The Balance, The balance has a 6-inch beam and was specially constructed for the experiment by Mr. Oertling. The general arrangement will be seen from Fig. 1. The base-plate is of gun-metal, as are also two sides and the top of the case. The front and back of the case are of thick plate-glass fixed to the metal by marine glue. In the experiments the base-plate was supported on levelling-screws on a slate slab, and between it and the slab was a gas-pipe with pinhole burners so that it could be warmed. When the case was to be fixed in position the jets were lighted and seahng-wax was smeared on to the area of contact of plate and case. When the wax was quite liquid the case was put down on the plate and the gas was turned off. When the metal was cool the joint was perfectly air-tight. The tube T in which the weight W hung was of brass, 4-1 cm. internal diameter and 62-5 cm. long. It consisted of three parts. The topmost was brazed to the base-plate and the two lower parts were attached to it and to each other by flanged joints ff. Between the flanges was placed a circular lead washer of diamond-shaped section. When the flanges were pressed together by bolts the joint was quite air-tight. Round the middle section of 152 AN EXPERIMENT WITH THE BALANCE TO FIND IF the tube was a water-jacket wj through which water flowed while an experi- ment was in progress, and round the lowest section was a steam-jacket sj through which either water or steam could be passed. This jacket could be removed and could be replaced by a vacuum- vessel 30 cm. long containing liquid air. Fig. 1. W, weight of which the temperature is to be raised, Tf counterpoise. T, tube in which it hangs, with a number of diaphragms with |-inch holes. sj, steam-jacket, replaced by hquid-air jacket. //, flanged joint with lead washer. wj, water-jacket. p, pipe to the exhausting pump. sc, scale read by a microscope not shown. rr, rider-rod passing through stuffing-boxes, sb, enlarged in Fig. 2. GG, gas-burners to heat the base-plate before seahng up. The weights W, W were turned from the same gun-metal bar. The length of each was 4-45 cm., the diameter 3 cm., and the solid weights were each 266-17 grammes, while the hollow ones were 57-86 grammes. They were hung directly from the end-plates of the balance by platinum wires, and any residual mequality was compensated by moving a centigramme-rider along CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT 153 the beam by the rider-rod rr. This rod passed through stuffing-boxes sh, designed for us by Mr. G. 0. Harrison, the mechanical assistant in the labora- tory, to whom we are much indebted for this and many other valuable suggestions, and for the careful construction of all the apparatus except the balance. These stuffing-boxes were perfectly air-tight when screwed up and the rod could still be rotated without any leakage. But to draw it in or out it was necessary to loosen the screws slightly, and in one case when this was done some leakage occurred. As the construction appears to give an efficient mode of moving apparatus inside a vacuum from without, we give in Fig. 2 a section of a stuffing-box. Fig. 2. rr, rider-rod. CO, side of case. ivw, two or three circular washers punched out of soft leather and soaked in oil. p, plunger driven in by screws, ss. oh, oil-hole through which valvoline, a thick lubricating oil, was inserted. The position of the balance-beam was read by a microscope viewing a glass scale sc, Fig. 1, interposed in the suspension of W . The scale was divided to 0-1 mm. and numbered in millimetres. The objective of the microscope was placed inside the case and the eye-piece with cross-hairs was fixed outside it. The axis of the microscope was horizontal and a lamp at the back illuminated the scale. The case was surrounded by felt and a tin cover was placed over the whole, small windows through the felt then allowing the scale to be seen. A thermometer placed between the felt and the case was taken to give the temperature of the case. A brass pipe f from the floor of the case led to the pumping apparatus. This pipe was connected to a branched glass tube, one branch going to a Fleuss pump and the other to a 4-fall Sprengel, made continuous in its action 154 AN EXPERIMENT WITH THE BALANCE TO FIND IF by a steel pump which was worked by a motor, and which raised the mercury again from the cistern at the base to the reservoir at the top. When the case was to be exhausted the Fleuss pump was first used and then sealed off and the exhaustion was carried on by the Sprengel. The degree of exhaustion was estimated by sending a discharge through a vacuum-bulb 10 cm. diameter connected with the tube to the pump, and usually the pumping was continued till the negative dark space was of the order of 4 cm. As a rule the vacuum held without serious change for days or even for weeks. Mode of Experiment. A large number of preliminary experiments was made with a pair of brass weights each about 187 grammes. These were only useful in bringing to the front the difficulties in obtaining good results and in suggesting means for overcoming them. We shall only record the final results with the 266 grammes and 58 grammes weights. The weights and the lower section of the tube were first cleaned by boiling in caustic potash solution and washing in distilled water. They were then suspended, being handled with gloves only, and the lowest section of the tube was screwed on. Stea?n- Heating. The jacket sj (Fig. 1) was fixed on the lower section of the tube and the balance was set free to vibrate, being left free during a whole series of experiments. The case was then sealed on and the value of a scale-division was determined by the rider. Any change in the value during a series could be determined from the change in period of the swing. The time of swing in different series ranged from 24 to 42 seconds. After the stuffing-boxes were tightened the case was exhausted till the pressure was estimated to be not more than j^jj mm. of mercury. The weight of air displaced by a weight was then of the order 0-001 milHgramme and the change in this with change of temperature was quite neghgible. Cold water was passed through the water-jacket tvj, and sometimes, while steam was being got up in a boiler at some distance well screened from the balance, through sj also. The centre of swing and the temperature of the case were observed, and before any heating occurred the balance was usually quite steady. Steam was then blown through sj, water still flowing through wj. After considerable changes, which will be described later, the centre of swing in the course of five or six hours settled down to a steady march which appeared to correspond to change in temperature of the case. KSometimes steam was turned off after eight or nine hours, but in some cases it was kept on for 24 and 48 hours and even longer. Then it was turned off and the CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT 155 jacket was allowed to cool. The centre of swing was observed when steady, several hours later or next day. The results in the first few heatings and cooHngs, after an exhaustion of the case, were rejected, as there was evidence that the earlier heatings drove gas from the weight. Only after successive heatings gave fairly consistent values were these taken into account. One effect of the steam-heating was always to raise the temperature of the case, probably through convection of the residual air. A rise of temperature in the case was always accompanied by a lowering of the scale-reading, corresponding to a diminution in weight. The effect was somewhat irregular, but an average value of the lowering per 1° rise was determined by observing the centre of swing of the balance at intervals through several days, when the balance was left to follow the varying temperature of the room and no steam was flowing. The value thus obtained was used to correct all the readings to 15° C. As an example of the method pursued, we give in Table I the series of readings used to obtain the temperature-correction for the hollow weights. Table I. Change of Centre of Swing with Change of Temperature of Case. Date Time Centre of swing in Temperature millimetres of scale of case 6.12.04 1.0 p.m. 14-34 14-75 ,, 3.0 „ 14-355 14-55 ,, 5.40 „ 14-35 14-65 7.12.04 12.35 „ 14-ai 12-70 jj 5.5 „ 14-465 13-60 8.12.04 11.5 a.m. 14-43 14-70 ,, 12.55 p.m. 14-20 14-95 9.12.04 ■ 9.51 a.m. 14-555 12-80 >> 2.27 p.m. 14-525 13-3 10.12.04 9.37 a.m. 14-46 15-0 The temperature-correction deduced from these numbers by the method of least squares is a decrease of 0-13 division per 1° C. rise, and as the sensibility was 1 division for 0-248 milligramme, there was an apparent decrease of weight of 0-032 miUigramme per 1° rise. Two similar series with the soHd weights gave a decrease of 0-044 division per 1° C. rise, and as the sensibility was now 1 division per 0-803 milligramme, there was an apparent decrease of weight of 0-035 milhgramme per 1° rise. Another series with the solid weights when steam was passing all the time for several days, gave a decrease of 0-052 division per 1° rise, but as the 156 AN EXPERIMENT WITH THE BALANCE TO FIND IF values were more irregular, the series giving 0-044 division were used. This series with steam sufficed to show that very nearly the same temperature- correction apphed when the weight was hot as when it was cold. The irregularity of the observations is only to be expected when it is remembered that the balance was subjected to some considerable vibration at times through machinery running in the same building, and that the observations extended over several days. Indeed it is remarkable that there was not more irregularity, and the fair consistency of the observations illus- trates once more the marvellous accuracy of a well-made balance. The following Table II will serve as an example of a complete experiment in which one of the hollow weights was cold initially, was then surrounded with steam for 24 hours, and was then allowed to get cold again. The obser- vations recorded are at about hourly intervals, but intermediate ones, not used, were frequently taken to be sure that there were no sudden changes. Table II. Experiment with Hollow Weight raised to 100° C. and then cooled, 1 mm. = 0-248 milligramme. Correction for temperature of case — 0-13 division per 1°. Condition Centre of Tempera- Centre of Date Time of weight swing, 1 = 1 mm. ture of case corrected to 15° C. Remarks 17.11.04 9.25 a.m. Cold 14-905 14°-75 14-872 Steam put on just after 9.25 ,, 4.0 p.m. Hot 14-40 15 -0 14-400 and kept on till 10 a.m. ,, 5.20 „ ,, 14-39 15 -0 14-390 next day >? 6.12 „ ,, 14-375 15 -0 14-375 ^, 7.8 „ ?j 14-365 15 -05 14-372 ,, 7.56 „ 14-36 15 -1 14-373 18.11.04 9.5 a.m. ,, 14-30 15 -05 14-307 „ 9.55 „ ., 14-295 15 -1 14-308 Steam turned off just after ,, 5.36 p.m. Cold 14-55 16 -00 14-680 9.55 ,, 6.39 „ " 14-56 15-95 14-684 ,, 7.56 „ j» 14-58 I 15 -75 14-678 19.11.04 9.40 a.m. 14-67 14 -30 14-579 1 11.38 „ " 14-665 14-20 14-561 Initial reading, cold at 15° 14-872 divisions Final mean reading, cold at 15° 14-636 „ Mean reading, cold 14-754 „ hot 14-360 Cold-hot 0-394 division The following Table III gives the results of the various experiments with the hollow weight, treated as in Table II, the readings of the centre of swing being at about hourly intervals when on the same day. CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT 157 Table III. Exferiments with Hollow Weight raised to 100° C. and then cooled, 1 mm. = 0-248 milligramme. Correction for temperature of case — 0-044 division per 1°. Date Condi- tion of weight Centre of swing corrected to 15° Number of readings from which centre of swing is found Greatest deviation from the mean Excess of cold above hot Remarks 16.11.04 ... 17.11.04 ... Hot Cold 14-749 14-872 4 1 0-0201 0123 Temperatures 14°-75 to 16°-3 The initial cold reading was rendered useless by a subse- quent shift of scale-reading, probably due to slight dis- placement of the eye- piece 17.11.04 ... 17-18.11.04 18-19.11.04 Cold Hot Cold 14-872 14-360 14-636 1 7 5 0-058 i 0-075J 0-394 Temperatures 14°-2 to 16° The set given in full in Table II. The last of the preceding used as the first of this 21.11.04 ... 22-23.11.04 23.11.04 ... Cold Hot Cold 14-218 14-091 14-294 1 6 3 0-095 i 0-040j 0-165 Temperatures 10° -45 to 14° -8 25.11.04 ... 1 26.11.04 ... Cold Hot Cold 14-421 14-029 14-010 1 2 2 0-001 0016 0-187 Temperatures 10°-95 to 14°-5 1.12.04 ... 2.12.04 '.'.'. Cold Hot Cold 14-500 14100 14-367 1 3 1 0-007 0-334 Temperatures 16° to 16° -9 2.12.04 ... 2-3.12.04 3.12.04 ... Cold Hot Cold 14-367 14-218 14-400 1 3 1 0-048 0-166 Temperatures 16°-9 to 17°-45 12.12.04 ... 13.12.04 ... Cold Hot Cold 14-379 14-106 14-390 1 3 1 0-018 0-279 Temperatures 13°-65 to 15°-75 Mean value cold - hot = 0-235 division = 0-058 milligramme. The following Table IV gives the results with the sohd weight. They are not so consistent as those with the hollow weight, probably because they were spread over a longer time on the average. This was done to secure that the weight should be more nearly at the temperature of its surroundings. A rough estimate shows that if heat be gained by radiation alone and the brass is taken as a full radiator, three hours will be required to bring it within 1° of the temperature of the steam. The last two experiments were incom- plete in that no final cold weighing was taken, but the results obtained were regarded as probably sufficient. 158 AN EXPERIMENT WITH THE BALANCE TO FIND IF Table IV. Exferiments with Solid Weight raised to 100° C. and then cooled, 1 mm. = 0-803 milligramme. Correction for temperature of case — 0-044 division per 1°. Date Condi- tion of weight Centre of swing corrected to 15° Number of readings Greatest from which deviation centre from of swing is ; the mean found j Excess of cold T> 1 above I Remarks hot 1 26.12.04 ... 27-28.12.04 29.12.04 ... Cold Hot Cold 16-295 16-032 16-046 1 2 1 0-017 0-139 Temperatures 9°-9 to ll°-0 30.12.04 ... 30-31.12.04 2.1.05 Cold Hot Cold 16-016 15-914 16-035 1 4 1 0-021 0-112 Temperatures 9°-l to 12°-05 2.1.05 3.1.05 4.1.05 Cold Hot Cold 16-035 16-022 16-047 1 2 1 0-018 0019 Temperatures 9°-l to 12°- 1 5.1.05 >9 Cold Hot 15-987 15-919 1 3 0-002 Temperatures ll°-95 to 12°-65 0-068 Experiment interrupted by stoppage of steam tubes 9.1.05 10.1.05 Cold Hot 16-048 16-039 1 4 0-020 0-009 Temperatures 13°-7 to 15°-65 Heating continued several days after this to obtain tempera- ture-correction. No final cold reading taken Mean value cold -hot = 0-069 division = 0-055 milligramme. From Tables III and IV we have : Solid weight, 266 grammes: cold — hot ... = 0-055 milligramme. Hollow weight, 58 grammes: cold — hot ... = 0-058 ,, For the difference, 208 grammes : hot — cold = 0-003 ,, Taking the rise in temperature as 85°, this gives a change of the order of 1 in 6 X 10^ per 1° rise. But evidently the smallness of the result is accidental, and probably all we can assert from the work is that any change of weight with change of temperature between 15° C. and 100° C. is not greater than 1 in 10^. Cooling with Liquid Air. Experiments were made in which heating by steam was replaced by coohng with liquid air. This was supphed to us by Sir Wilham Ramsay, and we desire to express our hearty thanks to him for his ready kindness in helping us to increase the temperature range so considerably. In these experiments the steam-jacket was removed and replaced by a vacuum- vessel 30 cm. deep and 6 cm. inside diameter, kept full of Hquid air. After the CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT 159 steady state was reached the hquid air was removed, the jacket was replaced and cold water was again passed round the tube. Owing to the evaporation of the air the experiments had to be carried out more rapidly than those with steam, but through the absence of convection- currents a steady state was more rapidly reached, and the variation in the temperature of the case was very small. The temperature-correction was not observed, but as in the subsequent observations with both sohd and hollow weights it was found to be about 0-03 milHgramme per 1°, this value was assumed to hold here. In any case its effect is very small, as the temperature varied so little. The centre of swing was observed nearly continuously from the time when the hquid air was apphed and again after it was removed. After a time in each case it became steady and only these steady values are recorded in the following Tables. Table V. Experiment with Solid Weight cooled by Liquid Air, 1 mm. = 0'315 milligramme. Correction for temperature of case — 0-1 division per 1°. Date Time Condition of weight Centre of swing Tempera ture of case Centre of swing corrected to 16°-6 Remarks 28.7.04 3.25 p.m. 5.50 „ 6.0 „ 6.10 „ 8.45 „ 9.15 „ Normal Cold Normal 11-085 11-07 11-07 11-07 11-095 11-095 16°-6 16 -65^ 16 -65 \ 16 •65j 16 -4 \ 16 -4 j 11-085 11-075 11-075 Liquid air applied just after 3.25 Liquid air removed and water applied just after 6.10 Normal -cold = 0-005 division = 0-0016 milligramme. Table VI. Experiment with Hollow Weight cooled by Liquid Air, 1 mm. = 0-343 milligramme. Correction for temperature of case — 0-1 division per 1°. Date Time Condition of weight Centre of swing Tempera- ture of case Centre of swing corrected to 16°-6 Remarks 9.9.04 9.40 a.m. 11.40 „ 11.50 „ 5.25 p.m. Normal Cold Normal 14-485 14-480 14-480 14-480 16°-3 16 -4 1 16 -4 / 16 -4 14-455 14-460 14-460 Liquid air applied at 9.43 Removed at 11.52 Steady The balance next morning read 14-48 at 16°-3 Normal - cold = - 0-002 division = - 0-0007 milligramme. 160 AN EXPERIMENT WITH THE BALANCE TO FIND IF From Tables V and VI we have : Solid weight, 266 grammes: normal - cold ... = 0-0016 milhgramme. Hollow weight, 58 grammes: normal — cold... = — 0-0007 „ For the difference, 208 grammes : normal — cold = 0-002 „ Taking the fall in temperature as 200°, this gives a change of the order of 1 in 2 X 1010 per 1° fall. These hquid-air experiments were not repeated. But the conditions are probably much less disturbed than with the steam experiments, and we may safely say that if there is any change of weight with change of temperature between 16°-6 C. and - 186° C, it is not so great as 1 in lO^^ per 1° C. Note on the Change of Af parent Weight on First Heating or Cooling. We have mentioned that while the changes in the temperature of the weight were in progress there were considerable apparent variations in weif^ht. These, in a few cases, amounted to as much as 0-6 milligramme. They were almost certainly due to radiometric forces or to other gas-action, for they were very dependent on the disposition of the diaphragms in the tube T (Fig. 1), and also on the way in which the steam was blown through the jacket. In the preliminary experiments with solid weights the lowest diaphragm was 5 to 6 inches above the weight, and the steam was blown into the top of the jacket. Under these circumstances the following variations occurred when the steam was turned on : At first the weight apparently increased, until in 15 to 20 minutes it reached a maximum, which was in some cases as much as 0-6 milligramme above the real weight. After reaching this maximum the weight apparently decreased, till in four hours it had reached a nearly steady value, which was a little less than the value at the temperature of the laboratory. If, now, the jacket was filled with cold water, the apparent weight first increased for about one minute and then decreased for about two hours to a minimum, which was a little lower than the final weight at 100°. After this the weight very slowly increased, till in five to six hours it had recovered the value which it had before the experiment. These changes did not vary very much with the pressure, but at lower pressures they took place more rapidly than at higher ones. On coohng the weight with liquid air, changes occurred exactly similar to those which occurred when the weight was cooled from 100° to the temperature of the laboratory ; and when the weight was warmed up from the temperature of liquid air to the temperature of the laboratory, the changes were similar to those when the weight was warmed from the temperature of the laboratory to 100° C. CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT 161 So long as the arrangement of the diaphragms and the weight remained the same, and so long as the steam was blown through in the same way, bhese changes were exactly similai:, but as soon as any alteration was made in these arrangements the character of the changes altered. In one series of experiments a sealed glass bulb containing mercury was used in place of the brass weight. In this case, immediately after the steam was turned on there was a rapid decrease in weight, and a minimum was reached in less than one minute. After this the changes were very similar to those occurring with the brass weight. On cooling, however, the changes were almost exactly the reverse of the changes on heating, and were not at all like the changes with the brass weight. In the final experiments, those recorded, the lowest diaphragm was within J-inch of the top of the brass weight. With this arrangement and with the steam blown into the top of the jacket, the following changes occurred : The apparent weight first increased rapidly, reaching a maximum in about one minute, then it rapidly decreased, reaching a minimum in about four minutes, and again increased to another and lower maximum in 8 to 10 minutes. After this it slowly decreased to a nearly steady value a little lower than the original value. On passing cold water through the jacket the apparent weight rapidly increased for about one minute, and then slowly decreased to its original value. Still another variation was arranged by blowing the steam in at the bottom of the jacket instead of at the top, all the other things remaining as in the last experiment. In this case, on turning on the steam, the apparent weight first decreased to a minimum in about one minute, then increased to a maximum in about six minutes, and finally decreased slowly to a nearly steady value a little below the original value. The cooling and the consequent changes were exactly similar to those in the last experiment. It is somewhat difficult to follow out exactly the changes which w^ould be caused by radiometer-action and by convection-currents in these different arrangements of the apparatus, but the fact that these changes depend entirely on the arrangement is sufficient evidence that they are caused by gas-action, and, as we have before said, we have some reason to believe that even the small final difference is due to air-currents. p. c w. 11 6. ON A METHOD OF DETERMINING THE SENSIBILITY OF A BALANCE. By J. H. PoYNTiNG and G. W. Todd, M.Sc. [Phil. Mag. 18, 1909, pp. 132-135.] [Read June 25, 1909.] In the method, as we have arranged it, a small frame (Fig. 1, end-view) is fixed at the centre of the beam of a 16-inch Oerthng balance. This carries two Vs about 2 cm. apart, and in the Vs hes a straight wire or fibre about 3J cm. long, parallel to the beam and level with the central knife-edge. This wire takes the place of the ordinary rider, and we shall call it 'the rider.' Its weight is determined before use as accurately as possible by weighing on an assay-balance. The sensibility is determined by moving the rider either to right or left a measured distance. If this distance is d, if the half-length of beam is b, and if the weight of the rider is R, the movement is equivalent to an addition of weight to one pan, Rd/b. In order to move the rider a definite distance a stout horizontal rod (Fig. 2) passes through the balance-case from side to side without contact with the case, and is supported at its ends outside, and independent of, the case. It is parallel to the beam and a little lower than the V frame. On the rod are fixed horizontally two Brown and Sharp micrometer-screws divided to 0-01 mm. and allowing an estimate of 0-001 mm. Their axes are in one fine coinciding with the axis of the rider, and they are fixed so that one can bear against one end and the other against the other end of the rider. Their ends are plane and the ends of the rider are bluntly pointed. Each micrometer-screw head has a cross-piece fixed on it ; and a fork, which can be rotated about an axis in the continuation of the axis of the screw by a pulley outside the case, can engage with the cross-piece and so advance or withdraw the screw. The pulley is worked by an endless string passing to a pulley at the side of the observer, who is about 2 metres in front of the balance. The micrometer-divisions are illuminated and each micrometer is viewed by its own telescope. The position of the balance-beam is read by a double- suspension mirror, telescope and scale. The scale is divided to miUimetres ON A METHOD OF DETERMINING THE SENSIBILITY OF A BALANCE 163 and is about 3 metres from the mirror. The double-suspension mirror is fully described in the Phil. Trans. A, vol. 182, p. 572*. It is of course not essential to the method, but was chosen because of the great magnification of the deflection which it gives. Let us suppose that the value of the scale-divisions of the deflection is to be determined by a movement of the rider from right to left. The two micrometer-screws are withdrawn so that neither is in contact with the rider, that on the left so far that the rider will not touch it in its subsequent travel. The beam is lowered and allowed to swing. Then the right-hand screw is advanced till it bears against the end of the rider and pushes it some small distance. The contact is seen to have occurred by the interference with freedom of swing, as watched in the telescope. The micrometer is then y/ qULJi> " I Fig. I. End-view of V frame fixed to balance-beam. Fig. 2. Arrangement of right-hand micrometer-screw. read. Let its reading be m^. It is then withdrawn a httle so as to leave the rider free, and the centre of swing C-^ is determined in the usual way from three successive turning-points. Then the micrometer is advanced again so as to push the rod a little further, and its reading m^ is taken. It is then withdrawn and the new centre of swing C^ is taken. If m^ — m^ = d, Ci — C2 divisions of deflection are due to an addition of Rdjh to the left pan. The right-hand micrometer may then be withdrawn and the left-hand micrometer may be brought into action in a similar manner, and so on, the two screws being used alternately. The balance-case was fixed on a shelf and was enclosed in a tin-foiled wooden box with wool loosely packed between box and case. The case and * [Collected Papers, Art. 3.] 11—2 164 ON A METHOD OF DETERMINING THE SENSIBILITY OF A BALANCE box were provided with plate-glass windows to view the mirror and the micrometer-divisions. The following abstract of some determinations of sensibihty will serve to show what accuracy may be attained: I. Rider German silver wire, 7-35 mgm. Half-length of beam, 20-272 cm. 10 determinations alternately left and right. Mean travel of rider ... ... 24850 mm. Mean deflection ... ... ... 21-26 divisions. Mean value for 20 divisions . . . 0-0848 mgm. The separate determinations range between 20 divisions - 0-0877 and 20 „ = 0-0824. II. The same rider. 10 determinations alternately left and right. Mean travel of rider ... ... 5-2713 mm. Mean deflection 45-47 divisions. Mean value for 40 divisions ... 0-1681 mgm. The separate determinations range between 40 divisions =-- 0-1722 and 40 „ = 0-1632. III. Rider German silver wire, 189-05 mgm. 7 determinations alternately left and right. Mean travel of rider 0-1764 mm. Mean deflection 38-70 divisions. Mean value for 40 divisions ... 0-1691 mgm. The separate determinations range between 40 divisions = 0-1709 and 40 „ = 0-1654. IV. The same rider. 7 determinations alternately left and right. Mean travel of rider 0-3004 mm. Mean deflection 64-84 divisions. Mean value for 60 divisions ... 0-2578 mgm. The separate determinations range between 60 divisions = 0-2612 and 60 „ = 0-2518. PAET II. ELECTRICITY. ON THE LAW OF FOKCE WHEN A THIN, HOMOGENEOUS, SPHEKICAL SHELL EXEKTS NO ATTRACTION ON A PARTICLE WITHIN IT. [Manchester Lit. Phil. Soc. Proc. 16, 1877, pp. 168-17L] [Read March 6, 1877.] If a homogeneous, thin, spherical shell of uniform thickness exert no attraction on a particle within it, then the law of the force is the law of nature. Professor Maxwell uses this proposition {Electricity, vol. 1, § 74) to deduce the law of the force between electrified bodies, and shows that it proves, far more conclusively than any direct measurements of electrical forces, that the law is that of the inverse square. It would therefore be an advantage to have a simpler proof of such an important proposition than that given by Laplace {Mec. Celeste, liv. ii, ch. 2) and followed by Maxwell. The following seems more simple, as it requires neither integration nor the solution of a functional equation : Let P be any point inside the spherical shell, C the centre of the sphere, DPCE the diameter through P, and APB perpendicular to CD. In Newton's proof of the proposition that, if the law of attraction be that of the inverse square, the force at P is zero, the surface is divided into an indefinitely great number of opposite elements by small cones having their vertices at P, and the attractions of each of these pairs of elements are shown to balance each other. We shall first show Fig. 1. that if the attraction at P is zero, then it follows inversely that, for at least 166 ON THE LAW OF FORCE WHEN A THIN, HOMOGENEOUS, SPHERICAL one position (if not for all positions) of the cone MPm besides the position APB, the attractions of the opposite elements balance each other ; and we shall thence prove that the law of attraction must be that of the inverse square. Let us suppose the cone, with vertex at P, to move round from the position where AB is its axis to any other position MPm. At AB the attractions of the opposite sections on P are e({ual, whatever the law of the force. As the cone leaves AB let us suppose the resultant attraction of the two opposite elements to be no longer zero, but to act, say, towards the centre side of APB. Then it will either continue towards that side as the cone moves all the way round from APB to BPA, or it will vanish at some position, and then act in the opposite direction. In the first case we should have a number of forces all acting from P towards the same side of APB, whose resultant is zero ; then each separate force must be zero. In the second case the resultant attraction of the opposite sections vanishes somewhere between AP and EP\ then for at least one position of the cone, besides the position APB, the resultant attraction of the opposite sections vanishes. Since this is true for any position of P, we can show that the law of the force must be that of the inverse square. In the position where the two opposite sections exert equal attractions, two sections of the same thickness perpendicular to the axis of the cone would also exert equal attractions ; for they would bear to each other the same ratio as the two oblique sections made by the sphere, since these two obhque sections make equal angles with the axis of the cone. Then what we have proved is, that for every position of P there are two different distances for which the attractions of the sections of a small cone of equal thickness on a point at its vertex are equal. N Fig. 2. Let us take VMX to represent the axis of a cone of very small angle of which F is the vertex. At any point M draw an ordinate MN to represent the attraction of a section of the cone at M of small given thickness on a point at the vertex. Then N will trace out a curve as M moves along VX. Now take a spherical shell of thickness equal to the thickness of the sections of the cone, and of radius nearly equal to VM, where M is any arbitrary point in VX. Take a point near the centre of this sphere. As a cone moves round with this as vertex, its sections by the sphere must be always at distances very nearly equal to the radius from the vertex; SHELL EXERTS NO ATTRACTION ON A PARTICLE WITHIN IT 167 and, by what we have proved above, for some position of the cone the attractions of the opposite sections must be equal. Therefore (in Fig. 2), for two distances very nearly equal to VM the ordinates must be equal to one another. Then the tangent to the curve near N must be parallel to VX. But M is arbitrary, for we can take the sphere of any size. Therefore at all points the tangent to the curve is parallel to VX ; and therefore the curve must be a straight line parallel to VX; or, the attractions by sections of the cone of equal thickness are constant, wherever the sections be taken. But the sections are proportional to the direct square of the distance; and therefore the law of the attraction must be that of the inverse square of the distance. 8. ARRANGEMENT OF A TANGENT GALVANOMETER FOR LECTURE ROOM PURPOSES TO ILLUSTRATE THE LAWS OF THE ACTION OF CURRENTS ON MAGNETS, AND OF THE RESIS- TANCE OF WIRES. [Manchester Lit. Phil Soc, Proc. 18, 1879, pp. 85-88.] [Read April 1, 1879.] Three coils of similar wire are arranged round the circumference of a circle with a compass-needle at the centre, each wire going only once round the circle. The six ends of the three wires are connected by thick copper w'ltes Avith the six binding screws A, B, C, D, E, F. On a concentric circle of twice the radius is arranged a coil of the same wire going twice round the circle and having its ends connected by thick wires with the binding screws G, H. On the sam.e circle is a single wire of twice the diameter, making only one turn round the circle, and having its ends connected with the binding screws K, L. The coils are denoted respectively by the numbers 1-5. The null method is adopted in each case. That is two forces acting on the needle and arising from different arrangements of the circuit are shown to balance each other, and from the arrangements necessary to produce this equihbrium the desired laws are deduced. I. If the current be reversed the force is reversed. Introduce the current at A. Join BD and lead away at C. Then the same current goes in opposite B A GALVANOMETER TO ILLUSTRATE ACTION OF CURRENTS ON MAGNETS 169 directions round the coils (1) and (2), and since the needle is not deflected the reversal of the force when the current is reversed is proved. II. The force is frofortional to the length of current acting. This is proved by the last, for the two coils (1) and (2) exert equal forces on the needle. If the current went round them in the same direction we should have twice the force which each exerted singly, with twice the length of current. This assumes that the current in different parts of the circuit is the same, which might be shown by sHghtly modifying I, thus : introduce between B and B various resistances and the equilibrium is not disturbed. III. The force is proportional to the strength of the current. Introduce at A, and connect A with C, and B with D; and connect D with F, and lead off from E. There will be two equal currents in the same directions in coils (1) and (2), for they are exactly similar to each other and similarly situated. These two currents unite to give a double current in the opposite direction in coil (3), and the double current in a single wire exerts twice the force exerted by each single current, since there is no deflection. IV. The force is inversely proportional to the square of the distance. Intro- duce at A, connect B with H, and lead off at G. Then since we have two turns to the coil (4) we have a current of four times the length at twice the distance acting in opposition to the same current through (1). Since there is no deflection, the two exert equal forces, and therefore the force is inversely proportional to the square of the distance. Resistance. I. The resistance is proportional to the length. Connect B with C, Z) with E, A with F. Introduce at A and lead off at E. We have then a divided circuit joining A and E, one branch consisting of the two coils (1) and (2), and the other the coil (3) only one-half the length and going in the opposite direction. But as there is no deflection the current through the first circuit of twice the length must be only one-half that through the other to exert an equal force, i.e., the resistance is doubled when the length is doubled. II. The resistance is inversely proportional to the cross-section. Introduce at A, connect A with L, connect B with C and D with K, and lead off at A'. Then we have two circuits connecting A with K, the first consisting of the two coils (1) and (2), the second of the coil (5) of the same length but of four times the cross-section and going round in the opposite direction. Since the needle is not deflected the two currents exert equal forces. But the coil (5) is at twice the distance and must therefore have four times as great a current through it as that through (1) and (2). That is, with equal lengths of wire of the same material connecting two points the currents conveyed are pro- portional to the cross-sections. 9. ON THE GKADUATION OF THE SONOMETER. [Phil Mag. 9, 1880, pp. 59-64.] [Read before the Physical Society, December 13, 1879.] It seems likely that such valuable results will be obtained by means of Professor Hughes's sonometer, that it is desirable that some method should be employed to turn its at present arbitrary readings into absolute measure, so that, for instance, the induced currents caused by different metals in the induction-balance may be measured and compared with each other. In Maxwell's Electricity, vol. 2, chap, xiv, the general formula is given for the coefficient of induction of one circular circuit on another. Adapting this to the case where two equal circular circuits are on the same axis at a distance apart greater than the radius of the coils, the following formula is obtained. Let a = distance between centres, h = radius of either circle, c = distance of either circumference from centre of other, M = coefficient of induction. a^ (2 2a^^l6a^ 32 a^^ 256 a^ etc.| . ...^^j Of these the latter uses directly the distance between the centres, the observed quantity — but is not nearly so convergent as the former, in which c may be at once deduced from c = Va^^b^ To obtain formulae which might be strictly applied to the sonometer, we should have to consider the more general case of two coils of unequal radii 6 and ^, for which I have found the formula corresponding to (2), viz. ^-^ _ 47762^2 ji ^ 3 52^_|_ ^2 15 54 ^ 352^2 _!_ ^4 + 16 a4 _35 6e+66_^^66^^_^^^^^ (3) ON THE GRADUATION OF THE SONOMETER 171 We should then have to take the finite integrals of each term between the limiting values of b and ^. But this would be exceedingly complicated and would require a knowledge of all the details of construction; and we may at least get a first approximation to the true result by replacing the coils by a single one of a radius intermediate between the greatest and least radii. In Prof. Hughes's paper {Phil. Mag. July 1879) he gives the internal and external radii of his coils as 15 miUims. and 27-5 millims. respectively. I have considered, then, that 25 millims. will give results not very far from the truth ; and as it makes the calculations considerably easier, I have taken that as the value of b and applied the formulae to the numbers given in the paper. The resultant current in the middle coil was zero when it was distant 47 millims. from one end and 200 from the other. This enables us to find the ratio between the number of turns in the two ends at least sufficiently nearly to apply to some of the results. Let M^ be the coefficient of induction of the larger coil on the moveable one, ilf 2 that of the smaller, the former having m turns, the latter n. When the moveable coil was 200 millims. from the large and 47 millims. from the small coil, since there was no induced current, mMi = nM^ . Applying formula (1), we have for the larger coil c = V2002+ 252 = 201-5, and for the smaller coil c = \/472+ 252 = 53-2, b being the same for both. Then _jri__ jl _ 3 / 25 y 15 / 25 y _ (201-5)3 [2 4 1201-5/ ^ 8 V201-5y ~ ^*^' 1 3/25 \2 15 / 25 \4 35 / 25 \6 2835 / 25 (53-2)3 (2 4 / 25 \2 15 / 25 y 35 / 25 \« 2835 / 25 y 153-2/ + 8 V53-2J 8 V53-2/ "^ 256^ 153-2; Multiplying each side by 2 and finding the successive terms, 122 m X ^3 {1 - -02308 + -00088 - etc.} = nx ^ {1 - -33123 + -18286 - -09422 + -02633 - etc.}, ^ = 43-6. n I have applied the formula to the results for various metals given by Prof. Hughes in a table in his paper. In the table below, in the second column are Prof. Hughes's numbers, i.e. distances from the point of no induction. In the third are numbers proportional to mM^ — nM^ ; where M^ , M2, are the coeiB&cients of induction of two simple coils calculated on the above 172 ON" THE GRADUATION^ OF THE SONOMETER hypothesis, m and n the number of turns in the two respectively. In the fourth column are the resistances for bars of the metal 100 miUims. long and 1 millim. in diameter (Jenkin, p. 249). In the last column are the products of the numbers in the two preceding columns. Metal Distance from point of no induction 7nMj^ - nM., , proportional to E (mif 1 - nM^) R Silver 125 178 •21 37-4 Gold 117 135 -27 36-5 Aluminium 112 116 •375 43-5 Copper 100 84 -21 17-6 Zinc 80 501 •72 36-1 Tin 74 44-6 1-70 75-8 Iron ... 45 22-46 1-25 281 Lead 38 18-87 2-5 47-2 Antimony 35 17-35 4-5 78-1 Bismuth 10 5-75 16-8 96-6 Mercury has been omitted, as it gives a very much higher value than any of the others. Were the induced currents in the induction-balance pro- portional to the resistances given in the table, the numbers in the last column would of course be all the same. The deviations from equality are far greater than could be accounted for by errors in the approximations I have adopted, especially for the metals not at the beginning or end of the list. Hence we are driven to conclude, either that the resistances of the metals given in the tables are not the same as the resistances of the metals used by Prof. Hughes, or that the induced current is not proportional to the conductivity of the metal. It should be noticed that the method of measuring currents by the sonometer assumes that the telephone integrates, as it were, the current; i.e. the loudness of the sound depends only on the total current, not on the time during which the current is passing, provided that the time be very short. I do not know whether this point has been investigated ; but if not, it would probably be easy to examine it by means of the sonometer. It would be advisable to modify the instrument in such a way that the formulae might be more easily employed, and that the approximations might be nearer to the truth. The formulae used in this paper may be obtained as follows, the method being adapted from that given by Maxwell. The potential of a circular unit current at any point is the same as that of a magnetic shell of unit strength bounded by the circuit. This, again, is the same as the attraction of a thin plate of matter of unit surface-density ON THE GRADUATION OF THE SONOMETER 173 in a direction perpendicular to the plane of the plate. If co be the attraction of a plate of radius b, at a point distant c from the plate along its axis, = 277 Cv'-i) ^ (162 1.3 6* 1.3.5 6« ) = ^" 12 ^-271 0-^ + 27176 c«-"*"t If we introduce zonal harmonics as coefficients, this becomes ^ fl 62 1 . 3 6* ^ 1 . 3 . 5 6« ^ " = ^" 12 ^^^"2-74 0^^3+ 27476 -B^5-etc. This is now the potential at any point in space where 6 < c. If there be a second circular circuit of radius j8 on the same axis, we may suppose it replaced by a magnetic shell bounded by the current and lying on the sphere, with centre at the centre of the first current, the radius of the sphere being c. This shell may be considered to consist of two layers of matter of equal and opposite densities, fi and — /x, at distances c and c + dc from the centre. The potential on the second layer is II' II fJLO) dS, where the integration is taken over the shell. The potential on the first layer is jLt f CO + -,- dc) dS, the sum being — j I [jl -^ dcdS ; but since the strength = 1, fjidc = 1, and we have the mutual potential Replacing the element dS by cH^zdcj), the Umits will be for ^ from to 277, and for jjl from 1 to fx. Integrating with respect to (f>, and remembering that c is constant in integrating for jjl, we have But we have the relation for zonal harmonics, 174 ON THE GRADUATION OF THE SONOMETER Substituting, we obtain The following are the values for the coefficients (Ferrers's Spherical Har- monies : p. 23), both in terms of /jl and when we substitute /x^ = 1 — ^ : djji dl^- 4 „,._.,. l(4_4:). dp, 'dl^- .i?|2V'-lV+l)-f(8-28f + ='«'.)^ dp, 3003/x« - 3465^4 + 945^^ _ 35 ^^^ " ^^^4 ^ + etc. di. 16 16 dPs 109395/x8 - 180180/x6 + 90090^* - 13860/x2 + 315 d^ 128 5760 + etc. 128 Substituting these values and putting c^ = a^ + ^^^ ^g obtain , 62/3M1 3 62 + R2 15 6*+ 362^2 _^ ^4 M = — 477 — ^ - — I - - - - - a3 ^^2 4 a2 -+-16 ^4 _ 35 66jf664^2 _!_ 652^4 _|_ ^6 I 32 a« +etc.|. The more useful form is obtained by retaining c. If we take the two circles of equal radius (i.e. 6 = ^), we obtain ,. , o &M1 3 62 15 64 35 ¥ 2835 6^ ] ^^-^""c3]2-4c2 + ¥c4-¥c«+^5^cS-^'4' 10. ON THE TRANSFER OP ENERGY IN THE ELECTROMAGNETIC FIELD. [Phil. Trans. 175, 1884, pp. 343-361.] [Received December 17, 1883. Read January 10, 1884.] A space containing electric currents may be regarded as a field where energy is transformed at certain points into the electric and magnetic kinds by means of batteries, dynamos, thermoelectric actions, and so on, while in other parts of the field this energy is again transformed into heat, work done by electromagnetic forces, or any form of energy yielded by currents. Formerly a current was regarded as something travelling along a conductor, attention being chiefly directed to the conductor, and the energy which appeared at any part of the circuit, if considered at all, was supposed to be conveyed thither through the conductor by the current. But the existence of induced currents and of electromagnetic actions at a distance from a primary circuit from which they draw their energy has led us, under the guidance of Faraday and Maxwell, to look upon the medium surrounding the conductor as playing a very important part in the development of the phenomena. If we believe in the continuity of the motion of energy, that is, if we believe that when it disappears at one point and reappears at another it must have passed through the intervening space, we are forced to conclude that the surrounding medium contains at least a part of the energy, and that it is capable of transferring it from point to point. Upon this basis Maxwell has investigated what energy is contained in the medium, and he has given expressions which assign to each part of the field a quantity of energy depending on the electromotive and magnetic intensities and on the nature of the matter at that part in regard to its specific inductive capacity and magnetic permeabihty. These expressions account, as far as we know, for the whole energy. According to Maxwell's theory, currents consist essentially in a certain distribution of energy in and around a con- ductor, accompanied by transformation and consequent movement of energy through the field. Starting with Maxwell's theory, we are naturally led to consider the problem : How does the energy about an electric current pass from point to 176 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD point — that is, by what paths and according to what law does it travel from the part of the circuit where it is first recognisable as electric and magnetic to the parts where it is changed into heat or other forms ? The aim of this paper is to prove that there is a general law for the transfer of energy, according to which it moves at any point perpendicularly to the plane containing the Hnes of electric force and magnetic force, and that the amount crossing unit of area per second of this plane is equal to the product of the intensities of the two forces, multiphed by the sine of the angle between them, divided by 477 ; while the direction of flow of energy is that in which a right-handed screw would move if turned round from the positive direction of the electromotive to the positive direction of the magnetic intensity. After the investigation of the general law several appHcations will be given to show how the energy moves in the neighbourhood of various current- bearing circuits. The following is a general account of the method by which the law is obtained. If we denote the electromotive intensity at a point (that is, the force per unit of positive electrification which would act upon a small charged body placed at the point) by @, and the specific inductive capacity of the medium at that point by K, the magnetic intensity (that is, the force per unit pole which would act on a small north-seeking pole placed at the point) by ^ and the magnetic permeability by yi, Maxwell's expression for the electric and magnetic energies per unit volume of the field is K^^I^TT + ilSy-I^TT (1) If any change is going on in the supply or distribution of energy the change in this quantity per second will be /.ef/4. + ,^f/4. (2) According to Maxwell the true electric current is in general made up of two parts, one the conduction-current ^T, and the other due to change of electric displacement in the dielectric, this latter being called the displace- ment-current. Now, the displacement is proportional to the electromotive intensity, and is represented by /i(S/47r, so that when change of displacement takes place, due to change in the electromotive intensity, the rate of change, that is, the displacement-current, is K~ Utt, and this is equal to the difference between the true current (5 and the conduction-current >T. Multiplying this difierence by the electromotive intensity ik the first term in (2) becomes — ^^=m-^.^ (3) ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 177 The first term of the right side of (3) may be transformed by substituting for the components of the total current their values in terms of the com- ponents of the magnetic intensity, while the second term, the product of the conduction-current and the electromotive intensity, by Ohm's law, which states that ^ = 6'@, becomes ^^/C, where C is the specific conductivity. But this is the energy appearing as heat in the circuit per unit volume according to Joule's law. If we sum up the quantity in (3) thus transformed, for the whole space within a closed surface, the integral of the first term can be integrated by parts, and we find that it consists of two terms — one an expression depending on the surface alone to which each part of the surface contributes a share depending on the values of the electromotive and magnetic intensities at that part, the other term being the change per second in the magnetic energy (that is, the second term of (2)) with a negative sign. The integral of the second term of (3) is the total amount of heat developed in the con- ductors within the surface per second. We have then the following result. The change per second in the electric energy within a surface = (a quantity depending on the surface) — (the change per second in the magnetic energy) — (the heat developed in the circuit). Or rearranging : The change per second in the sum of the electric and magnetic energies within a surface together with the heat developed by currents is equal to a quantity to which each element of the surface contributes a share depending on the values of the electric and magnetic intensities at the element. That is, the total change in the energy is accounted for by supposing that the energy passes in through the surface according to the law given by this expression. On interpreting the expression it is found that it implies that the energy flows as stated before, that is, perpendicularly to the plane containing the lines of electric and magnetic force, that the amount crossing unit area per second of this plane is equal to the product electromotive intensity x magnetic intensity x sine included angle while the direction of flow is given by the three quantities, electromotive intensity, magnetic intensity, flow of energy, being in right-handed order. It follows at once that the energy flows perpendicularly to the fines of electric force, and so along the equipotential surfaces where these exist. It also flows perpendicularly to the lines of magnetic force, and so along the magnetic equipotential surfaces where these exist. If both sets of surfaces exist their lines of intersection are the lines of flow of energy. The following is the full mathematical proof of the law : V. a. w. 12 178 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD The energy of the field may be expressed in the form (Maxwell's Electricity, vol. 2, 2nd ed., p. 253) 1 1 j(P/+ Qg + Ith) dxdydz + g- | U(aa + 6^ + cy) dxdydz, the first term the electrostatic, the second the electromagnetic energy. But since f = — P, with corresponding values for g and h, and a = fia, h = /x^, c ^ /xy, substituting, the energy becomes ^ \\\{P^ + Q^ + i?2) ^:r%6?0 + £- [J|(a2 + ^^ + ^2) dxdydz. ...(1) Let us consider the space within any fixed closed surface. The energy within this surface will be found by taking the triple integrals throughout the space. If any changes are taking place the rate of increase of energy of the electric and magnetic kinds per second is Now Maxwell's equations for the components of the true current are , df dg dh where f, q, r are components of the conduction-current. But we may substitute for ^ its value y- , and so for the other two, and we obtain KdP_ 477 dt ~ K dQ 4^dt^'-^ K dR in dt Taking the first term in (2) and substituting from (3) we obtain ,(3) 477 = i i |{^ i^* -~P)^Q{v-q) + R {w - r)} dxdydz = \\\(Pu + Qv + Rw) dxdydz - 1 1 j(P^ + §^ + Er) dxdydz. ...(4) Now the equations for the components of electromotive force are (Maxwell vol. 2, p. 222) : ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 179 Q = az-cx-^^ -^r = az-cx + Q'^, (5) „ , . . dH dih J . . -,,\ R = hx — ay — -J- — -^ = bx — ay + R ] where P', Q', R' are put for the parts of P, Q, Ji which do not contain the velocities. Then Pu + Qv + Rw = (cij — h'z) u -h {az — ex) v + (bx — ay) iv + P'u + Q'v + R' w = — {{vc — wb) X + {wa — uc)y -{- (ub — va) z) + P'u + Q'v + R'w = -(Xx+ Yy + Z'z) + P'u + Q'v + R'w, where X, Y, Z are the components of the electromagnetic force per unit of volume (Maxwell, vol. 2, p. 227). Now substituting in (4) and putting for u, v, w their values in terms of the magnetic force (Maxwell, vol. 2, p. 233) and transposing we obtain + I \j{{Xx +Yy + Z'z) + [Pv + Qq + Rr)} dxdydz = \\\(P'u + Q'v + R'w) dxdydz (Integrating each term by parts) = ^ \\(R'^ - Q'y) dydz + ^ [[(P^ - R'a) dzdx + ^ \\(Q'a - P'^) dxdy I [(({^dR' dQ' ^ dP' dR' ^ dQ' r,dP'\. . . (The double integral being taken over the surface) 1 ^^ ^{l (R'p - Q'y) + m (P'y - R'a) + n (Q'a - P'^)} dS - Mk S - f ) - * (f - f ) - r (f - SI "'"■■■■^" where 2, m, n are the direction- cosines of the normal to the surface outwards. 12—2 180 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD But from the values of P' , Q\ R' in (5) we see that dQ' dz dR' d^G d^ dm ^ d^ dy ~ dtdz dxdz dtdy dzdx d /dH dG\ dt \ dy dz ) "Tt^^Tt (^^^^^11' ^^^- 2> P- 216) dR' dP' db dp dx dz dt ^ dt' dP' dQ' _dc _ dy dy dx dt ^ dt' similarly Whence the triple integral in (6) becomes Transposing it to the other side we obtain (Xx + Yy + Zz) dxdydz + I \(Pf + Qq+ Rr) dxdydz = ~ j |{/ (R'P - Q'y) + m {P'y - R'a) + n (Q'a - P'^)} dS. ...(7) 477.' .' The first two terms of this express the gain per second in electric and magnetic energies as in (2). The third term expresses the work done per second by the electromagnetic forces, that is, the energy transformed by the motion of the matter in which currents exist. The fourth term expresses the energy transformed by the conductor into heat, chemical energy, and so on ; for P, Q, R are by definition the components of the force acting at a point per unit of positive electricity, so that Pp dxdydz or Pdxpdydz is the work done per second by the current flowing parallel to the axis of x through the element of volume dxdydz. So for the other two components. This is in general transformed into other forms of energy, heat due to resistance, thermal effects at thermoelectric surfaces, and so on. The left side of (7) thus expresses the total gain in energy per second within the closed surface, and the equation asserts that this energy comes through the bounding surface, each element contributing the amount ex- pressed by the right side. This may be put in another form, for if (S' be the resultant of P', Q\ R\ and 9 the angle between its direction and that of ^, the magnetic intensity, ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 181 the direction-cosines L, M, N of the Une perpendicular to the plane containing @' and «§ are given by @'§sin<9' r§sin6>' ($'^ sin ^ ' so that the surface-integral becomes ^ [fd'^ sin ^ (Z? + Mm + iV^n) cZ/Sf. If at a given point (?/S be drawn to coincide with the plane containing (S' and t§, it then contributes the greatest amount of energy to the space ; or in other words the energy flows perpendicularly to the plane containing ($' and ^, the amount crossing unit area per second being ^'^ sin ^/477-. To determine in which way it crosses the plane take ^' along Oz, § along Oy. Then P' = 0, g' = o, |!=1, a=0, 1 = 1, y = 0, and if sin ^ = 1 X - 1, M = 0, iV = 0. If now the axis Ox be the normal to the surface outwards, ? = 1, m = 0, n = 0, so that this element of the integral contributes a positive term to the energy within the surface on the negative side of the yz plane ; that is, the energy moves along xO, or in the direction in which a screw would move if its head were turned round from the positive direction of the electromotive to the positive direction of the magnetic intensity. If the surface be taken where the matter has no velocity, (S' becomes equal to d, and the amount of energy crossing unit area perpendicular to the flow per second is electromotive intensity x magnetic intensity x sine included angle 477 Since the surface may be drawn anywhere we please, then wherever there is both magnetic and electromotive intensity there is flow of energy. Since the energy flows perpendicularly to the plane containing the two intensities, it must flow along the electric and magnetic level surfaces, when these exist, so that the lines of flow are the intersections of the two surfaces. We shall now consider the appHcations of this law in several cases. Applications of the Law of Transfer of Energy. (1) A straight wire conveying a current. In this case very near the wire, and within it, the lines of magnetic force are circles round the axis of the wire. The Hnes of electric force are along the wire, if we take it as proved that the flow across equal areas of the cross- 182 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD section is the same at all parts of the section. If AB, Fig. 1, represents the wire, and the current is from A to B, then a tangent plane to the surface at any point contains the directions of both the electromotive and magnetic intensities (we shall write e.m.i. and m.i. for these respectively in what follows), and energy is therefore flowing in perpendicularly through the surface, that is, along the radius towards the axis. Let us take a portion of the wire bounded by two plane sections perpendicular to the axis. Across the ends no energy is flow^ing, for they contain no component of the e.m.i. The whole of the energy then enters in through the external surface of the wire, and by the general theorem the amount entering in must just account for the heat developed owing to the resistance, since if the current is steady there is no other alteration of energy. It is, perhaps, worth while to show independently in this case that the energy moving in, in accordance with the general law, will just account for the heat developed. Let r be the radius of the wire, i the current along it, a the magnetic intensity at the surface, P the electromotive intensity at any point within the wire, and V the difference of potential between the two ends. Then the area of a length I of the wire is 27Trl, and the energy entering from the outside per second is c. _) Fig. 1. area x e.m.i. x m.i. ^Trrl.P.a Att 477 27rm . PI 4i7T for the hne-integral of the magnetic intensity 277m round the wire is 47r x current through it, and PI = V. But by Ohm's law 7 = iR and iV =-- i^R, or the heat developed according to Joule's laAv. It seems then that none of the energy of a current travels along the wire, but that it comes in from the non-conducting medium surrounding the wire, that as soon as it enters it begins to be transformed into heat, the amount crossing successive layers of the wire decreasing till by the time the centre is reached, where there is no magnetic force, and therefore no energy passing, it has all been transformed into heat. A conduction-current then may be said to consist of this inward flow of energy with its accompanying magnetic ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 183 and electromotive forces, and the transformation of the energy into heat within the conductor. We have now to inquire how the energy travels through the medium on its way to the wire. (2) Discharge of a condenser through a wire. We shall first consider the case of the slow discharge of a simple condenser consisting of two charged parallel plates when connected by a wire of very great resistance, as in this case we can form an approximate idea of the actual path of the energy. Fig. 2. Let A and B, Fig. 2, be the two plates of the condenser, A being positively and B negatively electrified. Then before discharge the sections of the equi- potential surfaces will be somewhat as sketched. The chief part of the energy resides in the part of the dielectric between the two plates, but there will be some energy wherever there is electromotive intensity. Between A and B the E.M.I, will be from A to B, and everywhere it is perpendicular to the level surfaces. Now connect A and 5 by a fine wire LMN of very great resistance, following a line of force and with the resistance so adjusted that it is the same for the same fall of potential throughout. We have supposed this arrange- ment of the resistance so that the level surfaces shall not be disturbed by the flow of the current. The wire is to be supposed so fine that the discharge takes place very slowly. While the discharge goes on a current flows round LMN in the direction indicated by the arrow, and there is also an equal displacement-current from B to A due to the yielding of the displacement there. The current will be 184 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD encircled by lines of magnetic force, which will in general form closed curves embracing the circuit. The direction of these round the wire will be from right to left in front, and round the space between A and B from left to right in front. The e.m.i. is always from the higher level surfaces — those nearer A, to the lower— those nearer B, both near the wire and in the space between A and B. Now, since the energy always moves perpendicularly to the hues of e.m.i. it must travel along the equipotential surfaces. Since it also moves perpen- dicularly to the hnes of m.i. it moves, as we have seen in case No. (1), inwards on all sides to the wire, and is there all converted into heat — if we suppose the discharge so slow that the current is steady during the time considered. But between A and B the e.m.i. is opposed to the current, being downwards, while the m.i. bears the same relation to the current as in the wire. Eemem- bering that e.m.i., m.i., and direction of flow of energy are connected by the right-handed screw relation, we see that the energy moves outwards from the space between A and B. As then the strain of the dielectric between A and B is gradually released by what we call a discharge current along the wire LMN, the energy thus given up travels outwards through the dielectric, following always the equipotential surfaces, and gradually converges once more on the circuit where the surfaces are cut by the wire. There the energy is transformed into heat. It is to be noticed that if the current may be con- sidered steady the energy moves along at the same level throughout. (3) A circuit containing a voltaic cell. When a circuit contains a voltaic cell we do not know with certainty what is the distribution of potential, but most probably it is somewhat as follows* : — Suppose we have a simple copper, zinc, and acid cell producing a steady current. There is probably a considerable sudden rise in passing from the zinc to the acid, the place where the chemical energy is given up, a fall through the acid depending on the resistance, a sudden fall on passing from the acid to the copper, where some energy is absorbed with evolution of hydrogen, * It seems probable that the only legitimate mode of measuring the difference of potential between two points in a circuit consisting of dissimilar conductors carrying a steady current, consists in finding the total quantity of energy given out in the part of the circuit between the two points while unit quantity of electricity passes either point. If this is the case, it seems impossible that the surface of contact of dissimilar metals can be the chief seat of the electro- motive force, for we have only the very slight evolution or absorption of energy there due to the Peltier effect. I have therefore adopted the theory of the voltaic circuit in which the seat of at least the chief part of the electromotive force is at the contact of the acid and metals. The large differences of potential found by electrometer methods between the air near two different metals in contact are, in this theory, to be accounted for by the supposition that the air acts in a similar manner to an oxidising electrolyte. A short statement of the theory is given in a letter by Professor Maxwell in the Electrician for April 26th, 1879, quoted in a note on page 149 of his Elementary Treatise on Electriciiy. (See also § 249, vol. 1, Maxwell's Electricity and Magnetism.) June 19, 1884. ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 185 and then a gradual fall through the wire of the circuit round to the zinc again. There will be a sHght change of potential in passing from copper to zinc, but this we shall neglect for simphcity. The equipotential surfaces will probably then be somewhat as sketched in Fig. 3*, all the surfaces starting from where the acid comes in contact with the zinc, some of the highest potential passing through the acid, others passing between the acid and copper, and crowding in there, the rest lower than these cutting the circuit at right angles in points at intervals representing equal falls of potential. Fig. 3. If this be the actual arrangement, then it is seen that the current, which travels round the circuit from zinc through acid to copper, is in opposition to the E.M.I, between the zinc and acid, while the m.i. is related to the current in the ordinary way. The energy will therefore pass outwards from there along the level surfaces. In fact, the medium between the zinc and acid behaves Kke the medium between the plates of the condenser in case No. (2j, and it seems possible that the chemical action produces continually fresh * electric displacement' from acid towards zinc which yields as rapidly as it is formed, the energy of the displacement moving out sideways. Some of this energy which travels along the highest level surfaces will converge on the acid, and there be, at any rate ultimately, converted into heat. Some of it will move along those surfaces which crowd in between the acid and copper and there converge to supply the energy taken up by the escaping hydrogen. The rest spreads out to converge at last at different parts of the circuit, and there to be transformed into heat according to Joule's law. It may be noticed that if the level surfaces be drawn with equal differences * In this and the succeeding cases the circuit is alone supposed to cause the distribution of potential. In actual cases the surfaces would probably be very much deflected from their normal positions in the dielectric through the presence of conductors, electrified matter, and so on. 186 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD of potential, equal amounts of energy travel out per second between successive pairs of surfaces. For the amount transformed in the circuit in a length having a given difference of potential V between its ends will be 7 x current, and therefore the amount transformed between each pair of surfaces drawn with the same potential difference will be the same. But since the current and the field are steady, the energy transformed will be equal to the energy moving out from the cell between the same surfaces — the energy never crossing level surfaces. This admits of a very easy direct proof, but the above seems quite sufficient. This result has a consequence which, though already well known, is worth mentioning here. Let F^ be the difference of potential between the zinc and acid, V2 that between the acid and copper. If i be the current, V-^i is the total energy travelling out per second from the zinc surface. Of this Fg^ is absorbed at the copper surface, the rest, viz., (Fi — Fg) ^, being trans- formed in the circuit. The fraction, therefore, of the whole energy sent out which is transformed in the circuit is - ^w — -, a result analogous to the expression for the amount of heat which can be transformed into work in a reversible heat-engine. One or two interesting illustrations of this movement of energy may be mentioned here in connection with the voltaic circuit. Suppose that we are sending a current through a submarine cable by a battery with, say, the zinc to earth, and suppose that the sheath is every- where at zero potential. Then the wire will everywhere be at higher potential than the sheath, and the level surfaces will pass from the battery through the insulating material to the points where they cut the wire. The energy then which maintains the current, and which works the needle at the further end, travels through the insulating material, the core serving as a means to allow the energy to get in motion. Again, when the only effect in a circuit is the generation of heat, we have energy moving in upon the wire, there undergoing some sort of transformation, and then moving out again as heat or Ught. If Maxwell's theory of light be true, it moves out again still as electric and magnetic energy, but with a definite velocity and intermittent in type. We have in the electric Hght, for instance, the curious result that energy moves in upon the arc or filament from the surrounding medium, there to be converted into a form which is sent out again, and which, though still the same in kind, is now able to affect our senses. (4) Thermoelectric circuits. Let us first take the case of a circuit composed of two metals, neither of which has any Thomson effect. Let us suppose the current at the hot junction flows from the metal A to the metal B, Fig. 4. According to Professor Tait's ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 187 theory it would appear that the e.m.i. at the hot junction is to that at the cold as the absolute temperature at the hot is to that at the cold junction. If the current is steady there is probably then a sudden rise in potential from A to B at the hot junction, a gradual fall along B, a sudden fall at the cold junction — less, however, than the sudden rise at the other — and a gradual fall along A. The level surfaces will then all start from the hot junction, the higher ones cutting the circuit at successive points along B, several con- verging at the cold junction, and the rest cutting the circuit at successive points along A. The heat at the hot junction is converted into electric and magnetic energy, which here moves outwards, since the current is against the E.M.I. Some of this energy converges upon B and A, to be converted Fig. 4. into heat, according to Joule's law, and some on the cold junction, there producing the Peltier heating effect. Let us now suppose that we have a circuit of the same two metals, now all at the same temperature, but with a battery interposed in B, which sends a current in the same direction as before (Fig. 5). Then if C be the junction which was hot, and D that which was cold in the last case, we know that the current will tend to cool C and to heat Z). In going from A to B at C there will be a sudden rise of potential, and in going from B to A at D there will be a sudden fall. Then, since the potential falls, as we go with the current along A, there will be a point on A near C which has the same potential as B at the junction. From this point to C, A will have lower potentials, and points with the same potentials will exist on B between C and the battery. Then either the level surfaces passing through C are closed surfaces, cutting A or B, and not passing through the battery at all, or, as seems much more 188 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD probable, the surfaces from the battery which pass through C cut the circuit in three points in all outside the battery: once somewhere along A, once at C, and once somewhere along B. I have drawn and numbered the surfaces in the figure on this supposition. The heat developed in the parts of the circuit near C will thus be partly supphed from the junction C, where the Fig. 6. current is against the e.m.i. The energy therefore moves out thence, giving a cooling effect. The Thomson effect may be considered in somewhat the same way. Let us suppose that a metal BC of the iron type, and with temperature falhng from B to C, forms part of a circuit between two neutral metals of the lead type AB and CD, Fig. 6, and let us further, for simphcity, suppose that these 1 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 189 metals are each at the neutral temperatures with respect to BC, so that there is no E.M.I, at the junction. If we drive a current from A to D hj means of some external e.m.i., say at a junction elsewhere in the circuit, the potential will tend to fall from A to D. But a current in iron from hot to cold cools the metal, that is, the e.m.i. appears to be in opposition to the current, so that the energy moves outwards. The potential, therefore, tends to rise from B to C, and actually will do so if the resistance of BC is negligible compared with that of the rest of the circuit. In this case the level surfaces will probably be somewhat as indicated in Fig. 6, where they are numbered in order, each surface which cuts BC also cutting AB and CD, and the energy moving outwards will come into the circuit again at the parts of AB and CD near the junctions, where it will be transformed once more into heat. If the resistance of BC be gradually increased the fall of potential, according to Ohm's law, will tend to lessen the rise, and fewer surfaces will cut BC. It would seem possible so to adjust matters that the two exactly neutralised each other so that no energy either entered or left BC. In this case we should only have hues of magnetic force round BC, and no other characteristic of a current in that part of the circuit*. If this is the true account of the Thomson effect it would appear that it should be described not as an absorption of heat or development of heat by the current but rather as a movement of energy outwards or inwards, according as the e.m.i. in the unequally heated metal opposes or agrees with the direction of the current. (5) A circuit containing a motor. This case closely resembles the third case of a circuit containing a copper- zinc cell, the motor playing a part analogous to that of the surface of contact of the acid with the copper. Let us, for simplicity, suppose that the motor has no internal resistance. When it has no velocity all the level surfaces cut the circuit, and the energy leaving the dynamo or battery is all transformed into heat due to resistance. But if the motor is being worked the current diminishes, the level surfaces begin to converge on the motor and fewer cut the circuit. Some of the energy therefore passes into the motor, and is there transformed into work. As the velocity increases the number cutting the rest of the circuit decreases, for the current diminishes, and, therefore, by Ohm's law, the fall of potential along the circuit is less ; and ultimately when the velocity of the motor becomes very great the current becomes very small. In the limit no level surface cuts the circuit, all converging on the motor. * Perhaps this is only true of the wire as a whole. If we could study the effects in minute portions it is possible that we should find the seat of the e.m.i. due to difference of temperature not the same as that which neutrahses it, which is according to Ohm's law. One, for instance, might be between the molecules, the other in their interior, so that there might be an interchange of energy still going on, though no balance remained over to pass out of the wire. 190 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD That is, all the energy passes into the motor when it is transformed into work, and the efficiency of the arrangement is perfect, though the rate of doing work is infinitely slow. (6) Induced currents. It is not so easy to form a mental picture of the movement of energy which takes place when the field is changing and induced currents are created. But we can see in a general way how these currents are accounted for. When there is a steady current in a field there is corresponding to it a definite dis- tribution of energy. If there is a secondary circuit present, so long as the primary current is constant, there is no e.m.i. in the secondary circuit for it is all at the same potential. The energy neither moves into nor out of it, but streams round it somewhat as a current of liquid would stream round a solid obstacle. But if the primary current changes there is a redistribution of the energy in the field. While this takes place there will be a temporary E.M.I, set up in the conducting matter of the secondary circuit, energy will move through it, and some of the energy will there be transformed into heat or work, that is, a current will be induced in the secondary circuit. (7) The electromagnetic theory of light. The velocity of plane waves of polarised light on the electromagnetic theory may be deduced from the consideration of the flow of energy. If the waves pass on unchanged in form with uniform velocity the energy in any part of the system due to the disturbance also passes on unchanged in amount with the same velocity. If this velocity be v, then the energy con- tained in unit volume of cubical form with one face in a wave-front will all pass out through that face in l/t'th of a second. Let us suppose that the direction of propagation is straightforward, while the displacements are up and down; then the magnetic intensity will be right and left. If (S be the E.M.I, and «5 the m.i. within the volume, supposed so small that the intensities may be taken as uniform through the cube, then the energy within it is /i(£'-/87r -f- [mS^^/Stt. The rate at which energy crosses the face in the wave- front is l^"«§/477 per second, while it takes 1/vth of a second for the energy in the cube to pass out. Then p>^m^^f^^ 4.7TV 877 ^ 877 ^ ^ Now, if we take a face of the cube perpendicular to the direction of dis- placement, and therefore containing the m.i., the fine-integral of the m.i. round this face is equal to 477 x current through the face. If we denote distance in the direction of propagation from some fixed plane by z, the fine- integral of the M.I. is — -J- , while the current, being an alteration of dis- , ^ . K di^ placement, is j" "T~ • ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 191 Therefore -f = ^f (2) But since the displacement is propagated unchanged with velocity v, the displacement now at a given point will alter in time dt to the displace- ment now a distance dz behind, where dz == vdt. Therefore W^'^Tz (^^) Substituting in (2) -^ = Kv -^ , whence § = Kv^, (4) the function of the time being zero, since S^ and (S* are zero together in the parts which the wave has not yet reached. If we take the line-integral of the e.m.i. round a face perpendicular to the M.I. and equate this to the decrease of magnetic induction through the face, we obtain similarly ^ = iJ^v^ (5) It may be noticed that the product of (4) and (5) at once gives the value of V, for dividing out ($'*& we obtain 1 = ixKv'' 1 or _ v= -=^ . But using one of these equations alone, say (4), and substituting in (1) K for <^ and dividing by (E^, we have K^K^ IJiKV 47r " Stt Stt or 1 --= p.Kv'^, whence v = -^=^ . - This at once gives us the magnetic energy equal to the electric energy, for 877 877 877 It may be noted that the velocity is the greatest velocity with which the two energies can be propagated together, and that they must be equal when travelhng with this velocity. For if v be the velocity of propagation and 6 the angle between the two intensities, we have ^^ sin^ _ m^ fj.^ 4:7TV StT 877 ' 2 sin d 43 (i- 192 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD The greatest value of the numerator is 2 when ^ is a right angle, and the least value of the denominator is 2 V]JiK, when the two terms are equal to each other and to VijlK. 1 TT The maximum value of v therefore is . — , and occurs when 6 = ^ and The preceding examples will suffice to show that it is easy to arrange some of the known experimental facts in accordance with the general law of the flow of energy. I am not sure that there has hitherto been any distinct theory of the way in which the energy developed in various parts of the circuit has found its way thither, but there is, I believe, a prevaihng and somewhat vague opinion that in some way it has been carried along the conductor by the current. Probably Maxwell's use of the term 'displace- ment' to describe one of the factors of the electric energy of the medium has tended to support this notion. It is very difficult to keep clearly in mind that this 'displacement' is, as far as we are yet warranted in describing it, merely a something with direction which has some of the properties of an actual displacement in incompressible fluids or solids. When we learn that the ' displacement ' in a conductor having a current in it increases continually with the time, it is almost impossible to avoid picturing something moving along the conductor, and it then seems only natural to endow this something with energy-carrying power. Of course it may turn out that there is an actual displacement along the lines of electromotive intensity. But it is quite as likely that the electric 'displacement' is only a function of the true displacement, and it is conceivable that many theories may be formed in which this is the case, while they may all account for the observed facts. Mr Glazebrook has already worked out one such theory in which the com- ponent of the electric displacement at any point in the direction of x is ^- V^^, where | is the component of the true displacement {VMl. Mag. June 1881). It seems to me then that our use of the term is somewhat unfortunate, as suggesting to our minds so much that is unverified or false, while it is so difficult to bear in mind how little it really means. I have therefore given several cases in considerable detail of the apphcation of the mode of transfer of energy in current-bearing circuits according to the law given above, as I think it is necessary that we should reaUse thoroughly that if we accept Maxwell's theory of energy residing in the medium, we must no longer consider a current as something conveying energy along the conductor. A current in a conductor is rather to be regarded as consisting essentially of a convergence of electric and magnetic energy from the medium upon the conductor and its transformation there into other forms. The current through a seat of so-called electromotive force consists essentially ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 193 of a divergence of energy from the conductor into the medium. The magnetic Hnes of force are related to the circuit in the same way throughout, while the lines of electric force are in opposite directions in the two parts of the circuit — with the so-called current in the conductor, against it in the seat of electromotive force. It follows that the total e.m.i. round the circuit with a steady current is zero, or the work done in carrying a unit of positive electricity round the circuit with the current is zero. For work is required to move it against the e.m.i. in the seat of energy, this work sending energy out into the medium, while an equal amount of energy comes in in the rest of the circuit where it is moving with the e.m.i. This mode of regarding the relations of the various parts of the circuit is, I am aware, very different from that usually given, but it seems to me to give us a better account of the known facts. It may seem at first sight that we ought to have new experimental indica- tions of this sort of movement of energy, if it really takes place. We should look for proofs at points where the energy is transformed into other modifi- cations, that is, in conductors. Now in a conductor, when the field is in a steady state, there is no electromotive intensity, and therefore no motion and no transformation of energy. The energy merely streams round the outside of the conductor, if in motion at all in its neighbourhood. If the field is changing, energy can pass into the conductor, as there may be temporary e.m.i. set up within it, and there will be transformation. But we already know the nature of this transformation, for it constitutes the induced current. Indeed, the fundamental equation describing the motion of energy is only a deduction from Maxwell's equations, which are formed so as to express the experimental facts as far as yet known. Among these are the laws of induction in secondary circuits, and they must therefore agree with the law of transfer. We can hardly hope, then, for any further proof of the law beyond its agreement with the experiments already known until some method is discovered of testing what goes on in the dielectric independently of the secondary circuit. p. c. w. 13 11. ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD*. [Phil Trans. 176, 1885, pp. 277-306.] [Received January 31. Read February 12, 1885.] In a paper published in the Philosophical Transactions for 1884 (Part ii, pp. 343-361 )t, I have deduced from Maxwell's equations for the electromagnetic field the mode in which the energy moves in the field. The result there obtained is that the energy moves at any point perpendicularly to the plane containing the directions of the electric and magnetic intensities, and in the direction in which a right-handed screw would move if turned round from the positive direction of the electric intensity to the positive direction of the magnetic intensity. The quantity crossing the plane per unit area per second is equal to the product of the two intensities, multipHed by the sine of the included angle, divided by 47r t. Hence it follows that the energy moves along the intersections of the two sets of level surfaces, electric and magnetic, where they both exist, their intersections giving, as it were, the lines of flow. In the particular case of a steady current in a wire where the electrical level surfaces cut the wire * [Added July 15. Since the reading of the paper I have found a remarkable passage in Faraday's Experimental Researches, vol, 1, p. 529, § 1659, which I give below. The words I have put in italics might be regarded as the starting-point of the views which I have attempted to develop in this paper. '§ 1659. According to the beautiful theory of Ampere, the transverse force of a current may be represented by its attraction for a similar current and its repulsion of a contrary current. May not then the equivalent transverse force of static electricity be repre- sented by that lateral tension or repulsion which the lines of inductive action appear to possess (1304)? Then, again, when current or discharge occurs between two bodies, previously under inductrical relations to each other, the lines of inductive force will weaken and fade away, and, as their lateral repulsive tension diminishes, will contract and ultimately disappear in the live of discharge. May not this be an effect identical with the attractions of similar currents, i.e., may not the passage of static electricity into current electricity, and that of the lateral tension of the lines of inductive force into the lateral attraction of Hnes of similar discharge, have the same relation and dependence, and run parallel to each other?'] t [Collected Papers, Art. 10.] J I here adopt the simpler term 'Electric Intensity,' denoted by e.i., instead of 'Electro- motive Intensity,' for the force which would act on a small body charged with unit of positive electrification. The magnetic intensity, i.e., the force which would act on a unit north-seeking Pole, will be denoted by m.i. ON THE CONNECTION BETWEEN ELECTRIC CURRENT, ETC. 195 perpendicularly to the axis, it appears that the energy dissipated in the wire as heat comes in from the surrounding medium, entering perpendicularly to the surface. In that paper I made no assumption as to the transfer of the electric and magnetic inductions — the electric and magnetic conditions — through the medium, merely considering the movement of energy. I now propose to develop a hypothesis as to the transfer of the inductive condition in the medium, and its movement inwards upon current-bearing wires. The value of the electric induction at any point in an isotropic medium is equal to K x E.I./477, and the direction of the induction coincides with that of the intensity. Maxwell terms this electric induction 'displacement,' but I think that 'induction' is preferable, as it impUes no hypothesis beyond that of some alteration in the medium, which can be described by a vector. The value of the magnetic induction is equal to /x x m.i., and its direction coincides with that of the magnetic intensity. If we symbolise the electric and magnetic conditions of the field by induction-tubes running in the directions of the intensities, the tubes being supposed drawn in each case so that the total induction over a cross-section is unity, then we have reason to suppose that the electric tubes are con- tinuous except where there are electric charges, while the magnetic tubes are probably in all cases continuous and re-entrant. In the neighbourhood of a wire containing a current, the electric tubes may in general be taken as parallel to the wire while the magnetic tubes encircle it. The hypothesis I propose is that the tubes move in upon the wire, their places being supplied by fresh tubes sent out from the seat of the so-called electromotive force. The change in the point of view involved in this hypothesis consists chiefly in this, that induction is regarded as being propagated sideways rather than along the tubes or lines of induction. This seems natural if we are correct in supposing that the energy is so propagated, and if we therefore cease to look upon current as merely something traveUing along the conductor carrying it, and in its passage affecting the surrounding medium. As we have no means of examining the medium, to observe what goes on there, but have to be content with studying what takes place in conductors bounded by the medium, the hypothesis is at present incapable of verification. Its use, then, can only be justified if it accounts for known facts better than any other hypothesis. The basis of MaxwelVs Electromagnetic Theory. Maxwell's Electromagnetic Theory rests on three general principles. I. The first principle consists in the assumption that energy has position, i.e., that it occupies space. The electric and magnetic energies of an electro- magnetic system reside therefore somewhere in the field. It is an inevitable 13—2 * 196 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE conclusion that they are present wherever the electric and magnetic intensities can be shown to exist. For instance, suppose a small electrified body placed in a field where there is electric intensity ; then the body will be acted on by force and will receive energy which appears as the energy of motion, the electric energy at the same time decreasing. If energy has position, that which is now in the body must have come into it through the surrounding space, or it was present in that space before the body took it up. The alternative that it appeared in the body without passing through the space immediately surrounding the body need not be discussed. Hence the existence of electric intensity impUes the existence of electric energy in the place where the electric intensity is capable of manifestation. Similarly magnetic energy accompanies magnetic intensity. The inductive condition of the medium imagined by Faraday is due then to its modification when containing energy. Maxwell has shown that all the energy is accounted for on the supposition that the electric energy per unit volume at any point is K{B.i.yj87T, and that the magnetic energy is ju, (m.i.)2/877. He has given in his Elementary Treatise on Electricity, p. 47, another way of describing the distribution of energy which will be more useful for my purpose. If the field be mapped out by unit induction-tubes — either electric or magnetic — i.e., tubes drawn so that the total induction over every cross-section of a tube is unity, and if these tubes be divided into cells of length such that the difference of potential or the line-integral of the intensity between the two ends of each cell is unity, then each cell contains, if electric, half a unit of enero^v, if magnetic 5^ of a unit, the divisor 47r being introduced bv the 077 ''^ " difference in definition of the two inductions. Maxwell terms these unit cells. II. The second principle is in part experimental, viz. : — that the line- integral of the electric intensity round any closed curve is equal to the rate of decrease of the total magnetic induction through the curve. This is verified by experiment when the curve is drawn through conducting material. Maxwell supposes it to be true in all cases, that is, he supposes that electric induction can be produced in insulators by means of magnetic changes, without the presence of charges on conductors, and is therefore led to identify the growth and decrease of electric induction with current. III. The third principle is also in part experimental, viz. : — that the hne-integral of the magnetic intensity round any closed curve is equal to iv X current through the curve. This is verified by experiment when the current is in a wire, and Maxwell supposes it to be also true in the case where there is change of electric induction in an insulator. The supposition is justified by Prof. Rowland's well-known experiment. From these three principles Maxwell deduces his general equations of the ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 197 Electromagnetic Field. I have stated them in full as I propose to modify the second and third principles, and I wish to make quite clear the nature of the proposed changes. Modification of the Second Principle. I propose to replace the second principle by the following : Whenever electromotive force is produced by change in the magnetic field, or by motion of matter through the field, the e.m.f. per unit length or the electric intensity is equal to the number of tubes of magnetic induction cutting or cut by the unit length per second, the e.m.f. tending to produce induction in the direction in which a right- handed screw would move if turned round from the direction of motion relatively to the tubes towards the direction of the magnetic induction*. In order that the results obtained from this should agree with those obtained from Maxwell's statement of the principle, it is necessary that change in the total quantity of magnetic induction passing through a closed y curve should always be produced by the passage of induction-tubes through " the curve inwards or outwards. In some instances this is undoubtedly the case, as, for instance, where a part of a circuit moves so as to cut a fixed magnetic field, or where a magnet moves in the neighbourhood of a circuit. Here the e.m.f. is equal to the number of tubes cut by the wire per second, and its seat is that part of the wire cutting the tubes. In other cases, as, for instance, where the wire is between the poles of an electromagnet whose magnetising current is changing, we have no direct experimental evidence of the movement of the induction in or out. But the induction- tubes are closed, and to make them thread a circuit we might expect that they would have to cut through the boundary. The alternative seems to be that they should grow or diminish from within, the change in intensity being propagated along the tubes. This would be inconsistent with their closed nature, unless the energy were instantaneously propagated along the whole length, and is further negatived by the theory of the transfer of energy, which implies that the energy flows transversely to the direction of the tubes. I shall suppose, then, that alteration in the quantity of magnetic induction through a closed curve is always produced by motion of induction-tubes inwards or outwards through the bounding curve. * Taking the electric intensity as always perpendicular to the plane of motion of the magnetic tubes through a point, and equal to the number cut per second by unit length of the normal to the plane of motion, we can easily show that the component of the intensity in any other direction will be equal to the number of tubes cut by a Hne of unit length in that direction. For let OA represent a small length drawn perpendicular to the plane of motion, and let OP represent a line drawn in any direction making an angle 6 with OA. Draw AP perpendicular to OA, and meeting OP in P. Then the same number of tubes will cut both OA and OP, since AP is parallel to their plane of motion. If the number cutting OA he Ex OA, where E is the number cutting unit length, and therefore equal to the resulting intensity, the number cutting unit length of OP will OA be E . Yyp = E cos d, or the component of the intensity along OP. 198 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE Modification of the Third Principle. The third principle admits of similar analysis, according to which we may regard the magnetic intensity along a closed curve as due to the cutting of the curve by tubes of electric induction. If we regard the line-integral of the magnetic intensity round a tube of induction as measuring the magneto- motive force— employing a useful term suggested by Mr. Bosanquet — we may put the modification in the following form : Whenever magnetomotive force is produced by change in the electric field, or by motion of matter through the field, the magnetomotive force per unit length is equal to iir x the number of tubes of electric induction cutting or cut by unit length per second, the magnetomotive force tending to produce induction in the direction in which a right-handed screw would move if turned round from the direction of the electric induction towards the direction of motion of the unit length relatively to the tubes of induction. This is the most general form of the principle, but we shall only require the more special statement which immediately follows from it : that the line-integral of the m.i. round any curve is equal to 477 x the number of tubes passing in or out through the curve per second. We have reasons exactly similar to those given in the last case for supposing that any change in the total electric induction through a curve is caused by the passage of induction-tubes in or out across the boundary. The alternative, that change takes place by propagation from the ends, seems inconsistent with the theory of the transverse flow of energy. I shall postpone the discussion of the modifications of the general equations of the electromagnetic field following from these changes in the fundamental principles, and proceed to discuss the bearing which they have upon the nature of currents in conductors. A straight wire carrying a steady current. Let AB represent a wire in which is a steady current from A to B. The direction of the electric induction in the surrounding field near the wire, if the field be homogeneous, is parallel to AB. Let E be the value of the electric intensity, or the difference of potential per unit length perpendicular to the level surfaces, and let R be the resistance E of the wire per unit length. Then = ^ where C is the current, and C is uniform throughout the circuit. The magnetic intensity in the immediate 2C neighbourhood of the wire at a distance r from the axis of the wire is — . The hypothesis proposed as to the nature of the current is that C electric ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 199 c^) induction-tubes close in upon the wire per second. The wire is not capable of bearing a continually-increasing induction, and breaks the tubes up, as it were, their energy appearing finally as heat*. Let us see how this hypothesis accounts for known facts, when aided by the two principles just laid down. It accounts at once for the constancy of the current at all parts of the wire in the steady state, in so far as it reduces this constancy to a particular case of the law according to which there is the same total induction over all cross-sections of a tube. If, for instance, there were more induction entering at A than at B, then more tubes must be entering at A, and so there would be an increase in the number of tubes left in the medium about B, or the field would not be steady. Further, if we draw any closed curve embracing the wire once, we may apply the third principle to give us the fine- integral of the magnetic intensity round the curve. For this ^ . is a case where change is certainly going on in the electric field. ^jg, i, and the magnetomotive force is due to this change. The field being steady, if C tubes enter the wire and are there broken up, C tubes must cross through any encirchng curve to supply their place, or the line- integral of the magnetic intensity round the curve is equal to iir x number of tubes passing through the boundary per second, i.e., iirC. If the curve be a circle of radius r, with its centre in the axis and plane perpendicular thereto, the intensity at any point of this circle will be tangential to it, and equal to 477(7 ^ 2C 277r r The known constancy of the line-integral of the magnetic intensity round the wire, which the hypothesis thus accounts for, almost seems to force the hypothesis upon us, if we regard the field as caused by the inward flowing of the energy rather than by something propagated out from the wire. Assuming that the induction-tubes bring in their energy, the quantity is easily found. The number of unit cells per unit length is equal to the difference of potential per unit length, or E. Hence the energy per unit length of each E . tube is ^ , since each cell contains a half unit. If C tubes disappear in the CE Now the total energy dissipated per unit length is CE per second. Or the movement wire per second, they yield up -^ of energy per unit length. * May we not say that the tubes are dissolved ? The term seems to suggest that the induction is not destroyed, but only loses its continuity. Probably this is the case; for on the electro- magnetic theory of radiant energy, when the wire is heated, it sends out the energy it received, again in the electromagnetic form. 200 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE inwards of the electric induction will only account for half of the energy. The other half must be accounted for by the movement inwards of the magnetic induction. This movement of the magnetic induction is suggested by the existence of electric induction, which cannot be ascribed to statical charges. The electric intensity is E. Hence E tubes of magnetic induction must move in per second, cutting unit length parallel to the axis of the wire, in accordance with the second principle, and it will easily be seen that the inward motion gives the right direction of the electric intensity. The hne-integral of the magnetic intensity round a tube is ^ttO, the tubes being closed rings. Hence there are 4:7rC unit cells in the length. Since each of these contains 5— of energv, the quantitv per tube = -^ — = o • ^ tubes entering the wire 07T " " 077" Z OF' per second will carry in -^ of energy, the other half to be accounted for. We can in a similar manner trace the dissipation of the energy, which we must suppose taking place within the wire. The line-integral of the magnetic intensity round a circle, with its centre in the axis of the wire, is constant up to the wire, and equal to irrC. Within the wire it gradually diminishes as the circle contracts. At a distance r from the centre it is ^ttC -^ where a is the radius of the wire. If we assume this intensity to be still due to the passage inwards of the tubes of electric induction only, — ^ cross inwards per second at a distance r, the difference between this number and the C tubes entering the outer boundary being destroyed and their energy dissipated. The energy thus dissipated per unit length between the outer boundary and a coaxal cylinder of radius r will be -^ [l -A per second. H r = the whole of the electric energy is dissipated. It would appear, then, that we may represent the dissipation of the electric energy by the total destruction of the tubes all through their length. The value of the electric intensity being E throughout the wire the number of tubes of magnetic induction cutting unit length parallel to the axis is the same at all parts, viz., E per second. Hence the magnetic tubes are not destroyed as the electric tubes are. But the Hne-integral of the magnetic mtensity round the tubes diminishes as they approach the axis, being ^tt — ^ round that at distance r. The number of unit cells diminishes, and, therefore, the energy per^tube is less, the decrease being due to that dissipated. Thus 4- OF OF the energy entering in the E tubes at the outer boundary is "^ or -^ . 877 2i ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 201 That crossing in E tubes at a distance ri&^ E = -^ — - . The difference 877 a^ 2 a^ ^(1 -gj has been dissipated. Hence it appears that the energy dissipated per second may be repre- sented as half electric half magnetic, the electric energy being dissipated by the breaking up of the tubes and their disappearance, while the magnetic energy is dissipated by the shortening of the tubes and their final disappear- ance by contraction to infinitely small dimensions of the diameters of the rings by which we may represent them. At all points therefore outside and inside the energy crossing any surface may be represented as equally divided between the two kinds. As we know the value of the induction at any point, or the number of tubes passing through unit area, and as we also know the number of tubes cutting the boundary it is easy, on the assumption that the tubes move on unchanged, to calculate their velocity. Of course this velocity is purely hypothetical, as we cannot examine minutely into the medium and observe what goes on there. Probably, if we could observe with sufficient minuteness we should find unevennesses in the induction. If the velocity of the tubes has any physical meaning it is that these unevennesses are carried forward with that velocity. To illustrate this let us suppose that we have water flowing through a glass tube at a steady rate. We have nothing to show that the water is moving past any point in the tube beyond its disappearance at the entrance and its appearance at the exit, but knowing the cross-section of the tube, i.e., the quantity of water in any part of it, and the quantity entering and leaving, it is easy to assign a velocity to the water in the tube which shall account for the observed amount entering and leaving. This velocity is to a certain extent hypothetical. But if we examine the tube with a sufficient magnifying power to show particles of dust in the water the existence of the velocity receives a more direct proof. I do not know whether we should have any right to expect a similar proof of the motion of induction even if we had the means of observation. To find the hypothetical velocity of the electric induction-tubes let us calculate the number of tubes passing through a circular band with radii r and r -\- dr and centre in the axis of the wire, and lying in a plane perpen- KE dicular to the axis. The intensity being E the induction is -r— , and theretore 4:7T . .... the area of cross-section of each tube is -fv-^, since area x induction is unity. The number passing through the circular band is therefore ^ , KE KErdr ZTvrdr . -j— = — ^ — . 4:77 2 202 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE Since C tubes move in through the inner circle per second, — ^ — tubes move in in of a second, i.e., all the tubes passing through the band will have just moved in in this time. The outermost tubes therefore describe the space dr in time — ^r^:^ , or the velocity is -^-^ . Now we know that if E R be the resistance per unit length, C =- ^. Hence we may put the velocity in the form 2 1 KR' r' which is independent of the current. To take a special case, let us calculate the velocity just outside the boundary of a copper wire, the specific resistance of copper being 1642 in electromagnetic measure. Then if a be the radius of the wire ,2 and K = ^ where v is the ratio of the units, which in air may be taken as 3 X IQio. 2'U^77tt^ Then the velocity = v/> /^~ -^ 1642a 2 X 9 X IO^Ott^ 1642 - 345 X lO^S. At greater distances the velocity will be less, diminishing according to the inverse distance. The hypothetical velocity of propagation of the magnetic induction may be calculated in a similar manner. The intensity at a distance r from the 2C ^llC axis is' — and the induction is ^— . The area of each tube is therefore r r T ^— ^ , and the number lying in a ring of rectangular section with depth unity and interna] and external radii r and r + dr, will he 1 x dr -^ ^^ = — . 2/xC r But E tubes move in per second through the inner face of the ring, so that "^yiCdr . . . 2uiCdr —~ — tubes move m m time ^^ - , or this is the time taken by the outer- r Er ^ most tubes to move across the ring describing a distance dr. The velocity is therefore Er Rr '2iiC~~ 2jLt' which is again independent of the current. ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 203 If the current-bearing wire is copper, R = , and with ft = 1 the 7Ta' velocity becomes 1642r 277^2 We cannot assign a velocity to the electric tubes within the wire since the number is diminishing as their energy dissipates. But the magnetic tubes crossing unit length parallel to the axis are still unchanged in number, so that we may assign a velocity to them. This velocity means that with the known value of the magnetic induction this velocity will give the number crossing inwards required to produce electric intensity E, The velocity will be found equal to 2/xCr ^^ 2/xr' In the case of a copper wire this becomes 1642 2/x77r * Discharge of a condenser through a fine wire. Let us suppose that we have a condenser consisting of two parallel plates A and B and charged with equal and opposite charges. Then we know that there will be electric induction between the two plates, and that according to Maxwell's theory the energy of the system is stored there. We may form an idea of the distribution of the energy by drawing the unit induction-tubes, each starting from and ending in unit quantity of electricity, and dividing these into unit cells by the level surfaces, drawn at unit difference of potential (Fig. 2). If the dimensions of the plates be great compared with their distance apart, then nearly all the cells will be between the two plates, and since each cell contains half a unit of energy, nearly all the energy is there. There will, 204 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE however, be slight induction, and therefore some small quantity of energy in the surrounding space. Now let the two plates be connected by a wire. Discharge takes place, and we are fairly justified, from the heat in the wire and the transient magnetic effects, in saying that a current has been in the wire from the positive to the negative plate, or the wire was for the time being in the same relation to the surrounding medium as the wire in the case just considered, the condition of affairs, however, not being steady. Let us suppose the wire to have a very great resistance, in order that, at least in imagination, we may lengthen out the time of discharge. On the ordinary current- theory, combined with Maxwell's 'displacement' theory, the medium between the plates has returned from the strained condition, denoted by 'displacement' from the positive to the negative plate, causing displacement through the plates and along the wire, the displacement being in the same direction all round the circuit. This is generally, I think, supposed to take place by the recovery of the medium between the plates causing displacement in the metal immediately in front of it, the displacement being analogous to the forcing of water along a pipe corresponding to the plates and wire, by the recovery from strain of some substance placed in a chamber corresponding to the space between the plates. According to the hypothesis here advanced we must suppose the lessening of the induction between the plates — induction being used with the same physical meaning as Maxwell's displacement — to take place by the divergence outwards of the induction-tubes. We may picture them as taking up the positions of successive Hues of induction further and further away from the space between the plates, their ends always remaining on the plates. They finally converge on the wire, and are then broken up and their energy dissi- pated as heat. At the same time some of the energy becomes magnetic, this occurring as the difference of potential between the plates lowers, so that the tubes contain fewer unit cells. The magnetic energy will be contained in ring-shaped tubes which will expand from between the plates and then contract upon some other part of the circuit. To illustrate the movement of the electric induction-tubes let us suppose them to be represented by elastic strings stretched between the two plates. Then the motion of the tubes outwards would be roughly repre- sented by pulhng the elastic strings outwards and doubhng them back close against the wire, their ends being still attached to the plates. It is evident that if any ring surround the wire each of the strings must break through it in order to reach the wire. Hence the total number of strings cutting any ring surrounding the wire is the same wherever the ring be placed. Similarly the total number of tubes of electric induction cutting any curve encirchng the wire is the same, and therefore the fine-integral of the magnetic intensity ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 205 round the curve integrated throughout the time of discharge is the same, or the total magnetic effect is the same at all parts of the circuit. It is not necessary to suppose that a tube enters the wire at the same moment through- out its whole length ; indeed, the experiments of Wheatstone on the so-called velocity of electricity prove clearly that this is not the case, for in those experiments the tubes reached air-breaks near the two ends of the wire before they reached a break in the middle. We cannot by this general reasoning show that the energy entering any length of the wire will be proportional to the resistance of that length — the result obtained by Kiess. Indeed, this cannot always be the case. For instance, imagine a condenser discharged by two wires connected to the two plates of another condenser of greater capacity, whose plates are again connected by a fine wire of enormous resistance, through which the discharge can only take place slowly. Then the energy dissipated in the wires will not to a first approximation depend on their resistances but on the ratios of the capacities, that in the wire of high resistance bearing to that in the other wires the ratio of the less capacity to the greater. Probably Riess's results only hold when the discharge takes place in such a way that it may be looked upon at any one moment as approximately in the steady state. We have shown that the magnetic measure of the total current is the same all along the wire. Probably also the chemical measure is the same — meaning by the chemical measure whatever interchanging or turning round of molecules may occur when induction takes place in a conductor. For even if a tube does not enter the wire at the same time throughout its length, an end part, say, entering first, the point of attachment of the tube to the conductor being transferred from the plate to somewhere along the wire, this transference of the point of attachment from molecule to molecule imphes the same amount of chemical change within the wire as if the tube entered all at the same moment. It will not, however, take place equally throughout the cross-section as it does in the steady state. Probably we only have the simultaneous disappearance of all parts of a tube when the wire follows a line of electric induction, and has its resistance per unit length proportional to the intensity which would exist there if the wire were removed. The hypothesis here advanced is in accordance with Maxwell's doctrine of closed currents. For the induction dissipated at one part of the circuit has come there from another part where relatively to the circuit it ran in the opposite direction. The total result is equivalent to the addition of so many closed induction-tubes to the circuit, the induction running the same way relatively to the circuit throughout. If the two plates of the condenser are not connected by a wire but are discharged gradually by the imperfect insulation of the dielectric, then we 206 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE must suppose that the tubes of induction in this case are dissipated in situ, the induction simply decaying at a rate depending on its amount and upon the conductivity of the dielectric. We may still represent this process by a closed current by regarding the loss of induction (Maxwell's — j.) and the quantity of induction dissipated (Maxwell's f) as two different quantities. We have then p-{- -J- = ov we have two equal and opposite currents. But this seems artificial. It is more natural to look upon the process merely as a decay of electric induction without movement inwards of fresh induction- tubes, and therefore without the formation of magnetic induction. I have discussed the case of discharge of a condenser at some length, as we can here reahse more easily what goes on at the source of energy. The results obtained suggest that a similar action occurs at the source of energy or seat of the electromotive force in other cases where we do not know the distribution of induction, and are obhged to guess at the action. A circuit containing a voltaic cell. We may pass on from the discharge of a condenser to the consideration of the current in a circuit containing a voltaic cell. The chemical theory of the cell will be here adopted — in fact, the hypothesis I am endeavouring to set forth has no meaning on the voltaic metal-contact theory. Let us suppose the cell to consist of zinc and copper plates, a vessel of dilute sulphuric acid, and copper wires attached to each plate which on junction complete the circuit. For simplicity I shall disregard the effect of the air and suppose that it is a neutral gas causing no induction. We shall begin by supposing the circuit open. Then we know that on immersion there will be temporary currents in the wires, the quantities of these currents depending on the electrostatic capacity of the system composed of the wires. The currents last till the wires have received charges such that they are, say, at difference of potential V. If the terminals are connected to a condenser the temporary currents may be easily detected by a galvano- meter in the circuit. They are in no way to be distinguished in kind from the permanent current which will be estabhshed when the circuit is complete, except that they are of short duration and in general very small. There is no reason then to suppose that the action in the cell is different from that which takes place when the current is permanent, and I think we may safely assume that Faraday's law of electrolysis holds according to which the quantity of electricity flowing along either wire is proportional to the quantity of chemical action — or, in the form appropriate here, the number of tubes of induction produced is proportional to the quantity of chemical action. ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 207 Let Q be the total quantity of electricity upon the positive terminal ; then QV ^r- is the total energy thrown out into the dielectric. Let z be the quantity of zinc consumed per unit of electricity, then Qz is the total quantity consumed in the charging of the terminals. Let E be the energy set free by each quantity z of zinc consumed, after all actions in the cell have been provided for. E then is the e.m.f. which the cell will have on the closure of the circuit, as long as the chemical actions remain the same, for z corresponds to the passage of a unit of electricity or the production of one tube, and we know that the energy set free by C units is CE. Now while the charges are gathering and while the potential difference of the terminals is gradually increasing, the energy required to add equal increments of charge will also increase, and the charging will cease when the amount of energy given up by a given amount of chemical action in the cell is equal to the amount required to add the corresponding charge to the terminals. For to suppose the action to go beyond this is to suppose that the energy thrown out into the space between the terminals is greater than that yielded by the battery. Let dQ be the last quantity of charge added to the terminals. This requires energy VdQ. The corresponding quantity of zinc consumed is zdQ, giving up energy EdQ. The condition of equilibrium is that VdQ = EdQ or V^E, which agrees with the result of experiment that the difference of potential of the terminals in open circuit is equal to the e.m.f. of the cell immediately after closure. It may be noticed that the total quantity of energy extracted from the batterv is QE = QV, while the electric energy left in the medium is QV 2 ' or half the energy has been converted into heat in the wires. We will now consider the distribution of level surfaces in the field while the circuit is still open. There will be F — 1 surfaces between the terminals, dividing each tube into V cells. None of these will cut the homogeneous parts of the circuit, since the whole of each of these must be at one and the same potential. They can only cut the circuit by passing through the regions where there 208 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE is contact of dissimilar bodies. We will neglect the contact of the zinc and copper, as the difference of potential there is insignificant compared with that at the two surfaces, zinc-acid and copper-acid. Now we know that the energy of the cell is put out at the zinc-acid contact, but the amount is greater than that obtained from a consideration of the E.M.F. of the cell, for some energy is absorbed again, probably, at the copper- acid contact in the evolution of hydrogen. There is probably, then, induction between the acid and the zinc, and between the acid and the copper, these resembUng the spaces between the plates of two condensers, the acid being at a higher potential than either. But if a given amount of induction dis- appears from the zinc-acid contact and appears at the terminals, more energy is lost at the former than appears at the latter. Hence all the cells have not been transferred from one to the other, or the difference of potential zinc-acid Fig. 3. is greater than 7. Then more than V — I level surfaces pass between the zinc and the acid, the excess over 7—1 going round and passing between the copper and the acid, somewhat as in Fig. 3, where A, B are the metal plates. The surfaces are roughly sketched and numbered, on the supposition that the zinc terminal is at 0, the copper at 5, and the acid at 8. They have probably the same shape as those which would be produced by condensers at A and B with the wires attached, respectively, to one terminal of each, the other terminals being connected together and the charges adjusted so that the difference of potential of the two terminals at A was 3, while that at B was 8. Let us now suppose the circuit closed. Then the level surface will 'cut the circuit at various points, somewhat as in Fig. 4. The energy being dissipated in the wire, the cell will continually send out fresh energy, the induction-tubes, which proceed from the acid to the zinc. ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 209 diverging outwards in the same way as described in the discharge of a con- denser. They bend round, and finally go into the circuit, the energy they carry being used for the necessary molecular changes, and finally appearing as heat in the circuit — except at the copper-acid contact where there is a crowding in of level surfaces, and therefore a convergence of more energy, which is required to set the hydrogen free. Fig. 4. At the same time magnetic ring-shaped tubes will be continually sent out from the zinc-acid contact, expanding for a time and then contracting again on various parts of the circuit and also giving up their energy. There is, therefore, a convergence of tubes of electric induction on the circuit, running in the same direction throughout, viz., from copper to zinc outside the cell, and from zinc to copper inside, except between the zinc and acid, where there is a divergence of tubes in which the induction runs in the opposite way. But a divergence of negative tubes causes magnetic intensity in the same direction as, and may therefore be considered as equiva- lent to, a convergence of positive tubes. The current may therefore be said to go round the circuit in the same way throughout. The tendency to a steady state in which the current or the number of induction-tubes broken up per second is the same at all parts of the circuit, admits of simple explanation. We know, as the result of experiment given P by Ohm's law, that C' = p where R is the resistance per unit length and E the electric intensity. Until we can explain the molecular working of the current, i.e., the mode in which the induction-tubes are broken up, we must accept Ohm's law as a simple fact. Let us suppose that we have not yet arrived at the steady state, so that in some part of the circuit the electric intensity is less than in the steady state, while in another part it is equal p. c. w. 14 210 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE to it or greater. Let the steady value of the intensity be E, the actual value in the former part E' , and in the latter E" , By Ohm's law the number of tubes absorbed by the wire per second is given by C = E' jR, and C" = E" jR, in the two parts respectively, so that C"< C" since E' < E" or less tubes are being destroyed in the first than in the second part. But all the tubes are sent out from the source of the energy, and are only destroyed in the circuit, being otherwise continuous and with their two ends in the circuit. Hence, if more tubes are destroyed at one part than another, the parts of the tubes not yet destroyed will gather in the medium surrounding the part where fewer are destroyed, increasing the induction there, and so raising the intensity in the wire and therefore the number of tubes destroyed. The field can evidently only be steady when the number of tubes destroyed in all parts of the circuit is the same. But it does not follow that in the steady state each tube enters the wire along its whole length at the same moment. This would imply that the axis of the wire is a line of electric induction perpendicular everywhere to the level surfaces. If we draw the level surfaces due to the seats of induction at the contacts of acid and metal, they will probably be somewhat as drawn in Fig. 4. If now the wire is not so arranged as to follow with properly adjusted resistances a line of induction for these surfaces, but pursues an irregular course, then the level surfaces will be much distorted, and the distribution of the induction will be greatly altered. We may ascribe this alteration to a distribution of electricity along the wire, the quantity in any small area on the surface of the wire being equal to the difference between the number of tubes which have entered and the number which have left that area since the beginning of the system. We have a famihar example of this in the charging of deep-sea cables. Another example is afforded by a condenser with terminals connected to two points in the circuit. The plates of the condenser are then virtually parts of the circuit. The effect of a junction of two wires, say of the same diameter, but of different specific resistances, upon the level surface will resemble that of a charge upon the separating surface. This can be seen in a general way from the fact that the level surfaces must cut the wire with the higher specific resistance at intervals shorter than those at which it cuts the other wire. If there be an insulated conducting body, say a metal sphere, near the circuit, we know that in the steady state there is no electric intensity, and therefore no current within it ; consequently there is no movement of energy and no movement of induction through it. We can see how this condition is arrived at. As the first tubes of electric and magnetic induction come up to the sphere they will enter it, and the parts of the electric induction- tubes thus entering will be broken up, causing a transient current in the ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 211 sphere. The parts of the tubes left in the medium will end on the sphere giving a negative charge on the end nearer the regions of higher potential, and a positive charge on the end nearer the regions of lower potential. This will go on until such charges have accumulated that the sphere becomes itself a level surface. When this point is reached no more energy can enter the sphere, and the parts of the magnetic tubes within it cease to move. The charges formed on the wire or on neighbouring conductors are to be distinguished from ordinary statical charges in this : that their existence depends on the existence of the current, and therefore on the motion of magnetic induction. If the current is stopped by a break in the circuit, so that the motion of the magnetic induction ceases, the electric induction ceases and the charges are all lost. We should expect, therefore, to find that these charges can be described in terms of the magnetic motions which have occurred and are occurring in the system. Current produced by motion of a conductor iyi a magnetic field. We may explain by general reasoning the production of a current by motion of a part of a circuit so as to cut the tubes of magnetic induction. We will consider the simple case of a sHder AB, Fig. 5, running on two parallel rails, ^ ^ AC, BD, with a fixed cross-piece CD, the tubes of magnetic induction running from above downwards through the paper. Let AB move so as to enlarge the circuit. We know from experiment that this tends to Fig. 5. cause a current in the direction ACDB. As AB moves through the field its motion tends to cause electric intensity in the direction BA. At the same time its kinetic energy is being continually converted into electric and magnetic energy which travels to the rest of the circuit there to be dissipated, that is, there must be a divergence of energy from AB. Instead then of a convergence of positive tubes running from B to A, we shall have what is magnetically equivalent — a divergence of negative tubes or tubes running from A to B, their motion outwards being accompanied by tubes of magnetic induction running round in the same way as if there were an ordinary current from B to A. These magnetic tubes must be supposed to move outwards in order to account for the direction of the electric intensity*. When these electric and magnetic tubes converge upon the rest of the circuit they will evidently form a current running in the direction ACDB. * Note added July 15: The above must not be regarded as an attempt to explain the production of electric induction by the motion of a conductor in a magnetic field, but merely as an attempt to show how the induction arising in the moving part of a circuit finds its way into the rest of the circuit. 14—2 212 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE We have here taken, just as in the case of the condenser and the voltaic cell, the lessening of negative induction by its motion outwards, as equivalent to the increase of positive induction by its motion inwards, and we have con- sidered both of them to indicate the apphcation of electric intensity in the same direction in the conductor. If instead of considering AB as a whole we break it up into elements, each element will be a source of diverging negative tubes, and the remainder of AB will, to that element, be a part of the rest of the circuit. Hence some of the energy sent out from the element will converge on and be dissipated in AB, or AB will be heated just as the rest of the circuit. The general equations of the electromagnetic field. We can easily obtain equations corresponding to and closely resembling those of Maxwell by means of the principles upon which this paper is founded. The assumption that if we take any closed curve the number of tubes of magnetic induction passing through it is equal to the excess of the number which have moved in over the number which have moved out through the boundary since the beginning of the formation of the field, suggests a historical mode of describing the state of the field at any moment. Let a, b, c be the components of magnetic induction at any point 0. Consider a small area dy dz close to the point, then the number of tubes passing through the area dy dz will be adydz. This will be equal to the difference between those which have come in and those which have gone out. Let Ldx, Mdy, Ndz denote the numbers of tubes which have cut the lengths dx, dy, dz since the beginning of the system, those being positive which have tended to produce electric intensity in the positive direction along the axes, and those being negative, and therefore subtracted, which have tended to produce intensity in the opposite direction. Let us consider the number which has come into the area OB DC = dydz (Fig. 6). The number which has come in across OB is - Mdy (- because the movement of tubes passing through dydz in the positive direction must be outwards to produce e.i. along OB). The number which has passed out across CD is - (m + -j- dz) dy. The differ- ^ig- ^■ dM ^nce is -^ dydz. The number which has come in across OC is + Ndz {+ because the movement of tubes passing through dydz in the positive direction must be inwards to produce e.i. along OC). The number which has passed out across BD is (n + ^ dy] dz. The difference is - ^ ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 213 The number still passing through dydz is therefore (-^ -^-^ dydz. dy. Equating this to the actual induction through the area, viz., adydz and performing the same process for the corresponding areas dzdx, dxdy, we obtain dM dN\ dz dy dN_dL dx dz a = c = dL dy dM dx (1) Comparing these with Maxwell's equations (vol. 2, p. 216) we see that dM_dN^dH_dG dz dy dy dz ' with two similar equations, F, G, H being the components of the vector- potential. We should obtain Maxwell's equations if we defined F, G, H to be the number of tubes which would cut the axes per unit length if the system were to be allowed to return to its original unmagnetic condition, the tubes now moving in the opposite direction. According to Maxwell, the vector whose components are F, G, and H 'represents the time-integral of the electromotive force which a particle placed at the point {x, y, z) would experience if the primary current were suddenly stopped' (vol. 2, 2nd ed., p. 215). If the electric intensity is produced by the motion of magnetic induction, then our definition of F, G, H will by the second fundamental principle agree with Maxwell's statement. If u, V, w be the components of current — including, of course, under currents, growth of induction — we have from the third principle Maxwell's equations E (vol. 2, p. 233), which on multiplying by /x become when /x is constant , dc dh\ dy da do 4:7TIJLV = ^ -^ 4:7TfJLW db dx dx da dy >• (2) Combining these with equations (1) (as in Maxwell, vol. 2, pp. 236-7), d^ and writing V^ for i^ + i dx^ dy 2 + ^^2' we obtain 214 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE d /dL dM , dN\\ dx \dx d /dL „,,^ d (dL , dM , dJS\ dM^ dy dM dz J dN^ y. dN\ 4^ ^ = _ vw - - r + "^ + -- 1 ^^ dz \dx dy dz J (3) These equations only differ in sign from Maxwell's, and are therefore to be solved in the same way. It is easy to see by substitution that if we assume L' = — ^\\\- dxdydz M' = — ii\\\- dxdydz iV' = — jLt I - dxdydz dx dy then the following will be solutions L = U - M = M'- N = N' ■ dz dH dx dH dy dH dz , J - dxdydz (4) ,(5) It is evident that we may add to the right-hand side of equations (5) 7 ' 77 ' 77 respectively, where </> is any function of x, y, z, since these will disappear from (3) and also from (1). The electric intensity, in so far as it depends upon magnetic motions, will consist of two terms, one depending upon the motion of the material at the point (its components being found as in Maxwell, vol. 2, p. 227, note), the other upon the motion of magnetic induction about the point. We may add a third term, arising from any electrical distribution with a potential ijj. If there is no material motion we shall have d^\ dx dM di/j dt dy ^_# dt dz / dL dt Q y. R (6) dx dy dR , dz = '^ - dxdydz, 'ffdu - dxdydz ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 215 Substituting from (4) and (5) we get r> (ddu 1 ■, 7 7 Id fff/ d dL , d dM , d dN\ 1 ^ , , # ^^-^\\\dt'-r^''^y^'-^Tx\j\[Txlt+ryW^dz^^^^ \du \ . . . Id {[[(dP dQ dR\l ^ ^ ^ _ . _ dxdydz - ^^^j\\ (-^- + ^- + -^) - dxdydz substituting for -^ , etc., from (6). The last two terms cancel each other, and we get du 1 . , . Id [[{(dP dQ dR\l , -, . ,„, or if we put and jj.fr " ' with similar equations for Q and' 7?. If the system is steady 777^ jT^ ;7r are all zero, and then dx' dy' dz' The quantity />, of which 7 is the potential, will be zero within non- conducting homogeneous parts of the field, for there ,_ZP _KQ ._KR ^ 477 ' ^ " 477 ' '^ ~ 477 ' nd ^ ^ dR_i7Tfdl dg dJi\ _ dx dy dz K \dx dy dz) ' since no charges can reside within a homogeneous non-conducting medium. Or, stating it in another way, all the induction-tubes brought into any part of such a medium remain there without dissipation, a charge in a non-homo- geneous medium being due to unequal amounts of dissipation of induction in different parts of the medium. But p will have value at surfaces separating dissimilar substances either in the insulating or conducting parts of the medium. For in the former the induction is continuous, while the intensity is discontinuous, and in the latter the current or rate of destruction of induction may be continuous, but the relation between intensity and current changes discontinuously with the 216 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE conductivity. At surfaces separating insulators from conductors p may have value, as, for instance, at the surfaces of the plates of a condenser with its terminals connected with two points in a circuit, or at the surface of an insulated conductor near the circuit. It is also to be noted that p will have value at the seat of electromotive force. The values of the components of magnetic induction a, b, c are not in any way dependent on p. For taking the first of equations (1) and sub- stituting from (5) we have _dM _dN ^dM^_dN^_ d^ d^^dM;__dN^ ~ dz dy ~ dz dy dydz dzdy dz dy ' '"^ ' where M' and N' depend on the currents in the system and not on the charges. Comparing our equations with Maxwell's we see that the important point of difference is that we can no longer put the quantity corresponding to his 7 1 . 7 K • • . dF ^dG ^dH J equal to zero, J bemg given by -, — h -7 — ^ ~J~ ' This does not affect the determination of velocity of propagation of dis- turbance in a homogeneous non-conducting medium. For in such a medium we shall have df K d^ dt 477 dt ' with corresponding values for v and w. Substituting in (3) the first equation becomes dt dx\ dx dy dz differentiating with respect to t ^ dt^ dt dx [dx dt dy dt dz dt . and putting -^7 = ? + ^, since d^^dQ dR^i^^df dg dj.^ dx^ dy^ dz K \dx ^ dy^ dz)~ within a homogeneous non-conductor. This gives the velocity of propagation of electric induction equal to We can also obtain the corresponding equation for the magnetic induction. (10) ELECTBIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 217 Substituting in (3) for u, v, and w in terms of P, Q, and R, as above, differentiating the second with respect to z, and the third with respect to y, and subtracting dfdQ_dR\^_ fdM _ dN\ ^dt\dz dy)~ ^ \dz dyl' then from (6) ^ d/d^dMd^ddN d^ \_ fdM dN\ dt\dt dz dzdy dt dy dydzj~ \dz dyj' ^dt^Kdz dy) \dz dyj' or from (1) Kyi'^^ = -V^a, (]]) whence the velocity of propagation of magnetic induction is also equal to It would seem that in some cases, such as that of the field surrounding a straight wire with a steady current, the electric intensity may be regarded as entirely due to the motion of magnetic induction, and its components will ^. , . dL dM dN thereiore be -^ , -^r- , -^r - dt' dt ' dt But in other cases it would seem that the electric induction cannot be wholly due to the motion of magnetic induction, and we must therefore introduce the terms involving ifj. If, for instance, the electric and magnetic intensities were inchned at an angle 6, we should have to suppose the electric intensity E to be produced by the motion of the component of magnetic induction I perpendicular to E, viz., /xZsin^, the other component [jlI cos 6 being at rest. To produce intensity E, E tubes must cut unit length in the direction of E per second ; and since the value of the magnetic induction is ^I sin 6, this requires a velocity v, given hj v . [jlI sind = E oi v = E/fil sin 6. Now we can easily imagine a case where E and I coincide, as, for instance, a condenser with its planes parallel to the axis of a wire carrying a current, and its terminals connected with two points in the wire. Here 7 sin ^ = 0, and V is infinite. Or we have to suppose the electric intensity to be produced by the movement of tubes of induction of no intensity with infinite velocity, a statement without physical meaning. But it is, perhaps, worth noting that if we suppose that the electric intensity is produced by the motion of magnetic induction, and that the magnetic intensity is produced by the motion of the electric induction, each carrying its energy with it, the right quantity of energy crosses the unit area. 218 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE For E magnetic tubes, with / sin d unit cells per unit length, will carry p -n IT ^., EI sind 1 ,£ j_-, across unit area m the plane of E and / a quantity — ^ , or halt the energy which actually crosses the plane. If I sin 9 is due to the motion of electric tubes, then I sin d/i-n- tubes must cut unit length in the direction of I sin 6 per second. The number of unit cells per unit length is E, and therefore the motion of the tubes will carry a quantity ot energy — ^ , or the other half actually crossing. The equations which have been obtained in the foregoing manner by the aid of the hypothesis of movement of magnetic induction may also be obtained without any special hypothesis as to the motion of the induction- tubes, merely assuming that growth of induction through a curve is accom- panied by electric intensity round the curve. Instead of connecting L, M, N with the number of tubes which have cut the axes, we start with the following definitions : Let L, M, N denote the time-integrals of the components of the electric intensity parallel to the axes since the origin of the system, so that L = \pdt, M = JQdt, N = ^Rdt, ,. J. dL ^ dM -^ dN then p = Q = -,^ , R = ^^ , dt ^ dt ' dt If a, b, c be components of magnetic induction, since the growth of in- duction through a curve is equal to the Hne-integral of the electric intensity round a curve in the negative direction, we have (h^dQ_dR^d /dM _ ^\ dt dz dy dt \ dz dy j with corresponding equations for -j and -j . Integrating with respect to t from the origin of the system, when all the quantities were zero ^dM_dN dz dy '-S-S^ <>■) _ dL dM dy dx equations the same in form as equations (1). ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 219 As before we obtain equations (3), (4), and (5), while instead of (6) we have the simple equations P = -^rr and the two others. dt dL Substituting for -j- we obtain an equation of the same form as (7), which may also be put into the form (8). Equations (9) and (10) will also follow. Just as we have obtained equations by considering the growth of the magnetic induction to its present state so we may obtain corresponding equations by considering the growth of the electric induction. Let Adx Bdy 477 = Cdz be the algebraic sum of the number of electric induc- 477 ' 477 ' 477 tion-tubes which have cut dx, dy, dz drawn from a point in such a way as to create magnetic intensities in the positive direction along dx, dy, dz. The excess of the number of tubes which have passed in over those which have passed out through the boundary of any area will be equal to the time- integral of the total current through the area. The components of the total current are V + de dg dt' w r + dh dt' f, q, and r being the components of the conduction-current or the number of tubes dissipated per second, and/, g, h the components of the induction actually existing. As in the last case, if we put/'== \udt, etc., we at once obtain the equations 477/ dC dy dJB dz ^ dz dx ^h' = dB dx dA dy \ ,(12) Corresponding to the current-equations (2) we have three equations obtained from the condition that the rate of increase of magnetic induction through an area is equal to the integral of the electric intensity round it in the negative direction. These are da _ dQ dR dt dz dy dh _dR _dP dt dx dz dc_dP_dQ dt dy dx .(13) 220 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE If C^ is the specific conductivity we may by Ohm's law put the current- equations after integrating in the form KQ 4:77 ■ KR g'= C.JQdt J 47r whence in media where K is constant dg' dh' _ ^(dQ dR dz dy ' }\dz dy dt K(dQ_ 477 \dz dR dy. K da with two similar equations. Finding the values of the left-hand side from (12) we obtain 47rC,a + iiC^ = -VM ' dt d fdA dx \dx dB dC\\ dy dz) ' dt dy\dx dy dz) irrCn + K dc dt -V^C dy d fdA dy dB + dC If we assume 477 /// dz\dx ' dy ' dz da\ 1 iirC .a + X^j- dxdyd with corresponding values for B' and C and J^ [[[fdA dB dC\\ 4:7TjJJ\dx dy dz M 1 - dxdydz, then A' B' C dL^ dx dM dy dN dz (14) (15) are solutions of (13). We may obtain by substitution from (15) in (12) values for/, g', h' corresponding to the values of the magnetic induction in (9), viz. : dC _ dR dy dz ' w ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 221 and two others ; where A', B\ C are given in terms of the magnetic induction as above. It is only in special cases, such as that of a straight wire with a steady- current, that the magnetic intensity will be equal to 47r times the number of electric induction- tubes passing through unit length per second. In all cases the Une-integral of the magnetic intensity round a closed curve is equal to 47r times the number of electric tubes passing through the boundary, but the electric tubes may be more crowded in some parts than in others, while the magnetic intensity is not altered in a corresponding manner. For instance, the magnetic tubes will be continued through an insulated conductor in the field, while in the steady state no electric tubes pass through it. But each element adds to the hne-integral the quantity which, after Mr. Bosanquet, I have called the magnetomotive force, this being equal to hr times the number of electric tubes passing through the element. But it only adds it on integrating round the whole of the closed curve. The intensity at any point will therefore be the resultant of the intensities produced by the magnetomotive forces in the various elements. Perhaps the simplest mode of finding it is as follows. The components of the magnetomotive force produced in a cube dx, dy, dz parallel to the three edges will be dA . dB . dC . W^^' W^^' Tt^'^ for T- , -. T— , -r- -T- are by definition the rates at which electric tubes 47r dt ' i^ dt ' 477- dt -^ are cutting unit lengths parallel to the axes. But these magnetomotive forces would be produced by currents round the cube in planes perpendicular to the axes respectively, and equal to I dA , ^ dB . 1 ^C , 4.rW^^' SrW^^' i^Tt^'^ for the Hne-integral of the intensity round a curve threading a current is 47r X current. But the magnetic intensity at any point due to a current is equal to that of a magnetic shell of strength (i.e., intensity x thickness) equal numerically to the current bounding the shell. If we suppose the thickness of the shell equal to that of the cube, the effect is the same as if the cube were magnetised with intensity having com- ponents ]^dA l^dB \_dC 47r dt ' ^ dt' 47r dt ' 222 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE ^-m- The potential of such a distribution of magnetisation is (Maxwell, vol. 2, p. 29, equation (23)) dA dp dBdjp dCdjp\ Wdx^dtdy^ dt dz) ^''^^^''' where p = -, and the magnetic intensity is given by ""' dx' ^ dy' '^ dz' It may be noticed that in a steady field ~j-, -^, rr are all zero, so that 47rjj. d dM dp d^dM dp d^dM dp\, dx dt dx dy dt dy dz dt dz) ^ ' We may obtain equations of the same form as those given in (14) without any hypothesis as to the movement of electric induction- tubes, merely assuming that the total current through a curve is equal to 477 x line-integral of magnetic intensity round the curve. We start with the following definitions. Let A, B,ChQ the time-integrals of the components of magnetic intensity since the origin of the system. Then A=^jadt, B=l^dt, C^jydt, and a dA. n_dB _dC dt' ^~lt' '^~Tt' We have the equation ^ttu = ^ ^ dy dz and two others. Integrating with respect to t we have . r ,, , ., dC dBs J -^ dy dz \ also 4^.' = ^ _ ^ I dz dx I 4 y ._ ^-^ ^^ dx dy which are of the same form as (12). Hence exactly as before we obtain equations (14) and their solutions (15). The equations for the magnetic intensity are now ^_dA dB dC ""- dt^ p = iu' y^it' ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 223 If we differentiate (14) with respect to t, and substitute from these equations for magnetic intensity, we obtain with corresponding equations for ^ and y. Differentiating the second of these with respect to z, and the third with respect to y, and subtracting, we obtain with corresponding equations for v and w. These correspond to Maxwell's equations (7), p. 395. In conclusion it may be remarked that the equations found in this paper give the same expression for the rate of Transfer of Energy as that in my previous paper derived from Maxwell's equations involving F, G, and H. 12. DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR. [Birmingham Phil. Soc. Proc. 5, 1885, pp. 68-82.] [Read December 10, 1885.] Maxwell has shown that the phenomenon known as the Residual Discharge may be accounted for on the supposition that the dielectric is an imperfect insulator in which the conductivity varies in different parts. His theory is really quite simple and straightforward and free from any hypothesis beyond the fundamental one of electric displacement. But its very generality makes it, I beheve, difficult to grasp. The idea of a yielding of displacement in the dielectric, accompanied by a conduction-current in the opposite direction, gives us no help in forming a mental picture of the process actually going on in the dielectric. A hypothesis as to the nature of electric current, which will shortly be pubHshed in the Philosophical Transactions, seems to me to render the theory easier to follow, and I propose in this paper to arrange Maxwell's account of the Residual Discharge in accordance with it. I shall first give some account of the hypothesis referred to in the special case of the discharge of a condenser. Let us imagine that we have two conductors, A and B, which we may suppose to be the two plates of a con- denser, charged with equal and opposite amounts of electricity, that of A being positive. Then the lines of force will run from A to B through the medium, the condition of the medium being described by saying that there is 'electric displacement' from A to B. Or we may describe it without introducing the confusing term 'displacement' by returning to Faraday's term 'induction.' We may then say that tubes of electric induction pass through the medium, each tube starting from + 1 of electricity on A, and ending in — 1 on B. The total induction across any section of a tube is then always equal to L If we draw the level surfaces at unit differences of potential the tubes will be divided up into cells, and if we suppose each cell to contain half a unit of energy then the whole energy of the electrified system is accounted for. Maxwell has called these unit cells {Elementary Treatise on Electricity, p. 47). According to the views of Faraday and Maxwell, the charges on the conductors bounding the dielectric are to be regarded as the surface-manifestations of the altered state of the dielectric corresponding to DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR 225 the energy put into it, somewhat as the pressure on a piston in the wall of a closed vessel of compressed water might be regarded as the surface-mani- festation of the strained condition of the water. In order to follow out the process of discharge in the medium, i.e., the mode in which it is relieved from its strained condition, we will first take a simpler ease in which we connect the two plates, A and B (Fig. 1), of one condenser to the two plates, C and D, of another condenser previously- uncharged, and so far from A and B that there is no appreciable direct induc- tive action on C and D. When equilibrium is again restored the + charge is shared between A and C, the — charge between B and D, while the difference of level has decreased. There is the same total number of tubes of induction, but each contains fewer unit cells than before, the energy corresponding to the decrease having been transferred to the wires, where it has been dissipated as heat. I shall use the term energy-length to indicate the line-integral of the electric intensity along its axis, this being the same as the difference of WIRE lllllllllimiTTTT ""™=n ■■""■ i""iijij f imimiijiiiiimirn potential when there is equilibrium. We may say then that the energy- length of the tubes has decreased. During the change some of the electric energy was converted into magnetic energy in the medium. This might be observed if sufficiently delicate means were used. If we confine our attention to the charges on the conductors we must say that equal quantities of + and — have moved respectively from ^ to C and from B to D along the wires. But taking into account the condition of induction in the medium, described by the induction-tubes, we must say that the induction- tubes move sideways out from the space between A and B into the space between C and D, the motion of the charges along the wires being really the motion of the ends of the induction- tubes. (See Fig. 1, where 1-6 may be taken as successive positions of a tube.) During the motion of the tubes some of their energy was converted into the magnetic form, the co-existence of the two forms, electric and magnetic, being a necessary condition of motion. We may illustrate this from the analogous case of a strained incompressible solid which can be sheared. If there is any mode of escape given to the strain- 15 p.c.w 226 DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR energy by a slipping of the surface against the constraint, then the state of strain will be propagated outwards from the interior of the sohd, but some of the strain- energy will be converted into kinetic energy, and the presence of the two is a necessary condition for the propagation of the strain. Since the energy-length of a tube diminishes as its ends move along the connecting wires, we may represent this by supposing that parts of the tubes move into the wire. If a similar motion of electric induction took place into a dielectric it would remain, and the dielectric would become electrically strained, but in the wire the strain breaks down rapidly, the energy being converted into heat. I think there is good reason tO suppose that it is the electric energy which thus breaks down, the magnetic only being dissipated after it has been reconverted into the electric form. We may now consider the case in which total discharge of a condenser takes place through a connecting wire. Considering merely the conducting plates and the wire, we say that the charges move along them towards each other and finally unite, neutralising each other and producing heat in the wire. Regarding the medium we must suppose the tubes of induction to move sideways towards the wire, shortening as their ends, which are repre- sented by the charges, approach each other, and finally disappearing into the wire. Faraday describes the process by saying that 'when current or dis- charge occurs between two bodies, previously under inductrical relations to each other, the lines of inductive force will weaken and fade away, and, as their lateral repulsive tension diminishes, will contract and ultimately dis- appear in the line of discharge.' [Exf. Res. vol. 1, p. 529, § 1659.) The so-called velocity of electricity is merely the velocity of the ends of the tubes, and this may evidently vary according to the nature of the circuit. It is quite conceivable that if the wire be in a neutral medium, i.e., one in which there is no surface-difference of potential, say gold in air, and if it follow the direction of a tube of induction, then a tube may move into the wire throughout its whole length at once. In this case the 'velocity of electricity' would be infinite. We know from experiment that if a galvanometer be inserted in the connecting wire then the same magnetic impulse is observed wherever in the circuit the galvanometer be placed, the impulse depending on the galvanometer-constant and on the total discharge. The same experimental result may be stated in an equivalent form, viz., that the line-integral of the magnetic intensity round a closed curve encircHng the wire if integrated for the time of discharge is the same for all positions of the curve. On the hypothesis here described all the electric induction-tubes of the system finally pass sideways from the medium into the wire. They must, therefore, on their way pass inwards across any curve encircling the wire, so that the total number of induction-tubes cutting such a curve is the same for all DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR 227 positions of the curve. In the paper above referred to I have sought to connect these two constants by supposing that the magnetic effect is due to, or more correctly accompanies, the motion inwards of the condition of electric induction. As soon as motion commences some of the electric energy is converted into magnetic, and the magnetic induction may be represented by ring-shaped closed tubes surrounding the wire. The two inductions, electric and magnetic, co-existing, will propagate the energy onwards till it finally arrives in the wire and is dissipated as heat, the induction there losing its directed condition. The flowing of electric charges along the wire, which is usually considered as the essential part of the phenomenon, or at least that to which attention is to be chiefly directed, becomes on this hypothesis merely the last stage in the process, which consists of a propagation from the surrounding dielectric towards the wire of electric and magnetic induction, which we may symbolise by the motion inwards of two sets of tubes, the electric tubes being, on the whole, more or less in the direction of the wire, the magnetic tubes being closed rings surrounding it. The wire plays the part of the refrigerator in a heat-engine, turning the energy it receives into heat — a necessary condition for the working of the machinery. Let us now take the case of a condenser in which the dielectric, though homogeneous, is imperfectly insulating, so that the charge gradually dis- appears. According to Maxwell, in this case 'induction and conduction are going on at the same time.' Though Maxwell gave no precise account of the process of discharge, his theory and the mechanical illustration accom- panying it are based on the supposition that two processes are going on at the same time in every part of the medium, viz. : (1) a yielding of the electric strain or 'displacement' in the dielectric, equivalent to a displacement- current from the negative towards the positive plate, and (2) a conduction- current from the positive plate to the negative equal to (1) in amount. This latter is accompanied by dissipation of energy. The two equal and opposite currents being superposed have no external magnetic effect. But it seems to me that we may equally well and more simply represent the facts by considering the first process alone, viz., the yielding of the electric strain, the medium being incapable of bearing it permanently. The electric energy is gradually converted into heat in the same part of the dielectric where it was previously electric, i.e., there is here no transfer of energy. The decrease of induction in the medium is accompanied by a corresponding decrease of charge on the plates, not by conduction of + or — electricity either way through the medium, but simply because there is a decrease of the induction in the medium of which the charges on the plates are the surface- manifestations. The induction decreases equally through the whole length of a tube, so that the tube 'weakens' at the same rate throughout its length. 15—2 228 DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR There will be no magnetic effect in the surrounding space for there is no movement inwards of electric induction-tubes to supply the place of those which decay. Perhaps we may take the following as illustrating the two modes of regarding the process. Suppose that a solid is submitted to some strain and kept in the strained position, but that the energy of the strain gradually dissipates ; then we may confine ourselves simply to the statement that owing to some rearrangement of the molecules they cease to have molecular strain- energy, the energy in each portion of the mass being transformed into heat in that portion, or we may imagine that there is a continual return from the strained towards the original position, accompanied by an equal reverse flow of the matter towards the strained position, this latter not storing up energy but dissipating the energy given up by the yielding of the strain. The ultimate result according to each is the same, but the latter account is purely hypothetical. We may at once obtain the equation giving the value of the charge at any time in terms of the initial charge when the condenser is left insulated. Let a be the charge per unit area, this being equal to the electric induction across unit area in the dielectric. Let K be the specific inductive capacity. Let X be the electric intensity in the dielectric, i.e., force per unit electricitv on a small electrified body. We have ^ "^ ^ (1) Now we know that the rate of decrease of charge on the ends is proportional to the charge a and therefore to X. The decrease of charge or of induction in the medium is therefore (it r' ^-^ where r is a constant, which we may term the specific resistance. Hence from (1) ^ + ^ ^^ 0, (3) or CT = GToC ^^'^ (4) If we use p to denote the decrease of induction per second, _ da _ _KdX KX^ The energy per unit volume is -^— ; its rate of decrease is therefore KX dX "l^TW (^> DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSLTLATOR 229 Substituting from (2) and (5) we get the expression which here corresponds to Joule's law for the heating effect, viz., rate of decrease of electric energy per unit volume = pV. If at any moment the two end-plates be connected by a wire, transfer of induction will at once take place into the wire, and the whole system will be completely discharged. During this discharge there will be magnetic energy accompanying the motion of electric induction. We will now investigate the more complicated case of a stratified dielectric in which the different layers have different specific resistances. Before proceeding to the mathematical account we shall consider the process generally, taking the simple case in which K is the same throughout. Let the con- denser be charged very rapidly and then insulated. At the first moment there will be equal and opposite charges on the two end- plates, and the number of induction-tubes running through unit area parallel to the plates will be the same in each layer. But decay of induction, and dissipation of energy, at once sets in, the rate of decay varying in different layers, so that after a time the number of induction-tubes in contiguous layers will differ and there will be charges on the separating surfaces. In those layers where the rate of decay is most rapid there will be negative charges on the surface nearer the + plate, and + charges on the surface nearer the — plate. But still the induction in all is in the same direction. Now let the two end-plates be connected by a wire. At once induction is propagated into the wire and transference takes place from the space between the plates until they are at the same potential, i.e., until the line-integral of the electric intensity, or, since K is constant, that of the induction, from plate to plate is zero. The same number of tubes must have entered all parts of the wire, otherwise there would be charges at points along its length. Hence the same number of tubes running in the positive direction must have passed out from each of the layers. The result must be a reversal of the induction in some of the layers, viz., in those in which the induction decayed most rapidly. This, of course, means that after their positive induction has all flowed out and they are quite discharged, tubes from the other layers have bent round and entered them, now charging them in the opposite direction. We may imagine the process to be somewhat as in Figs. 2 and 3, representing a condenser with three layers, A, B, C, the decay having been most rapid in the middle one, so that it has become completely discharged, while there is still positive induction in A and C. 1, 2, 3, 4 (Fig. 2) represent successive positions of a tube moving out from A towards the wire ; T, 2', 3', 4', successive positions of a tube moving out from C. When they have taken up the positions 4, 4' they come in contact, and where they overlap they will neutraHse each other and break up into two portions, the outer part of each forming one 230 DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR positive tube, as 5, Fig. 3, which will move off to the wire, inner parts uniting to form a negative tube 6 in B. When the difference of potential between the end-plates is zero, suppose the wire to be removed. The induction still remaining decays. If it decayed in the same proportion throughout, the difference of potential would always remain zero. But it decays in greater proportion in the negative layers, since in these the dissipation is, by hypothesis, most rapid. Hence in the line- integral of the induction from plate to plate the negative terms decrease more rapidly than the positive, and so the total value becomes positive. Then on WIRE Fig. 2. Fig. 3. again connecting with a wire another positive discharge occurs, may evidently be repeated, the discharge always being positive, until finally it becomes insensible. The process p r r A r r r B r r Fig. 4. The analogy between the residual dis- charge and the phenomenon of elastic recovery in strained solids, pointed out by Kohlrausch, suggests a simple illus- tration. Suppose that we build up a cube with successive layers of substances with the same instantaneous rigidity but with different viscosities. Let this be placed between two plates, A, B, Fig. 4, the lower plate being fixed. Let rigid transverse partitions, r, r, be passed through the layers and attached by hinges to the two plates, and then let the upper plate be acted on by a force in a direction perpendicular to the partitions, so that a shearing strain is given to the whole cube. The partitions, r, r, are merely put so that the distortion from the original position shall always be the same throughout. When a given strain has been produced let the upper plate be also fixed. Now if the rate of dissipation of strain- energy were the same throughout the layers the stress would also be the same throughout, though gradually DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR 231 decreasing, and on removing the constraint the upper plate would return by a certain amount and then remain in its new position. But the dissipation is not uniform and after a time the stress in some of the layers is greater than in others. Hence, on removing the constraint from A and allowing it to return, when those in which dissipation has been most rapid have become entirely free from strain-energy, there is still some remaining in the other layers. These latter will, therefore, strain the former, and we shall have a reverse stress in some of the layers. Thus A will come to a new position of equilibrium, not so far, however, as its first position. Suppose that it is now again fixed. At first no force is necessary to keep it in position, but the stress exerted by the negative layers decays more rapidly than that exerted by the positive, and soon, on being released, A will return still further towards its original position. The process may be repeated, the successive discharges of momentum imparted to A being always in the same direction. (Added April 16th, 1886. The supposition of stratification made by Maxwell is, no doubt, very artificial, and was made for the sake of simplicity in the mathematical treatment. He states that 'an investigation of the cases in which materials are arranged otherwise than in strata would lead to similar results, though the calculations would be more complicated, so that we may conclude that the phenomena of electric absorption may be expected in the case of substances composed of parts of different kinds, even though these individual parts should be microscopically small. ' It by no means follows that every substance which exhibits this pheno- menon is so composed...' (Electricity and Magnetism, 2nd ed., vol. 1, p. 419). Probably in the case of blown glass or any dielectric made up of hetero- geneous parts, which has been flattened by rolling, there is more or less approach to the stratified condition, but in other cases, such as shellac or paraffin, we might fairly expect the dielectric to be similarly constituted in all directions. We can only, therefore, take Maxwell's investigation as showing in a general way that heterogeneity would introduce absorption phenomena, and we cannot expect the results obtained on the supposition of such a special arrangement to agree with those of experiment. We may regard the stratified arrangement as giving a superior limit, as it were, this being the constitution most favourable to the production of the phenomena in the way supposed. The inferior limit would be given by an arrangement in which each portion of the substance of the same kind stretched from plate to plate with the same cross-section throughout. In this case there would be no residual discharge produced. Using Maxwell's notation (see below), the resistance per unit cross-section may be shown to be ^ + -'+ ... 232 DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR instead of R = a^r^ + a^rz + ... and R' is always less than R. In any intermediate composition in which portions of more conducting matter are insulated from each other by less conducting matter we shall have residual discharge. It appears probable from experiments of Dr. Schulze-Berge {Nature, March 4th, 1886, p. 432) that the resistance of certain dielectrics is not pro- portional to the thickness, but is much less for thin layers than might be expected. May this not possibly arise from the size of the heterogeneous portions being comparable with the thickness of the dielectric, so that the more easily conducting portions may stretch in some parts from plate to plate? If so, we approximate more nearly to the inferior limit.) The mathematical account of the residual discharge on this hypothesis is practically the same as Maxwell's, but it may, perhaps, be worth while to give it with the necessary alterations, as these seem to make it somewhat more straightforward and evident. We shall suppose with Maxwell, ' for the sake of simplicity, that the dielectric consists of a number of plane strata of different materials and of area unity,' and that the induction is in the direction of the normal to the strata. Let a^, a^, etc. be the thicknesses of the different strata. Let Zi , Z2 , etc. be the electric intensity within each stratum. Let 2^1, 2^2 5 ®tc. be the amount of decay of induction per second in each stratum. Let/1,/2, etc. be the induction in each stratum. Let %i-^, ^^2 5 ^^^- b^ ^tie total number of tubes of electric induction entering each layer sideways, i.e., crossing in through its boundary, per second. Let r^, r^, etc. be the specific resistance referred to unit of volume. Let K^, K2, etc. be the specific inductive capacity. Let ^1, ^2 5 ^tc. be the reciprocal of the specific inductive capacity. Let E be the electromotive force due to a voltaic battery placed in the part of the circuit leading from the last stratum towards the first, which we shall suppose good conductors. Let Q be the total number of induction-tubes which have left the battery and entered the wires and dielectric up to the time t. Then since the same number of tubes enter all parts of the circuit in a given time, u^ = u^^ u^= ... ^ u say (1) These tubes tend to increase the induction in the layers. But at the same time decay is going on so that we have % = Vi + §, ^h = V2 + 5, etc., (2) DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR 233 whence 2?i + ^ = 2^2 + -^^ = etc (3) We also have by Ohm's law Vi = ^> :P2=v^> etc., (4) and by the relation between induction and intensity X,= ^7rKh, (5) whence '^=-^+ir^ tt (6) Let us suppose that at j&rst there is no charge and that suddenly the E.M.F. E is made to act. Then if at once Q tubes enter the dielectric, Zi= iTTk^Q, etc., (7) and since £■ = a^Zi + 0^2X3 + ..., (8) E ^ 4:77 (^1% +^52^2+ ...)Q. The instantaneous capacity C which is equal to ^ is given by C= . .. J. — -. (9) i7r{k^ai + k^a^-i- ...) But dissipation at once sets in, and if the electromotive force E be continued uniform a steady state will ultimately be reached in which the dissipation in each layer is equal to the number of fresh tubes reaching that layer. The number of tubes entering being the same throughout, the dissipation p is also the same throughout. We have then p = ^ = ^ = etc., (10) and substituting in (8) E = (/!% + r2^2 + •-•) V- Hence if R = r^^a^-^- ..., P-R <^^' In this state we have the induction given by '^^^ ^7Tk^~ ^irkj ^irk^R ^ ^ If we now suddenly connect the extreme strata by means of a conductor of small resistance, E will be suddenly changed from the value Eq to zero and Q' tubes will pass out from each layer of the dielectric into the wire. If then X' be the new value of the intensity, O' = ^1 _ ^1 ^ iTTk^ iirk^' whence * Z/ = Zi - 477^^1^' (13) 234 DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR Since then the difference of potential is zero a^X^ ^a^X^ -V .,. = 0, substituting from (13) we get ^iZi + a^X^ + ... = 477 («!% + ajk^ + ...)Q' or ^ %Zi + a,Z, + ... Q^^Q (14) from (8) and (9). Hence the instantaneous discharge is equal to the instantaneous charge. By (10) and (11) we may put (13) in the form Zi' = fV^ - iTT^Q = ^^-i^k,Q (15) Let us next suppose the connection broken immediately after the dis- charge. No fresh tubes enter any layer, so that putting t^ = we have from (6) = Zi 1 dX^ ^1 477A;i dt ' 4TrA;i , or Zi = Z/6 where Zi is now the value of the electric intensity at any time t after the connection is broken. Substituting from (15) and putting Eq for the initial value of E, X,==E,(^^-i7Tk,Cy~'^ (16) The value of E at any time is E = ^iZi + a^X^-^ ... = Eo (f"^'^ - ^^a,k,c) e-'-f' + (^^^^- - 477^2^1^26') e i-rrko E, {^-^ - 477;l^iCJ a,e r. + Q - 477y^2C) a,e -^ ' + ..\ EM-.^CnY-^^e-"^^' ^\ (17) A J R The instantaneous discharge obtained at any time t will be, as before, CE. If the terms be arranged in descending order of magnitude of ~\ then the exponentials are also in descending order of magnitude, or the negative terms decrease more rapidly than the positive, and E is positive. 13. ON THE PKOOF BY CAVENDISH'S METHOD THAT ELECTRICAL ACTION VARIES INVERSELY AS THE SQUARE OF THE DISTANCE. [British Association Report, 1886, pp. 523-524.] The proof of the law of electrical action depending on the fact that there is no electrification within a charged conductor was first given by Cavendish. His proof was made more general by Laplace, who has been followed by other writers, including Maxwell. Maxwell and MacAlister have also verified the experimental fact, repeating an investigation of Cavendish only recently pubHshed in Maxwell's edition of the Cavendish papers. The proof may be analysed in the following way : Take the case of a uniformly charged sphere. The action at a point within it may be considered as the resultant of the actions of the pairs of sections of the surface by all the elementary cones, with the point as vertex. If, then, the resultant action is zero for all points and for all sizes of the sphere, it follows that the action of the pair of sections by each elementary cone is zero ; and, since the sections of the surfaces are directly as the squares of the distances, the two sections neutraHsing each other, the force per unit area must be inversely as the squares of the distances. There appear to be two objections to this proof. (1) That it takes no account of the always existing opposite charges. When the sphere, for instance, is positively charged, an equal and opposite negative charge is on the walls of the room, and the action of this should be considered. Probably this objection could be removed. (2) There is a solution still simpler than the inverse square law — viz., that no element of the surface has any action within the closed conductor. If we suppose that a conductor is a complete screen to electrical action, then, whatever the law of the force exerted across an insulator, there will be no action within the conductor. In any null proof it is not sufficient merely to show that there is no action in the null arrangement, but it is also necessary to show that on disturbing the null arrangement some action is manifested. Now, in the case here considered it is impossible to obtain any action within the conductor in any statical arrangement; it is only during changes of the system while charging or discharging that we can get a disturbance of the null arrangement. But here new phenomena come 236 ELECTRICAL ACTION VARIES INVERSELY AS THE SQUARE OF DISTANCE in, for we have currents, and therefore electromagnetic action. But, even disregarding the different kind of action occurring, the only experiment which I know of on this point was that of Faraday with his electrified cube. While the most violent charges and discharges were taking place on the outside of the cube, so that the null arrangement was probably disturbed, he found no action on his electroscope within. Possibly the actions were alternating, and so rapid that no electroscope of ordinary construction would reveal them. But he himself went into the cube, and he would probably be sensitive to rapidly alternating electromotive forces. It appears to me, then, that we cannot accept this proof, and must fall back upon the more direct proof of Coulomb*. I do not know whether Maxwell was aware of this objection ; but it is worthy of note that in the remarkable fragment published since his death, as An Elementary Treatise on Electricity, he returned to Coulomb's proof, and was apparently building up the mathematical theory of electricity in a way quite different from that followed in his larger work. * [In his lectures on electrostatics Poynting used to give an experimental proof somewhat differing from that of Coulomb, and simpler. A brief account of Poynting's apparatus will be found in Electricity and Magnetism by Poynting and Thomson, vol. 1, pp. 65 and 66. Ed.] 14. ON A FORM OF SOLENOID-GALVANOMETER. [Birmingham Phil Soc. Proc. 6, (1888), pp. 162-167.] [Read May 10, 1888.] The instrument described in this paper is a form of solenoid-galvanometer in which the iron core is still far from saturation, so that the attraction of the core by the coil is nearly proportional to the square of the current. The peculiarity consists in an arrangement by which a pointer moves over a scale a distance not very far from proportional to the current. The moveable core of the solenoid consists of an iron rod or bundle of wires, and is suspended by a silk fibre, which is wrapped on to the circumference of a small wheel with a horizontal axis turning in bearings as free from friction as possible. The wheel has an arm (Fig. 1) with a moveable bob on it, and the bob is so adjusted that the weight of the iron core just balances it when the arm is horizontal. The equilibrium is of course unstable, and a stop S is necessary jiist above the arm when in the horizontal position. The arm ends in a pointer moving over a divided quadrant. The solenoid is placed above the iron core so as to act against its weight, and in the position of maximum pull. The coil is moveable up and down by means of a screw, so that it may always be put in this position of maximum pull. The current passing through the coil does not saturate the iron, and the upward attraction is therefore nearly proportional to the square of the current. Let it be equal to KC^ where K is a constant for the particular instrument. If W is the weight of the core, a the radius of the wheel, w the weight of the wheel and bob, and b the distance of its centre of gravity from the axis, the condition for equilibrium when no current passes is Wa = wb (1) If now a current C passes, the down pull of the core is lessened and the arm falls into a position in which the bob has a less moment. If it moves through an angle 9, (W -KC^) a = wb cos (2) 238 ON A FORM OF SOLENOID-GAL VANOMETEE Substituting from (1) for Wa, KC^a = wb{l- cos 6) = 2wb sin^ e or From this ^ / 2wb . 6 de = 2 Ka dC cos which only very gradually increases with 6, and when 6 = 90° it has a value '\/2 or 141 times its value at 0°. Fig. L ON A FORM OF SOLENOID-GALVANOMETER 239 It is very easy to construct an arc divided to give readings proportional to sin ^ as follows : — Describe a quadrant, and mark off points on it with ordinates increasing by equal amounts. On the radius from which these ordinates are measured describe a semicircle. Drawing the radii of the quadrant to the successive points marked, they will intersect the semicircle in points with equally increasing values of sin ^, 6 being the angle subtended at the centre of the semicircle. In practice it would no doubt be better to graduate by trial, having a standard instrument in the circuit. The instrument shown, though faulty in several points and far from frictionless, works fairly well. The range is limited by the fact that unless the adjustment is very perfect, the readings cannot be trusted below 10° or 20°, but I think the principle might be usefully adopted for voltmeters of small range, or for ammeters, to give a correct value for a current within a small range. It might be useful to extend the range by a counterpoise to part of the weight of the core, on the other side of the wheel. The instru- ment has the advantage that, when the current is passing, the pointer very rapidly comes to rest. A Suggestion for a Wattmeter. The above instrument has suggested to me a possible form of wattmeter which I have not seen described before. I have not yet constructed an instrument on this plan. A soft iron core is fixed vertically at one end of a steelyard, with a moveable counterpoise as usual on the arm beyond the knife-edge. Two co-axial solenoids, one of high and the other of low resistance, are fixed in the position of maximum pull on the core when the arm is horizontal. For stability they should be above the core. The ends of the high-resistance-coil are connected to the two ends of the circuit in which the rate of working is to be measured, a commutator being interposed so as to reverse the current in the coil. The low-resistance-coil forms part of the main circuit. When the currents pass in the same way through the two coils, the pull on the core will be {aC + hEf, where a and h are constants for the solenoids. The counterpoise is to be adjusted for equilibrium. The current now being reversed in the high- resistance-coil, and the counterpoise being again adjusted, the pull on the core will be (aC - hE)\ 240 ON A FORM OF SOLENOID-GALVANOMETER The distance through which the counterpoise has been moved will be pro- portional to the difference between these two pulls, or to 4:abCE, i.e., to the rate of working CE. A Square-Root Steelyard. Some years since another arrangement occurred to me for obtaining an equally divided scale, giving directly the square root of the pull on a soft iron core or on a moveable coil. After recently constructing a model, I found it was only a particular case of the very remarkable machine for solving equations, devised and constructed by Mr. Boys (Philosofhical Magazine, vol. 21, 1886, p. 241). Being, however, a very special case, it is less com- plicated than the general instrument, and as the model works easily and correctly, it may be worth while to describe it. ABC (Fig. 2) is a lever balancing on a knife-edge at B, and the pull W, of which the square root is to be measured, is applied at the end A. GE is C D B A H G F y^^^.y^yy.^y^yf/yZ'yyy/^y^^yyy^/y^.^^ Dp ^ ^ w Fig. 2. another lever balancing on a knife-edge at G, the arm GE being about equal to the arm BC. The plane upon which G rests is, in the model, a plate of glass, about equal in length to BC, and so arranged that GE may be moved until E is under any point of BC. E is connected by a link DE with BC, and from E , exactly under C, hangs a fixed weight P. If the down pull of the link at I) is T, and w is its weight, the up pull dXEi^T — w. The equations of equilibrium of the two levers are and or if GR be made equal to ^ GE. W . AB ^ T . BD (T-w)GE = P. GF, T .GE^P.GF = p(gf = P.HF w (1) .GE ge] (2) ON A FORM OF SOLENOID-GALVANOMETER 241 Multiplying (1) and (2) together, T is eliminated and W .AB.GE = P .BD.HF, Making HE equal to BC, and keeping P always exactly under 0, BD is equal to HF, and W .AB.GE = P,BD^: BD y AB,GE Vw, If then BC is equally divided, the equilibrium position of D gives a reading proportional to the square root of W. In the model a lever, not shown in the figure, fixes ABC, and at the same time Hfts P up so as to release GE. GE and the link DE can then be moved along to a new position. On moving back the lever, ABC is released and P is dropped again into position on GE, exactly under 0. p.o.w. [6 15. ON A MECHANICAL MODEL, ILLUSTRATING THE RESIDUAL CHARGE IN A DIELECTRIC. [Birmingham Phil. Soc. Proc. 6, 1888, pp. 314-317.] [Read November 8, 1888.] The model is designed to exhibit a phenomenon analogous to the residual charge which gathers in a condenser after it has been charged and then discharged, when the dielectric is not a perfect insulator. Its mode of action is similar to that which Maxwell supposes to occur in the dielectric. According to his theory the residual charge is due to the breaking down of the state of strain (or, perhaps, more correctly, of the stress) in the dielectric corresponding to the original charge, but in an uneven manner in different parts of the dielectric, so that just before the discharge the stress is greater in some parts than in others. On discharging, it is impossible, from the nature of electric discharge, to remove all the strain by connecting the two plates of the con- denser, and the condition of equilibrium which is arrived at consists in an actual reversal of the strain in the parts where the breaking down has been most rapid, the reversed stress in these parts balancing the remnant of the original stress in the other parts. On insulation, the strain breaks down again, and at the greatest rate in the same layers, now reversed. Consequently the reversed stress is no longer able to balance the direct stress, and, on the whole, there is a preponderance of strain in the original direction, or a gathering of charge the same in kind as the original charge. The model consists of a trough (see figure) of semicircular cross-section, 24 ins. long, 6 ins. diameter, and divided into eight equal compartments by a middle partition along the axis and three cross-partitions. It is supported at the two ends, so that it can rotate about its axis 00, a pointer P attached to one end moving in front of a scale S. Four pipes, with taps t, t, t, t, connect the opposite compartments when the taps are turned on. The trough is balanced by the weights w, w, so that when empty it is in neutral equihbrium. Turning the taps off, and pouring in water to the same depth in all the com- partments, the equilibrium at once becomes stable, and the trough, if displaced, stores up energy. It may be considered as analogous to a 'tube of force,' connecting charges ± q on the surfaces of two opposite conductors, the axis A MECHANICAL MODEL, ILLUSTRATING THE RESIDUAL CHARGE 243 of the trough representing the axis of the tube of force, the angle of displace- ment the charge at either end, or the induction along the tube. A clockwise rotation at the pointer-end might signify a positive charge at that end. As long as the taps are off, the trough represents a perfect insulator, a displacement through a given angle, and fixture at- that angle, corresponding to the com- munication of a charge and subsequent insulation. The energy remains in the trough undissipated. Discharge, of course, corresponds to release of the trough, and we have oscillations corresponding to the electrical oscillations brought recently into such prominence. It may be noted that a decrease in the quantity of water corresponds to an increase in specific inductive capacity, while a decrease ,in the weight of the trough corresponds to an increase in magnetic permeability. We might, perhaps, obtain an analogy to the spark- discharge by completing the cylinder, of which the trough forms half, and carrying the partitions up through the added half. On turning the trough through anything more than a right angle it would fall over and oscillate about a new position 180° from the original one, the discharge of energy occurring now with an increase of strain, not with a return to the unstrained condition. If the taps are turned on, but all to the same extent, the trough corresponds to a 'leaky' dielectric in which the conductivity is uniform. Turning the trough through a given angle and holding it, the water begins to flow back from the higher to the lower compartments, thus dissipating the energy, and if after a short time the trough is released it returns to a position short of the original position and remains there, the level of the water in the two sides of the middle partition being the same. But if the taps are turned on by different amounts — if, for example, the two end-taps are turned off while the two middle ones are turned on, — then on turning the trough through a given angle and holding it, the energy of the two middle pairs of compartments is gradually lessened, and on release the trough moves part way back. But now it is only the mean level which is the same on the two sides. In the two pairs of compartments with no communication there is still a positive difference of level, while in the other two there is now a negative difference. Holding the trough in its new position for a short time, the negative difference is reduced by leakage from one side to the other, and on release the trough returns by another amount towards its original position 16—2 244 A MECHANICAL MODEL, ILLUSTRATING THE RESIDUAL CHARGE —and this may be repeated several times, until finally the original position is sensibly regained. The first model I made, for ease of construction and without sufiicient consideration, with rectangular instead of circular cross-section; and with this the phenomenon of residual charge is obtained, even though all the taps are turned on equally. For consider what happens if the trough is turned through an angle and held. The water comes to a level after a time, but still its centre of gravity is not in the lowest possible position, and on release the trough returns part way, making a negative difference of level between the two sides. Again holding it, this difference is reduced, and on release there is another return, and so on. This suggests that possibly residual charge may occur not only when the substance is heterogeneous, but also when it is homogeneous, if with electric induction or strain there is both energy of the molecules as a whole and internal energy between the parts of each molecule. If the latter dies away after the bounding conductors are charged, the former may still remain, and on discharge it is possible that it may not all be dissipated, but may partly go to renew the internal energy. If this renewal accompanies a reversal of the direction of electric strain, we shall have the phenomena of residual charge. It is hardly necessary to point out that the model serves equally as an illustration of a possible explanation of elastic after-action. It is evident that the phenomenon of residual charge will always occur when a body strained is such that the stress dies away unequally in different parts, while at the same time its constitution is such that on release from strain an equal amount of strain is taken from each part. From this illustration of residual charge we may pass to a possible analogue of conduction in a metal wire. Let us suppose the trough replaced by a hollow cylinder, with its axis horizontal, ends closed, and without partitions. If the cylinder is only partly filled with water, a small couple applied to it will produce continuous rotation, but with a limiting angular velocity, attained when the water is dragged up in one side so far that the moment of its weight about the axis is equal to that of the applied couple. The energy put in by the couple is all ultimately converted into heat in the water. Thus the angular displacement increases indefinitely, though the stress always remains small. Similarly, as I believe, the 'electric strain,' or 'induction,' or ' displacement ' in a wire carrying a current increases indefinitely, as induction IS continually coming into it from the outside, although the stress always remains small. 16 ELECTRICAL THEORY. LETTERS TO DR. LODGE. [Electrician, 21, 1888, pp. 829-831.] to the editor of ' the electrician.* Sir: I have prevailed on Prof. Poynting to let me send you the enclosed two letters, wherein he continues the discussion of electrical theory begun in Section A at Bath. It must be understood that the letters are merely hasty epistles, not intended for publicity ; but Prof. Poynting's ideas are so original and weighty that one is glad to extract from him, when possible, a casual contribution to a discussion, as well as one of his sledge-hammer communications to the Royal Society. I hope that this may be the means of extracting a reply or a criticism more competent than anything of mine would be. Yours, etc. Oliver J. Lodge. Dear Lodge: I thank you very much for the copy of your exceedingly interesting account of Electrical A. Perhaps my gratitude would be best shown by silence, but I am tempted to show my appreciation by asking you to help me with some difficulties. My first difficulty is as to the interpretation of Hertz. You say — though, I think, FitzGerald is responsible for the statement — that ether is a demon- strated fact. I do not see how Hertz adds to our certainty. Is not our belief in ether due to the fact that light takes time to travel in interplanetary spaces, where we cannot put enough matter for it to travel by. so that we have to imagine something else for it to use. The fact that the velocity of light is nearly the same in vacuo and in gases, and not widely different in denser substances, of course supports the view that on the earth it also uses ether. Hertz shows that there is an 'interference' in electromagnetic disturbance which we can only (at least, with our present knowledge) put down to wave-motion traveUing with a definite velocity which he finds equal to that of light. Hence these disturbances probably make use of the same ether. Does this prove its existence any more? I should expect a sceptic 246 ELECTRICAL THEORY. LETTERS TO DR. LODGE to ask why may not electromagnetic disturbance make use of air, since Hertz carried out his experiments in air. I could only reply to the sceptic that he was a very disagreeable person. Secondly, Thomson's [Kelvin's] 'Simple Hypothesis' Paper* appears to be a very serious attack on Maxwell's theory ; in fact, on reading it over carefully, I can only come to the conclusion that it would lop off not only Maxwell's excrescences but his whole theory. According to the concluding sentence of § 4, (/ each component of electric current at any point is equal to the electric conductivity multiplied into the sum of the corresponding component of electrostatic force and the rate of decrease per unit of time of the corresponding component of velocity of liquid in our primary') the current pf_d^_ duj\ \ dx dtj' which = if C the conductivity = 0, so that Maxwell's / (his ' displacement- current ') goes altogether. The x component of Maxwell's e.m.f. will contain a term ^ ' ' ' -r-^ dxdydz, since Maxwell's u 477 Thomson's u-{- -. — ^ , where 477 dt P = E.M.F. along X. This has no representative in Thomson. Thus with a homogeneous but leaky condenser with no connecting wire, we have, according to Maxwell, total current = 0, for leak is made up for by yield of displacement ; •• dx' where V is potential due to electrification on the plates. According to Thomson, u is to be taken as rate of leak, and is positive ; P = du dt , , , dV — axaydz r- • r dx According to Maxwell there is no magnetic effect, since total current = 0. According to Thomson^ — using his notation — x component of magnetic dtVi dv-i _ ^_2 fdw dv^ dy dz \dy dz, d force u if V-2 and But dz \dy etc., are transposable u = KP, V :. u, = - 477V-2 from Thomson's equation (7), i.e.. dy KQ, w KB; dR dQ dy 477V- [dy\ dz) ' dw di ff, dxdydz T " dz * Reprinted in The Electrician, vol. 2L Sept. 14, 1888, p. 605. — dxdydzi ELECTRICAL THEORY. LETTERS TO DR. LODGE 247 the term in V disappearing. This is awful ; but I see no reason to suppose that it vanishes. If it does not vanish, then the existence of magnetic effect would decide against Maxwell. Thirdly, I note on p. 10 of your sketch that Rowland and FitzGerald consider that electrostatic potential is not propagated by end-thrust, and you remark that it is the magnetic potential which travels, generating the electro- static potential as it goes along. I think I remember that you have expressed the view that potential energy must undergo a kind of conversion, and that unless it be born again as kinetic energy it can in no wise go forward on its journey*. With strained solid waves it looks as if it were so, though it is, I think, possible to regard both energies as going forward linked together, yet retaining their individuality. And if potential energy is, after all, kinetic, but of another kind, it is conceivable that they should keep their separate identities. But with electrostatic and electromagnetic strains, which is potential and which kinetic? I know it is usual to call the magnetic kinetic ; but if we had started with permanent magnets, and travelled by means of magnetoelectric machines to our present knowledge of electric phenomena, I expect we should now^ be discussing the propagation of magnetostatic potential, and magnetoelectric potential, and we should, perhaps, consider the former generated by the latter. This is really the view I take, or, rather, I think both are true. It seems to me that the sideway propagation of electric induction is accompanied by (let us drop 'generated by') magnetic induction, and equally the sideway propagation of magnetic induction is accompanied by electric induction. The two go together when a disturbance is propagated. In a steady state they do not, i.e., if we can separate them from each other ; at least I do not see how otherwise to interpret the results I have obtained (see pp. 284-5 of paper referred to below) f. This brings us back to the old point whereon we have differed before. It would be better to give in like the unjust judge for the sake of peace and quietness, and to ward off any more such letters as this. I have been going again through a paper ' On Connection between Electric Current and Electric and Magnetic Induction' (Phil. Trans. 1885) {, in which I tried to work out the equations to the magnetic field on the supposition of this sideway propagation. "^ ceases to be troublesome, and both electric and magnetic inductions are propagated at the same rate; indeed, they are by Maxwell's equations, though I do not think he ever definitely worked this out. I cannot see any point, in the assumption to begin with, or in the subsequent reasoning, where I have gone astray. If you have any time to * Phil. Mag. October, 1879, p. 281, § 11; June, 1881, p. 534; and June, 1885, p. 486. I do not regard this as a 'view,' however, but as a proved truth, — 0. J. L. t [Collected Papers, pp. 201-2.] J [Collected Papers, Art. 11.] 248 ELECTRICAL THEORY. LETTERS TO DR. LODGE spare, would you look at the Paper, pp. 277-281, and 294-300? The rest is not essential, though on 301 I show that Maxwell, pure and simple, gives the same results*. Yours, etc., J. H. POYNTING. Mason College, Birmingham, October 12, 1888. [The following extract from my note in answer may be inserted, in order to make the next letter clear : ' As regards the proof of the ether, I confess I did not quite see FitzGerald's point as to why Hertz's experiments rendered the existence of ether any more certain; but knowing that I had felt it thoroughly estabhshed long ago, T supposed I was not a good judge. Of course he must appreciate all the stock arguments about air, etc., not transmitting transverse disturbances, and about neither it nor glass transmitting anything at the speed 3 x lO^^, etc. And Hertz's experiments only seemed to me to prove that electro- magnetic waves existed and travelled at the same speed, thus practically proving that Ught is electromagnetic waves, and estabhshing Maxwell's theory. ' This seems to me far more important than proving once more the existence of ether. At the same time I feel sure FitzGerald has some point. It may be only that an electromagnetic ether has been proved, and thus the action- at-a-distance-Germans confounded. He spoke as if he meant more than this. Perhaps I have to that extent misrepresented him in my " sketch." How does it strike you? 'With regard to Thomson's Paper, it is certainly very anti-Maxwellian, but I believe it only represents a transition stage through which he was somewhat rapidly passing, and through which he may now have almost passed. I certainly do not know now where he is. ' Is it not that, finding that displacement-currents have no magnetic effect, therefore he ignores them? But, then, have they no magnetic effect? In some cases they cannot have, for electrostatic displacement is of the nature of an '' expansion," as Clifford called it ; there is no '' spin " about it. ' I cannot find Thomson's Paper this minute to refer to, but I have it some- where. I will look it up again in the light of your remarks.' 0. J. L. After referring to my reply, Prof. Poynting writes, in a second letter :] About the ether, I should entirely accept your interpretation of Hertz. I should think, as you say, that FitzGerald was aiming at believers in action- at-a-distance, and probably he knew where to have them in showing that there is an electromagnetic medium. You say, with regard to Thomson's Paper, 'Is it not that, finding that dis- placement-currents have no magnetic effect, therefore he ignores them? But, then, have they no magnetic effect? In some cases they cannot have.' * [Collected Papers, pp. 194-198, 212-217, and 218 respectively.] ELECTRICAL THEORY. LETTERS TO DR. LODGE 249 It is just here that I find the supposition of transverse propagation of electric strain accompanied by magnetic induction so clarifying to my ideas. If the magnetic effect is the whirling of the machinery which is sending electric strain energy onwards, the machinery cannot know whether the U + + + + 4) Ai "C_ ni\\\ _J>" ' -• ] r- , V ' , ' ' ' ' \ / 1 . * - . J \A .' / / / I HiW energy is going to be dissipated in a Prony-brake-like wire, or whether it is going to increase the electric strain, so producing a 'displacement-current.' To use an illustration, I regard a re-entrant line of magnetic force as a kind of ring of Custom House officers registering the amount of electric strain sent in per second (they are porters as well), and they do not know what will happen to the energy. They will register just the same whether the imports are for immediate consumption or go to add to stock. Of course, they will take no account of shooting stars, balloons, or destruction of stock already within the ring ; which, being interpreted, is that the line-integral of the magnetic force (47^^■, isn't it?) will not be affected by Knes of electric force other than those coming in or going out through the boundary. For instance, if A moves through the ring RR to A^ (see figure) conveying a charge, hardly any Unes of electric force will cut the ring RR, yet the number of lines through it is increased. And it will not be affected by induction which dissipates itself in situ as in a leaky Leyden jar, where the energy changes to heat without moving. I do not know of any other case, but, being utterly ignorant of pyro-electricity, I imagine it might supply a case of establishment of electric induction without motion of the energy and without magnetic effect. Yours, etc., J. H. POYNTING. October 18, 1888. 17. AN EXAMINATION OF PROF. LODGE'S ELECTRO- MAGNETIC HYPOTHESIS. [Electrician, 31, 1893, pp. 575-577, 606-608, 635-636.] The leaders in Physical Science, impressed perhaps with the responsibility of their position, and fearing that their weaker followers will distort their views, are, as a rule, very cautious in giving us their vaguer speculations as distinguished from the more exact hypotheses which can at once be put into working shape. Yet these less-formed speculations are often helpful, even if they only arouse our minds to attempt to disprove them. Still more are they helpful if they aid us in thinking of facts in a more connected way until the finished working hypothesis is ready to take their place. We owe a special debt of gratitude on this ground to Dr. Lodge for his well-known book on the Modern Views of Electricity, in which he describes not only the more definite beliefs which he firmly holds, but gives us also any suggestions rising in his mind which seem to give promise of light to guide us in the dark ways of the science. He talks as it were confidentially to us, and though the speculative character of the book makes it by no means the easy reading which the absence of mathematical treatment might lead us to expect, and even perhaps unfits it for beginners, the bold attempts at explanation give it great value to more advanced students. Such students will be brought face to face with many difficulties which they may hitherto not have recognised through haziness of thought. And even where they are unconvinced by Dr. Lodge's attempts to solve the difficulties they will be gainers by the orderly review of their knowledge necessary before they can form a judgment. The book is built round a central hypothesis of the nature of electric action, which I propose to examine. I shall first give an account of the hypothesis as it appears to me to stand, freed from the details and the wealth of illustration, which, though appropriate, and even necessary, in the original work, make the argument at times rather difficult to follow. I shall then examine the evidence for or against the hypothesis. We start with that which everyone accepts as the result of experiment and observation, that there are two kinds of electrification with oppositely- AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS 251 directed qualities, and that they make their appearance always in equal amounts. Hence, on their union, the net electrification is zero. Compare this with the case of momentum. According to our experience, as summed up in the third law of motion, we may regard the mutual stress between two bodies as consisting of a transfer of momentum from one to the other. A gun at rest is fired. The momentum gained by the bullet and powder may con- veniently be regarded as derived from the gun, which, having none to begin with, now has an amount of negative momentum equal to the positive possessed by the charge. In other words, positive momentum is transferred from gun to charge. Or compare with a case of material transfer, as when A lends B a sum of money. Then A and B, after the transfer, are oppositely affected, so that if they both assign their share in the transaction to a third person C the effect on C is zero ; or if 5 re-transfers the sum borrowed to A the net result is zero. Such cases as these suggest that positive and negative electrification are merely the creditor and debtor sides of a single transaction, the sending out and the reception of something transferred (p. 9). What kind of transfer we must imagine is best gathered from the ' ice-pail ' experiment, a particular case of the general principle that when induction occurs + and — always /ace each other in equal quantities with an insulating medium between. We may suppose that we have a nearly closed hollow insulated conductor, and that through an orifice we introduce a body having on it a charge + Q. Immedi- ately — Q gathers opposite to it on the inside surface, and + Q is on the outside surface, facing — Q on the walls of the room in which the experiment is made. If, instead of carrying out this electrical experiment, we imagine an indefinitely extended incompressible liquid, and think of merely mathe- matical surfaces occupying the positions of the surfaces of the conductors, the introduction of Q of liquid within the inner surface would force Q of hquid through each of the two surfaces, and relative to the space between the surfaces, inwards towards the inner surface, and outwards from the outer. Now, suppose that such an incompressible hquid, to which for the present we need not ascribe gravitation, has an actual existence, that it fills all space with which we are concerned, and that it permeates matter. Let us suppose that in bodies which we term electrical conductors it is free to flow with nothing worse than frictional loss of energy, but that in insulators it has some kind of attachment to the matter, so that in the displacement of one relative to the other — that is, in the strain — energy is stored, and in such a way that it can be regained when the strain is relaxed. We shall call this liquid Electricity. When a displacement occurs, so that some of the liquid is pushed into or out of any conductor, the quantity flowing through the surface from or to the insulator is that which we have hitherto called the charge of electricity on the surface. We must now term it Electrification to 252 AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS distinguisli it from the general body of the hquid, which is, according to Lodge, all Electricity. It is important to notice this distinction in studying Lodge's hypothesis. Electricity is the generic name for the fluid all through space, Electrification the specific name for that part of it which happens to have flowed in any given disturbance through the surface of a conductor. Thus, if we imagine a conducting sphere A within a hollow conducting sphere B, B having an orifice through which we can introduce a wire to charge the inner sphere, say, positively, the gathering of a charge Q on ^ is to be regarded as a flowing of some of the all-extensive electricity along the wire into the sphere, which is, however, already full of electricity. Hence Q must be pushed through the surface of the sphere out into the insulator or dielectric. This outward displacement manifests itself as the + electrification of ^. As the dielectric is also initially full of electricity, Q must be pushed out through every surface in it completely enclosing A. The displacement relative to the air-particles stores energy — the energy of the charge. When we come to the inner surface of B, Q is pushed into the substance of B, an inward displacement which we term a negative charge. It is also pushed through the substance of B, but as this substance is conducting, no energy is stored, and only a little is dissipated by the frictional rub, or, perhaps better, the viscous cling. At the outer surface of B there is another displacement of Q outwards into the dielectric — i.e., another positive charge, and energy-storing begins again. The pushing out will take place through the second dielectric till we come to the walls of the room in which the action is occurring. Here it will probably end, for the generator of the original charge has probably sucked in the fluid from the walls. There is therefore a confined circulation, and not an infinitely extended pushing-out. Sources of electrification with their connecting wires are evidently to be regarded as turbines working in pipes or channels laid in space, incompressible and fluid electricity filHng both the pipes and the space outside them. When the turbines work, the fluid runs along the pipes, forming what we call an electric current. When a pipe ends in a reservoir bounded by a dielectric, the fluid presses out into the dielectric, and there stores the energy put into it by the turbine, minus that dissipated by the viscous resistance in the pipes and conducting channels. Prof. Lodge illustrates the connection between the electric incompressible fluid (why not the electric Hquid?) and the molecules of matter by a series of ingenious and suggestive models either with cord running through beads or with Hquid to represent the electricity. It is hardly necessary here to describe these. Any reader who is not yet acquainted with them should study the original account. It is enough to say that Prof. Lodge can make his models behave Hke Leyden jars and give the phenomena of charge, residual charge, and oscillating discharge as perfectly as if they were thorough beHevers in his hypothesis. AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS 253 I find it somewhat easier to form a picture of the hypothesis by supposing matter to have a sponge-Hke constitution, i.e., to be permeated in every direction by passages, the pores so dear at one time to the writers on elemen- tary science. These passages are filled with Hquid electricity. In every passage or tube, however short, we must imagine a Httle turbine turning round with the flow through it and never letting any fluid pass without duly turning. In dielectrics there is a spring, Hke the mainspring of a watch, attached to the spindle of each turbine so that when the wheel turns energy is stored. In conductors this spring is wanting, and there is only a viscous resistance to rotation. We may think of the source of electrification, machine battery or induction coil, as a large turbine somewhere in the system with a supply of energy behind it dealt out by a motor of some kind or other. When the large turbine works, a flow takes place in the system dissipating energy in the conductors and storing it in the dielectrics. When the charging turbine is removed or disconnected from the motor, so that the way is clear for a return, all the wound-up mainsprings return and drive the liquid back through the sponge. We have only to make the turbines with different moments of inertia, with different qualities of lubricator in the conductors, and with different strengths of spring, and different firmness of attachment in the dielectrics, to get varying permeability, electric resistance, specific inductive capacity, and residual charge. We may,- perhaps, simphfy the arrangement of affairs by supposing that the molecules themselves are the turbines, and then we get a kind of inversion of the hypothesis described hereafter, which Dr. Lodge develops to account for electromagnetism. In the displacement- hypothesis, with a single electric Hquid flowing past matter, there is a very serious difficulty. The energy being stored by the flow past the molecules of matter, we might reasonably expect the electricity to pull, or to tend to pull, the molecules with it, and there should, therefore, be a motion of matter along the lines of force in one direction. But instead of this we have a tension, both ways as it were, along the lines of force or flow. There is displacement of matter only in the case of electrolytes, and here it is both ways along the lines of flow, one set of atoms going one way and another set the opposite way. A modification of the hypothesis is suggested by Dr. Lodge to meet this difficulty and to account for the double electrolytic procession. He supposes that there are two constituents of the electric fluid, each in general filling half any space, intimately mixed and evenly distributed. The molecules of matter are made up each of two constituents in accordance with the usual view, and one of these is attached to one kind of electric fluid the other to the other. When electric displacement occurs it is really a double flow, the two constituents of the fluid travelling equal distances in opposite directions past each other. The atomic constituents of each molecule move in opposite directions, but not in general very far, so that there is in dielectrics no displacement of the matter as a whole. In electrolytes the displacement 254 AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS continues till separation and re- pairing occur, and thus we get a double procession. We see that in dielectrics, the pulls of the two constituents of electricity in the molecules, one on one atom, the other on the other, balance each other. These electric pulls lead to internal stresses within the molecule about which Dr. Lodge does not say anything very definite, but it seems to me that we have to introduce chemical forces here to account for the pull of the atoms on each other, distinct from the electric forces or the pulls of electricity on matter. This dualism is hardly in accordance with the late exposition of Dr. Lodge's ideas, where he appears to identify electrical and chemical forces (p. 84). Perhaps we might as well, while we are inventing a constitution for the ether, make a third or neutral constituent to which all the atoms of matter are attached. We will suppose this neutral electricity to resist extension and compression. We then have electro-positive atoms attached to positive electricity, electro-negative atoms to negative electricity, and all of them to neutral electricity. When an electric displacement occurs, positive ether tugs at one set of atoms, negative ether at the other set, and neutral ether prevents their separation, so that all our forces are of one kind, insomuch as they are forces between atoms and ether. I rather Hke this neutral ether, but I am afraid Dr. Lodge will not adopt a strange infant into a family already sufficiently large. But taking the hypothesis as set forth by its author, the mere duahty does not much affect the general notion of the nature of electric charge. We must remember that motion of negative fluid inwards equally with that of positive charge outwards gives a positive electrification, so that the explana- tion of the electrification of a sphere within a conductor already given has only to be amplified by supposing that there is another ethereal fluid displaced in the opposite direction at the same time and throughout the system. I can imagine the agnostic in ethereal matters protesting here against the multiplication of unknowns and unknowables. I can imagine him saying that his sense-organs are only excited by material motions and affections, that his instruments are all material, and only appear to undergo changes of shape, colour, sound — i.e., affections of matter, and that he cannot with any certainty get beyond these material affections. He will argue that, as we have no sense affected by ether alone, we can form no adequate conception of the ether ; we can only suppose it endowed with material properties, and conceive of it as some form of matter. And it would appear possible to imagine various material ethereal constitutions or connecting machineries between the different portions of matter evident to our senses, all equally accounting for all known facts. When a new fact turned up he would own that probably some of the ethereal machinery would fail to account for it, and so would have to be taken off to the lumber-room for worn-out hypotheses ; AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS 255 but the new fact, he would argue, would very Ukely enable us to imagine new types of machinery to replace some, at least, of the old rejected ones. And probably, till the whole range of physical phenomena was known, it would always be possible to imagine more than one kind of machinery to account for the phenomena known. Probably only when nothing remained to be discovered would there be a single solution, and only then would it be possible to give a single answer to the question. What is ether? And even then we might be wrong, for the ether might have properties in its action on matter quite different from any of which we have material types. Though our agnostic, when following this train of thought, may be un- convinced by Dr. Lodge's preface, and may urge that the ether is and will probably remain a hypothetical medium, he will, no doubt, adopt some form of hypothesis for working purposes. If he is an ordinary human being, when he studies such actions as we term actions-at-a-distance, he will prefer to think of the different parts of the acting matter as connected by something continuous, with material properties. Following Boscovich and Faraday, he may extend the atoms throughout space, and give this extension material properties (a special case of this type of hypothesis is presented to us in the ring- vortex theory of the Universe) ; or he may limit the atoms and put in some new connecting machinery to fill up the vacuum he abhors, and this he may as well call 'ether.' While, therefore, he may protest against Dr. Lodge's ' cocksureness ' about any particular constitution for the ether, he is bound to examine any hypothesis reasonably presented to see if it is likely to form a good working hypothesis to account provisionally for the observed facts. He would, no doubt, admit that Dr. Lodge's hypothesis is reasonably presented, and it would only be a question with him whether so complicated a constitution for the ether enables him to think sufficiently easily of the phenomena for which it is to account. Leaving him to consider this, we may pass on to the further development of the hypothesis. So far, we have only been thinking of the properties of electricity at rest. It is true that we have thought of the electricity as being pumped along conductors and as wasting energy in the passage, but this was only a step onwards to a final statical distribution. We are now to concentrate our attention on the pumping stage. When electricity is in motion in sufficient quantity and for sufficient time, a new set of phenomena come into prominence. Among these are the heating of the conductor, the heating or cooHng of junctions, the opposite ionic pro- cessions in electrolytes, and the creation of a magnetic field. The heating of the conductor is to be explained, according to Dr. Lodge, as something analogous to frictional, or rather viscous, dissipation of energy. We may think of the two streams of electricity flowing past the atoms of matter, and continually catching hold of them and letting them go again, as a fiddle-bow 256 AN- EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS catches hold of and lets go a fiddle-string. Thus, some of the energy of flow is converted into vibrational energy of the atoms, that is into heat. The junctional heat- phenomena still wait for complete explanation. We may consider that the facts imply that at a junction of dissimilar metals there is a tendency for positive electricity to move more easily in one direction than the other, and, of course, the reverse with negative electricity. A compound bar with free ends will thus tend to be positively electrified at one end and negatively at the other. Suppose, further, that the tendency to separation of electricities varies with the temperature, and we have at once the thermo- electric current in a closed circuit. Returning to the fiddle-bow and string used to illustrate the development of heat by conduction, we can see how it ought to work to illustrate thermo-electricity. Suppose a set of parallel strings in a horizontal plane, one half tuned to one note and the other half to another. The one set may represent one metal with its atoms vibrating in given modes, and the other set another metal in contact with it and with its atoms vibrating in other given modes. Now, laying the fiddle-bow across the strings after they are set in vibration, if the motion of the bow is always in one direction it illustrates the pushing of one of the electricities from one metal to the other. I have found that when light bits of paper are laid across two vibrating strings of different pitch there is frequently a movement of translation. Unfortunately for the illustration it is sometimes in one direction, sometimes in the other. The paper was not part of Prof. Lodge's book, and knew nothing of the hypothesis it was expected to support. The opposite procession of ions we may think of as the transport of the atoms by the positive and negative electric streams respectively, the connections between the pairs of atoms being broken down and renewed with fresh partners all along the line and continually. The most evident phenomenon characterising the electric current is the magnetic field around it. This we may regard as manifesting the existence of so much magnetic energy in the neighbourhood of the conducting wire. Let us see how the hypothesis will account for this energy. The first step is to reduce permanent magnetism and current-magnetism to one species by adopting Ampere's theory. If we consider a small closed current-bearing circuit, observation tells us that, at a distance from the circuit, the field is indistinguishable from that due to a small steel magnet, with centre in the plane of the circuit and axis perpendicular to it. Starting from this, we know that we can deduce an arrangement of permanent magnets, equivalent, as regards the outside field, to any current-bearing circuit. The circuit differs magnetically from the steel in two respects only, viz., that it requires a con- tinual supply of energy to maintain it, and that we can get into its inside. This last difference may be merely due to the large size of any apparatus at our command, and it is quite thinkable that, if we could make ourselves or our apparatus smaller than molecules, we could explore the inside of the steel AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS 257 molecules. The other difierence is possibly not one of the kind ; for suppose the resistance to be diminished till it disappears, the rate of energy-supply, C^R, disappears also, and we have a current-circuit — never mind how the current was started — which is as permanent a magnet as a steel bar. It is a short and inevitable step from this to Ampere's hypothesis that a magnetic molecule — a molecule of steel, say — is essentially a small closed perfectly- conducting circuit with a current of electricity in it ; or, in terms of Dr. Lodge's hypothesis, either two equal and opposite currents whirhng round at equal speeds in opposite directions in each molecule, or a positive whirl in one direction in one molecule, accompanied by a negative whirl in the other direction in the next molecule. We may dismiss this duaUty for the present, on condition that it comes up for sentence when called upon, and return to the single circuit. Such a circuit, when placed in a magnetic field, behaves, doubtless, Hke finite circuits, and tends to set itself perpendicular to the lines of force, and with its own lines parallel to the lines of the field, and in the same direction through the circuit. It tends to move from weaker to stronger parts of the field — tends, in fact, to include as many positive and exclude as many negative lines as possible. But the disappearance of resistance has a pecuHar effect. Even a circuit of the resisting kind, with which alone we have practical acquaintance, would protest against the inclusion of foreign fines of force in addition to its own, and the current would diminish while they were being included. An Amperean circuit would not merely protest, but would absolutely prevent any change in the total number included, for any increase would be accompanied by a finite negative e.m.f. proportional to the rate of increase, and therefore by an infinite current, since R is zero. This is not to be accepted as possible, so that all that can happen is that some of the current's own fines of force shall be replaced by those of the field, and there is a consequent weakening of the current. When a mass of iron consisting, we suppose, of such Amperean currents is brought into a magnetic field, all the circuits tend to turn round to include the fines of the field, and at the same time the Amperean currents decrease in strength. The circuits thread themselves like beads on to the fines of force of the external field, their currents falling as they thread on. But their own surviving lines of force are added to those of the field, so that the total field is greater, and the iron has greater permeabifity than a vacuum ; or perhaps it will be better here to say that the iron conducts the lines of force better than a vacuum would, for permeabifity has an exact significance, not quite describing the property now under discussion. As new fines of force are added to the field the iron will conduct them better than a vacuum, until every molecule has all its lines brought into service, and its current, therefore, reduced to zero.' After this point is reached, since the molecules either will p. c w. 17 258 AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS not take any more lines of force, or if they take any more will establish negative currents to neutrahse them, they are worse conductors than a vacuum, for they fill up part of the space uselessly or injuriously. They will therefore tend to move from stronger to weaker parts of the field, as if diamagnetic. Some indication of such a change in conducting power has been detected by Ewing in the magnetisation of iron in exceedingly strong fields, for though the permeabihty or induction produced -f- the field producing it — both reckoned from zero — was always greater than 1, the value of increase in induction -^ increase in field producing it, or the conducting power, as I have called it, fell ultimately below 1. For some reason, not yet explicable, the magnetic chains in iron become unstable, and break up when the temperature is raised to the neighbourhood of 800° C. The iron then above this temperature is practically equivalent magnetically to any other substance. Assuming then that we have some notion of what we mean by currents of electricity whirhng in channels of no resistance. Ampere's hypothesis gives us a fair explanation of iron and steel magnetism. If we accept the view that the interior of a magnet does not differ from the exterior in kind but only in degree and in permanence, we may attempt to extend the hypothesis to explain the magnetic quahties of all other substances, i.e., their power of carrying the Hnes of force, and of carrying them in sHghtly different degrees. Let us think of the lines of force in air circling round a current or passing from pole to pole of a magnet. We may think of these as passing through a number of electric whirls, or at any rate through perfectly conducting rings ready to exist as whirls. Before the passage of the lines of force these rings are turned in all directions. After the passage they tend to set perpendicular to the lines of force. If the medium is paramagnetic we may suppose the rings to have initial currents in them; if it is diamagnetic we may with Weber suppose that they have no currents initially, and when no Hnes of force pass through. On the estabhshment of the field negative currents are excited of such value as to make negative Hnes of force thread each ring equal in number to the positive Hnes sent through by the field. Thus each ring acts as a part of the field through which no lines of force pass, and the per- meabihty is thereby diminished. At the same time the diamagnetic substance will tend to weaker parts of the field, and we have the main facts of dia- magnetism explained. There is a serious difficulty in the nearly constant permeabihty, differing only by a very small amount for a diamagnetic soHd Hke bismuth and a magnetic gas like oxygen, the one with its molecules crowded together, the other with its molecules comparatively wide apart. Perhaps we can strengthen this weak point by supposing the conducting rings of very different diameters in the two substances, or we may think of the electric channels as different altogether from the molecules and the AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS 259 same in number per c.c. in bismuth and in oxygen. Another difficulty Hes in the non-existence of permanent magnetism except in iron, nickel, and perhaps cobalt, but it is no greater than the difficulty with iron above 800° C. Perhaps both will ultimately find the same explanation, and until this is forthcoming we need only say that Ampere's hypothesis is merely silent on the point and is not necessarily unable to explain it. Now as to magnetic energy. Dr. Lodge points out a number of facts which suggest that magnetic disturbance is of the nature of spin round the Hues of force. The Amperean circuits at once present themselves as being the seat of this spinning, and the electric fluid or fluids in the channels as the spinning material. These whirHngs of electricity, either in themselves or in the accompanying motion of the entangled matter (Maxwell, by the way, thought it was the matter), possess, according to Dr. Lodge, the magnetic energy of the system. In fact, they are themselves magnetism. So far we have been dealing with magnetism and its relation to electricity, the space-filling fluid, and we have come to the conclusion that magnetism consists of vortices in this fluid. It remains to explain the nature of the ordinary electric current, and the way in which its accompanying magnetic field is maintained. Dr. Lodge uses for the purpose a mechanical analogue or mechanical model, a modification of Maxwell's well-known model, which is described in his Scientific Papers, vol. 1, page 451, and I believe by Dr. Garnett in Maxwell's Life. Let us imagine the two fluids, which are to be regarded as jointly filling space, to have a cellular construction, each consisting of spheres or little india-rubber bags, or what you like, in contact with each other. In dielectrics we think of contiguous cells as gearing in some way. Let these cells, when in a magnetic field, be spinning round the fines of force, the positive in one direction, the negative in the opposite; and let positive and negative be alternated so that the opposite motions may be possible without sHp of gearing. Fig. 37 from Dr. Lodge's book illustrates this idea when the fines of force pass into the paper from above, the axis of spin, therefore, being perpendicular to the paper. We may further materiafise the conception by inserting teeth round the edges of the cells; and here the family likeness to the parent (Maxwell's model) becomes stronger. We then have Fig. 36. If we suppose Fig. 37 to represent the unstrained state in a dielectric, then electrostatic strain will be presented by some such deformation as that represented in Fig. A ; not, I think, in the manner represented by Fig. 46 in Dr. Lodge's book. If we take Fig. 36 as our type, it is easier to think of the wheels as being attached to some kind of framework. We may think of all the positive 17—2 260 AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS wheels in Fig. 36 as arranged flat against one series of parallel rods going down the page, and the negative wheels on another intermediate and parallel series. We may think of the wheels, or their teeth, as not quite rigid, so that when the positive rods are pulled down and the negative up, there is a slight Dr. Lodge's Fig. 36. Rows of cells alternately Dr. Lodge's Fig. 37. Section of a magnetic positive and negative, geared together, field, perpendicular to the lines of force; and free to turn about fixed axes. alternate cells rotating oppositely. (Another mode of drawing Fig. 36.) Fig. A. Electrostatic strain. The cells displaced slightly in opposite directions. C + 1 +1 2 Dr. Lodge's Fig. 46. Fig. B. Cells shpping in a wire carrying a uniformly distributed current. relative displacement and energy is stored, and this will represent an electro- static strain. Probably, if we arranged all the forces properly, we should be able to do without the framework of rods. Meanwhile I find I cannot think of the model clearly without it. AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS 261 The distinction between a dielectric and a conductor is to be represented by the abohtion of the teeth on the wheels in the conductor. The surfaces of the wheels are in contact, and are only more or less rough — viscously rough, rather than frictionally rough. It appears rather difficult to think of the layer of wheels separating a dielectric and a conductor, for on the dielectric-side they must have toothed gearing, and on the conductor-side they must only rub ; but we may get over the difficulty by supposing these wheels double, one toothed to gear on the dielectric-side, and the other untoothed to rub on the conducting side, and both keyed on the same axis. Dr. Lodge regards a current as represented by a slip of one row of wheels against the next. He says (p. 206), 'Notice that in a medium so constituted and magnetised — that is, with all the wheel-work revolving properly — there is nothing of the nature of an electric current proceeding in any direction whatever. For, at every point of contact of two wheels, the positive and negative electricities are going at the same rate in the same direction; and this is no current at all.... A current is nevertheless easily able to be repre- sented by mechanism such as that of Fig. 36 or 37 ; for it only needs the wheels to gear imperfectly and to work with shp. At any such shpping- place the positive is going faster than the negative, or vice versa, and so there is current there. A line of sHp among the wheels corresponds therefore to a linear current.... Understand: one is not here thinking of a current as analogous to a locomotion of the wheels — their axes may be quite stationary. The shp contemplated is that of one rim on another.' Thus in Fig. B, let the row of wheels represent the cells across the dia- meter of a wire carrying a steady current, ABCD representing the section of the wire. Let the speeds of rotation of the wheels be as marked on each. The resultant positive shp is + 1 at each surface of contact, a total of 6. It is perhaps presumptuous to quarrel with a parent such as Dr. Lodge as to his mode of developing the faculties of his own offspring, but I must venture here to differ from him entirely in the way in which he seeks to make his model represent current. It appears to me that he has grafted on to his own model the representation of current in the entirely different model of FitzGerald, and so obtains something quite inconsistent with his previous ideas. It is only by stopping short at the centres of the bounding wheels that he can obtain a resultant flow in one direction. Obviously, if he took in a whole number of wheels, each entire, he could get no resultant flow, for the flow on the opposite sides of each wheel is equal and opposite. Thus, in the figure, if he took into account the outer sides of the two last wheels in the wire, he would have — 6 neutralising the previously obtained -{- 6. Or, to put it in another way, if we draw a plane, say, above the line of centres, a tangent to all the wheels, there is evidently no resultant flow across that plane. And we can think of cases of shp when we have no reason to suppose 262 AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS I ABCDOdcba there is current. Thus, if a bar of soft iron is placed axially in a magnetic field, near the centre of the bar the hues of force are parallel to its length within and without the bar. According to Lodge, the electric whirls are very rapid within the iron, and comparatively slow in the air outside. There must, therefore, be slip at the boundary, and yet we have no dissipation of energy to indicate the existence of the current. Another objection is that the model worked thus would make a diflterence in kind between the process of displacing in a dielectric, the equal and opposite motions of the two fluids which are leading to an electrostatic strain, and the current in a conductor. This is rather setting back the clock. Perhaps we have gone on too fast, and the difference may exist, but I think we should hardly accept the evidence of the model on the point. What Prof. Lodge calls current appears to me then to be merely sudden change of magnetic intensity. If this is just criticism let us see if the model can be made to represent a true current. I shall take the case of a steady current in which the condition of affairs is not altering. It is always better to begin with such a case, just as it is better to begin with statics, hydrostatics, electrostatics, than with dynamics, hydrodynamics, and electrodynamics. When matters, or ethers, get into changing motion they are, like the celebrated pig of the Irishman, difficult to count. In the model we shall suppose that the current in the wire is represented by two equal and opposite processions of cells or wheels along the wire, and to help us in thinking of these processions we shall, as before, suppose two sets of rods parallel to the axis of the wire, the positive wheels on one set with their spindles perpendicular to the rods, and the negative wheels on the other set. In Fig. C the rods alone are represented, directions of motion of the wheels on each being shown by arrows. The + rods are to be regarded as moving down the page, and the — rods up. The wheels on the bounding rods, B, b, gear with the wheels on A, a in the dielectric, but sRp on those on C, c in the conductor. Now, if the conductor extended through all space I do not see that the motions of the rods would involve rotation of the wheels. For any rod would be as it were surrounded by a symmetrical system, its wheels would not know which way to turn, and so would merely rub against their neighbours. But here the conductor is bounded by an insulating medium in which the rods can only move a very httle way, and that only by straining the teeth of the wheels. Fig. C. Rods on which the positive and nega- tive wheels are sup- posed to be fixed in the model of a wire carrying a current. AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS 263 The result is that as B moves up past A its wheels behave as pinions moving on a rack in the place of A. They tend to turn the wheels of C in the opposite direction. These in turn tend to send round the wheels of B, and so on to 0, the middle rod. Now, starting from the other side, a is at rest, and as h moves past it its wheels must rotate. This rotation is trans- mitted inwards, as before, to 0. The result is that the wheels on are urged in opposite directions from the two sides, and so remain at rest, while those of D and d rub against them. Those of C and c rotate faster than those of D and d, the friction due to the sUp balancing that due to the sHp of D and d on 0, and so on, the rotation increasing outwards. When we come to B, b, their wheels have the frictional resistance against the C, c wheels on the inside, which must be balanced — since the motion is steady — by pressure on the outside against the A, a wheels. Hence the B, b wheels are not merely rolHng on the A, a wheels, but are pressing against them, and so turning them round. In other words, the rotation extends out into the dielectric. Thus, at the middle of the wire, there is no rotation — no magnetic intensity. It increases as we go outwards to the boundary, being in opposite directions on the two sides, and it exists in the surrounding medium. The rubbing of the surfaces, perhaps of the rods, perhaps of the wheels, dissipates energy which represents the heat appearing in the wire. The model is probably indeterminate as to the way in which the energy finds its way into the various parts of the conductor, where it appears as heat. We might imagine the rods moved by end- thrusts along the wire, their wheels setting the outside wheels spinning, or we might imagine a spin propagated along the outside wheels to the wire, and there setting the rods in longitudinal motion. I have no doubt that the model would be able to represent the phenomena of the current-induction when the motion is accelerating or unsteady, for Maxwell's model does this, and the differences of construction would hardly affect the point. I have now given an account of the main features of Dr. Lodge's hypothesis as to the nature of electric charge, magnetism, and current, and I have done this at considerable length because I propose to discuss the foundations for the hypothesis, and to see if they are sufficiently firm to bear the superstructure. To recapitulate the main features, we have a double ether consisting of equal -f and — portions attached in some sort of way to matter as a framework — elastically in dielectrics — loosely, with a kind of viscous connection in con- ductors. The two ethers always move in opposite directions along the fines of electric force, and in dielectrics they store energy by their displacement past the material framework, while in conductors they only dissipate it during the displacement. The ethers are cellular in construction, and the cells are capable of rotation, this rotation constituting magnetism, and the accompanying 264 AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS rotational energy being magnetic energy. The axes of rotation are along the hnes of magnetic force. The cells roll on one another as if quite rough in dielectrics, and as if lubricated with viscous oil in conductors, and they are alternately + and — , and moving in opposite directions, a clockwise moving + cell giving the same direction to the magnetic force as a counter-clockwise — cell. Current is, according to Lodge, shp of these cells against each other, but according to the interpretation I have given of his theory, current is a double and continuous procession of the cells in a conductor, with the consequent rotations. The hypothesis is avowed by its author to be an attempt to obtain a mechanical explanation of electric and magnetic phenomena. He uses the main idea of Maxwell's well-known model, but replaces Maxwell's duality of magnetic wheels and electric 'idle' wheels by a duahty of electric wheels. It is, perhaps, open to question whether this is really a simplification, but the attempt was very well worth making, for it is only by variation and natural selection that the mechanical model will be suited to its environment in the electric world. It is, I suppose, useless to look for any other than a mechanical hypothesis as final. Probably because we are able to picture mechanical processes, able to think of ourselves as seeing what goes on, seeing kinetic energy manifested in the moving parts, able to think of ourselves as part of the connecting machinery, feehng the stresses, and helping to make the strains, we have come to regard mechanical explanations as the inevitable and ultimate ones. Thus, though the old mechanical hypothesis of light is for the present discarded in favour of an electromagnetic one, I suppose no one is content with the present position ; but we are all looking forward to the time when, by mechanical explanation of electromagnetism, Ught shall once more become mechanical. Nevertheless, ifis well to bear in mind that all such explanations are merely hypothetical, and may at any time have to be discarded, as the solid-ether Hght-hypothesis has been discarded. Indeed, they are solely of value as a scaffolding enabling us to build up a permanent structure of facts, i.e., of phenomena affecting our senses. And inasmuch as we may at any time have to replace the old scaffolding by new, more suitable for new parts of the building, it is a mistake to make the scaffolding too solid, and to regard it as permanent and of equal value with the building itself. It is on this point that I find most cause for disagreement with Prof. Lodge. He appears to me to regard his hypothesis as inevitable and permanent, or at least as approximating to the permanent and inevitable. The scaffolding, in fact, is made as important as the building. It behoves us, therefore, to examine into the security of its foundations. We may take the hypothesis as based on two statements : I. Kegarding electricity at rest, 'Whenever we perceive that a thing is AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS 265 produced in precisely equal and opposite amounts, so that what one body- gains another loses, it is convenient and most simple to consider the thing not as generated in the one- body and destroyed in the other, but as simply trans- ferred' (p. 9). II. Kegarding electricity in motion: 'Magnetism is nothing more nor less than a whirl of electricity' (p. 171). Let us examine some cases which should come under the first statement. We may regard the upper and under surfaces of a vessel of liquid as respectively positive and negative, and if we measure them by their projections on a horizontal plane they are equal and opposite element by element. We clearly cannot create a positive without an equal negative. Bring the two elements together, and they neutrahse each other, for both cease to exist. We have no idea here of something transferred. Or, take the case of a rotating cord. Its two ends, as viewed from the outside, have equal and opposite properties, as may be seen at once if we think of them as brought close beside each other. If the rotations were transferred to one and the same body the net result both to themselves and to the body would be no motion. Or if the cord were cut away in slices, bit by bit, when it is all cut away the two ends have come together, and have, in a sense, neutrahsed each other. Here again we think of nothing transferred. As a third case take a magnet. We always create equal and opposite poles at the same time with mutually neutralising properties, and to bring this within Dr. Lodge's statement we have only to consider the magnet as made up of two bodies joined together in the middle. Or perhaps we might take the case of the creation of N. and S. poles by the breaking of a magnet. No one has yet made a displacement- or transfer-hypothesis for magnetism except for the purpose of illustration. This, by the way, is rather remarkable, for magnetism with its re-entrant tubes of induction is so much more ready to lend itself to such a hypothesis. We seem to have finally made up our minds that electric is strain-energy, and that magnetic is rotational energy, and we do not even consider the alternative of magnetic as strain and electric as kinetic energy. No doubt the existence of magnetic rotation of the plane of polarisation of light strongly suggests the rotational nature of magnetism, but we can hardly claim any explanation of the phenomenon yet put forth as complete, and certainly none is exclusive. The other facts would possibly be equally well explained if we thought of electric tubes of induction as strings of wheels with their spindles along the axis of the tubes, and magnetic tubes as fines of flow of ether or of two ethers, with a certain displacement- coefiicient, greater in diamagnetics, less in paramagnetics. I do not put this forward as worth following out, but merely to illustrate the contention that we need not necessarily think of -f and — electrification as due to a longi- tudinal transfer of something. It appears to me quite possible then to 266 AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS symbolise the positive and negative charges facing each other across a dielec- tric in other ways than by the transfer of something from one towards the other. Turning now to the second fundamental idea, Ampere's hypothesis, that a magnetic molecule is a small closed electric current, this is really closely connected with the idea that there is something moving in the direction of an electric current and constituting the current. It is in fact dependent on our acceptance of the first idea. For consider what an Amperean current is phenomenally. There is no resistance, and therefore no heat developed, and no fall of potential; there is no junction, and therefore no Peltier effect; there is no break, and therefore no chemical effect. The one effect left is the magnetic one. If, then, we take away the idea of some substance whirling round and round in a channel, all that we have left is a little permanent magnet. In other words, we explain the constitution of a magnetic molecule by supposing that it is a molecule having a magnetic constitution. It is true that if we get a bundle of Hues of magnetic force and tie them together with a perfectly conducting cord with its ends joined up we have a permanent magnet, for no more and no less Hues can pass through the ring of perfectly conducting stuff. But this does not really explain. It does not show that the unknown is a case of the known. For a perfect conductor is far more difficult to think of than a permanent magnet. It is getting time that these so-called 'perfects' were abohshed from Physics. We have to deal with matter as it is and not as we should have had it if we had been consulted as to how it should be made in order to simplify our equations. The perfect gas has nearly gone and I shall be glad to see the perfect conductor preparing also to depart. If, however, we grant substantiality to the electric current, we are met by a difficulty in connection with Ampere's hypothesis which, at any rate, requires examination. Let us suppose that we have a circuit of self-induction L and resistance R, and let N be the number of outside lines passing through the circuit. If E is the E.M.F. of the battery kind in the circuit the current- equation is li E = and i? = we have an Amperean circuit for which 01* LC + N = constant. In words, the total number of Hues of force threading the circuit is constant. AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS 267 If, now, N increases, either L or C must diminish. In order that L may- diminish the circuit must contract and pinch in, as it were, the Hues of force. But meanwhile, since the total is constant, the number of hues passing through the same area is increasing, and we should expect the spin, therefore, to increase and tend to make the tubes of force widen out. That is, there would be resistance to the contraction. The contraction is contrary, too, to our ordinary experience of a circuit with a current in it, for such a circuit tends to expand. If, on the other hand, the circuit is rigid, L is constant and C naust diminish. If, as I gather from the general tenor of Dr. Lodge's work, the Amperean circuit is one of the ether cells peculiarly constructed or at least of the same order of magnitude, a diminution of C means a diminution of spin of this cell. But the total number of lines of force through it is constant, which seems to imply that the spin remains the same. This seems to force us to suppose that the Amperean circuit is not of the order of the ether cells, but larger, so that it may include within its contour a number of these cells. Then when N increases, C round the boundary may diminish, and still LC + N he constant, for the internal wheels may be spinning at the same rate as before, but the external wheels more slowly, the current being a function of their difference. It would require further examination to see whether the total spinning energy could thus be accounted for. At first sight it would appear that that energy is diminished by the lessening of C just when we want it to increase. If we are to suspend our judgment as to the substantiality of electricity, and are not as yet to conclude that a current is a rushing round of a thing of some kind, we are bound also to suspend our judgment as to Ampere's hypothesis, and if we are in this state of suspense, and, for my own part, I must confess to being so, Dr. Lodge's hypothesis and its accompanpng models are to be used rather as illustrations — as analogies — than as ultimate solutions. For the purpose of illustration they are of the greatest value. Like a bank- reserve, ready in case of emergency to cash the symboUc bank-note, they are in the background in the mind ready to turn into mechanical reahty the ordinary electromagnetic symbols, such as hues of force, which are, I think, much more easy to deal with, but which are, doubtless, wanting in reahty. Perhaps Dr. Lodge may consider that my criticism is somewhat carping, in that I have nothing constructive to put in the place of the hypothesis to which I object. But I beheve that the time has hardly come for ultimate mechanical construction, and that, at present, progress is more Ukely to be made if we are content with an electromagnetic explanation — if we merely carry down to the molecules and their interspaces the electric and magnetic relations which we find between large masses and round large circuits, and leave the ether out of account. I beheve that we may symbolise electric and magnetic actions by means of fines of force and their motions in a way 268 AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS which allows us to think clearly of the phenomena, and though the ultimate nature of the lines of force is unknown, we can only say the same of the ether. In the application of these lines of force to the molecules, and their constituent atoms, there appears to be hope for some kind of explanation of the chemical phenomena of the circuit and of the distinction between electrolytic and metalhc conduction. The difficulties in the way are by no means small, as everyone knows. I may, at some future time, attempt to set forth some of the difficulties as they occur to me, in the hope that a plain statement of them may lead some reader to help in their solution. 18. MOLECULAR ELECTRICITY. [Electrician, 35, 1895, pp. 644-647, 668-671, 708-712, 741-743.] In a Paper in The Electrician of October 13, 1893*, after some criticism of the 'wheelwork' hypothesis of electromagnetic actions, I concluded by saying that I believed that ' the time has hardly come for ultimate mechanical construction, and that, at present, progress is more hkely to be made if we are content with an electromagnetic explanation — if we merely carry down to the molecules and their interspaces the electric and magnetic relations which we find between large masses and round large circuits, and leave the ether out of account.' I propose to follow out this idea, to see where it leads us, and what difficulties we have to face. The idea is in many minds, and we have examples of its use in the papers of Prof. J. J. Thomson and in the articles by Mr. Chattock in Dr. Lodge's Modern Views of Electricity and the Philosophical Magazine. The fullest account yet published is, I think, in Prof. Thomson's Recent Researches, Chapter i, but there are some difficulties not there brought to the front. I believe there will be some advantage in a new statement, starting with the very alphabet of the subject, and following it up by full examination of the consequences. In the belief that some way may yet be found to remove the difficulties, and that even now it gives us a valuable picture of some electrical actions, I venture here on a full description of the shape it has taken in my own mind. The hypothesis with which we start is that electrical and chemical forces are identical; that electrification is a manifestation of unsatisfied chemical affinities, and that chemical union is a binding together of oppositely-charged atoms or groups of atoms. Before descending to the atoms, let us briefly consider what we observe on the large scale coming within the range of experiment. The foundation- stone of our electrical knowledge is the experimental result that the two kinds of charge are always found in equal amounts with opposite or neutrahsing mechanical properties, and that they always face each other on the two sides of the insulating matter between them. We add to this the idea that the * [Collected Papers, Art. 17.] 270 MOLECULAR ELECTRICITY energy of the charges accompanies some kind of strain — alteration, perhaps, of atomic or molecular configuration — in the insulating medium. The two charges, in fact, always join hands through the dielectric; or, putting the same idea in another form, the charges are manifestations on the bounding surface of a state of strain within the dielectric, somewhat as hydrostatic pressure is a manifestation on the surface of the vessel containing a Hquid of a strain to which the liquid is subjected. When this idea of strain was enunciated by Faraday and rendered precise by Maxwell we had only the variation of inductive capacity to support it. But now the double refraction of dielectrics in the field discovered by Kerr, and the wave phenomena dis- covered by Hertz, give as near a positive proof of the existence of the electric strain as we can ever hope to get. In a system at rest the electric strain ends at the surface of a conductor and the electric stress or pull on the surface 2tto'^IK per sq. cm. is resisted on the other side of the surface by ordinary mechanical stress accompanying ordinary mechanical strain in the matter of the conductor. Some day, perhaps, we may be able to manufacture a hypothesis identifying mechanical and electrical strain ; but at present we are obliged to separate them and think of them as different in kind, since mechanical strain in moving off or dying away does not give rise to the magnetic or chemical actions which characterise the moving off or dying away of electric strain. We symbolise the relative value of the strain at different points of the field by unit tubes of induction each beginning at + 1 and ending at — 1 of electrification. The strain varies inversely as the cross-section of a tube, and we regard 1/^, the reciprocal of the specific inductive capacity, as a kind of modulus of electric elasticity. Though tubes of induction give us a better description of the imagined physical condition of the dielectric, it is easier to use lines of induction, one for each tube, when we have to draw figures. These ideas of duality of charge, and of strain between them, which we owe to Faraday and Maxwell, though now accepted by everyone, have hardly yet so saturated our minds that we instinctively use them in every case. We still too often find descriptions of elementary phenomena which entirely leave them out of account. Like charges are still described as repelling each other, as when the gold leaves of an electroscope diverge, without a hint that the apparent repulsion is really a pull on each body along the fines of induction stretching out to the opposite charges induced on the surrounding conductors. The earth is still looked upon as a big conductor which will hold any reasonable amount of electricity without showing signs of it, and we speak of discharging a body into it instead of saying that the opposite charge is usually on the earth or the walls of the room in which we work, and that when a conducting bridge is made the two charges can come together so that the inductive strain is refieved, and its energy is dissipated. Or to take another example where MOLECULAR ELECTRICITY 271 common language and thought lag behind the more precise ideas given to us by science: the earth, as the 'return' circuit of a telegraph wire, is still described as a reservoir from which, say, + can be pumped up at the sending end and into which it can be emptied out at the receiving end, just as if a telegraph wire corresponded to a pipe running above a water reservoir with a pump at one end and a spout at the other. Sometimes even we find the process described as if the charge going to earth at the receiving end knew its way back through the earth to the particular battery from which it started. Whereas the earth is really not a return circuit at all, but a parallel out-going circuit for the opposite charge, enabUng the two electrifications to travel in company and to face each other — one on the wire, the other on the earth — whether the earth be the bare ground under an aerial line or the sheath of a cable in the bed of the ocean. This travelling of opposite charges to meet each other is to be regarded, I am convinced, as the essential electrical part of the ordinary current. The lines of induction connecting the charges sweep through the air or other surrounding insulator, and this motion of the induction, or travelling onwards of the electric strain, is accompanied by — indeed, is rendered possible by — the magnetic induction which surrounds the conducting wire. The only case of current without motion of charge is the gradual cessation of charge in an imperfectly insulating condenser. Here the charges simply die away in situ — their strain decays and their energy is dissipated ; and as there is no motion of induction there is no magnetic field. To realise how ordinary current may be described in terms of motion of lines of induction let us suppose that, instead of the more usual dynamo or battery as source, we have a charged condenser with the — terminal to earth and the + terminal insulated, but capable of connection by a key with a wire earthed at the farther end. Up to the instant when the key is put down the two charges are almost entirely in the condenser, joining hands in the dielectric between the plates. The still prevalent mode of describing the current on contact at the key is equivalent to supposing that, as soon as the + found a door open to it, it freed itself from the embrace of its — , put its hands in its pockets, ran along the wire, setting all the neighbouring magnetic machinery spinning as it rushed past, and finally plunged into the earth at the farther end. The — , no longer kept up or 'bound' by the +, sank into the earth at the condenser end, and both were lost in that vast electrical abyss, the globe. But let us see what is really implied in the process of discharge if we keep to our fundamental principle that + and — always have induction-tubes or lines between them. Let us try the various suppositions which seem open to us, and let us first imagine as a possibility that the charges are able to start on their travels by breaking their induction-tubes somewhere in the dielectric between the condenser-plates, as in Fig. 1. At the broken ends we must 272 MOLECULAR ELECTRICITY have opposite charges (just as in a broken magnet we have opposite poles), for a charge is the end on matter of an induction-tube. If each + unit setting out along the wire drags the broken half of its tube after it into the wire, we shall have on the whole equal + and — travelling in the same direction along the wire and no external magnetic effect. Indeed, as soon as the leading + has drawn its piece of tube completely into the wire there does not appear to be any reason against the union of the + and — at opposite ends of the piece, and then nothing need occur in or around the rest of the wire. But as we have magnetic and other efiects in and around the whole length of wire, and everything is against the passage of + and — in the same direction, we are bound to reject this supposition of breaking tubes. As another supposition let us imagine the tubes as remaining in the dielectric of the condenser but growing longer where they stand, and as it were pushing their end-charges before them, the one through the wire, the other into the earth. But this Fig. 1. will leave the tubes still in full strength in the dielectric, and when the dis- turbance in the wire has died away the condenser will still be fully charged, which is absurdly at variance with experience. The only possible supposition still left to us is that the tubes of induction spread out sideways from the dielectric of the condenser into the surrounding air, connecting the -|- and — charges as these move respectively along the wire and the earth. The induction— that is, the condition we call electric strain — is propagated by the working machinery from point to point through the air, and the motion of this machinery is manifested as magnetic induction, symbolised by lines of magnetic force which form closed circuits round the wire. We cannot explain this mode of transfer, but the mechanical models of Maxwell and Lodge and FitzGerald help us to accept it as possible. We must now introduce the idea that conductors are dielectrics — of a special kind, no doubt — admitting of electric strain, though allowing it to decay rapidly and dissipating the energy they receive in the form of heat. We may take as a helpful analogue to this behaviour of a conductor the behaviour of a liquid under ordinary elastic shear-strain. A liquid can receive shear-strain just as much as a solid, but the strain decays, that is, rapidly loses its energy, and the lost energy is transformed into heat. But MOLECULAR ELECTRICITY 273 the still undecayed strain remaining at any instant gives a tangential resistance to shear, which we recognise as viscous resistance to the motion of one layer relative to the next. So in a conductor such as a wire, electric strain may be. produced and electric energy may be taken in ; but the strain decays almost as fast as it is received and the energy is changed to heat. The e.m.f. along the wire is proportional to the strain remaining still undecayed at any instant^ and this e.m.f. corresponds to the elastic stress — the viscous resistance — of a sheared liquid. In the case of the condenser which we are considering, as the induction- tubes spread out from it their end-portions are continually moving sideways into the wire, and to a less extent into the earth, and there melting away and dissipating their energy as heat. New ends are continually being formed at the junctions of the decayed and undecayed portions, and these ends travel further and further along the circuit until the tubes are entirely propagated into the conductors, when the discharge is complete. A full account of this mode of regarding current, and the relations involved between electric and magnetic induction, with an attempt to show that it applies to the voltaic circuit as well as to such a circuit as we have here con- sidered, was given in the Philosofhical Transactions for 1885*. Prof. J. J. Thomson has modified and elaborated the theory in the Philosophical Magazine for March, 1891, and in his Recent Researches in Electricity and Magnetism, Chapter i. The full theory shows that if we assume that the magnetomotive force round a closed curve is equal to 47r x number of electric tubes cutting the curve per second, then the magnetic field is accounted for. If, then, we make the electric tubes move so as to account for the right electrical quantities, the magnetic properties will follow as a matter of course, and we may therefore concentrate our attention on the electric tubes, as we have done in the pre- ceding account of one case of current. I have given this case at length, by way of introduction, to show that here at least the only way in which we can think consistently of current in a wire is in terms of motion of + and of — , and of tubes of electric induction connecting them and sweeping through the surrounding medium into the wire. We shall see later how such tubes may be supposed to be furnished by the recombinations which occur in the voltaic circuit, and I have no doubt that in all other cases of current the same kind of explanation may be given. There is another reason, perhaps, for giving here the foregoing account of a condenser- discharge, in that the hypothesis which I am going to describe may not inaptly be termed a condenser-hypothesis of electricity. The properties which we find in condensers will be carried down to the molecules, which we shall suppose to be small condensers with equally and oppositely charged atoms forming the two plates. Of course, this will not explain * [Collected Papers, Art. 11.] p. n. w. is 274 MOLECULAR ELECTRICITY electricity. It is a purely electrical hypothesis, and it shifts all responsibility of further explanation on to the shoulders of whatever atomic hypothesis we adopt, just as Weber's magnetic hypothesis gives no explanation of magnetism, but assumes the molecules to be ready-made magnets, and leaves to them the burden of accounting for themselves. We are naturally led to make the hypothesis from the consideration of the chemical action of current. When a voltameter is included in a circuit a perfectly definite quantity of electrolyte is decomposed for a given number of induction-tubes moving sideways into it. All parts of the liquid between anode and cathode are concerned in what goes on, and there are two opposite processions of ions. This is all experimental fact. Kepresenting the action atomically we may obtain the net result by the old Grotthus chain method of picturing the process, and each atom set free requires the supply of a definite constant amount of electric induction, calculable if we know the number of atoms per gramme of the substance. Let us now consider how electrolysis may occur. We shall imagine a somewhat abstract kind of electrolyte, one consisting of molecules, each with a pair of atoms, an abstraction, a simplification, which we shall have to discard later. To account for the exact chemical equivalence of electrolysis in different cases we shall have to suppose each atom to be supplied with the same amount of electric induction, to have the same minute fraction of an induction-tube starting from it if it is of one kind, ending on it if it is of the other kind. To bring the induction up to a thinkable size we must choose a new unit — I suppose miUions of billions of times less than the ordinary unit, and it will be convenient to picture the atom as supplied with two, four, or some such small number of tubes in terms of this new unit. Here, then, is our molecule, represented in Fig. 2. A is the positive and B the negative atom. The molecule is, in fact, a little condenser with fixed charges, and the distance — ' g between A and B is usually small compared with their dis- pig. 2. tances from neighbouring molecules. But we must suppose that in general A and B do not come in contact, perhaps through motion of vibration or of rotation relative to their centre of gravity. There is therefore persistence or conservation of their tubes. The electrical attractions sym- bolised by these tubes are absolutely identical with the chemical attractions of the molecules, and the electrical energy they contain with the chemical energy of the molecule. It is easy to see, however, that two molecules colHding may become connected, and form a more complex group, and may even effect an exchange of atoms. Thus, in Fig. 3, (a) to (/) represent successive stages in the process of exchange. Just as, when two condensers are brought near with their tubes of induction running in opposite directions, cross-connections are formed by MOLECULAR ELECTRICITY 275 the coalescence and neutralisation of parts of the tubes, so here we may suppose that in a molecular collision the tubes straying out coalesce, and the parts which run in opposite directions destroy each other. In (c) we have a more complex molecular group. If the process is continued we may arrive at (/), which gives the original molecular configuration but with exchange of partners. If we are ever to attempt an electrical hypothesis of elasticity and cohesion we shall have to suppose that straying of tubes is always going on in solids and liquids, and that complex groups like (c) are always being formed, so that each molecule is attached to its surroundings. In gases not in an electric field the molecules may be regarded as much more self-contained. (rt) >' '' /k j\ (L) X ,(/) Fig. 3. When a line or tube of induction moves sideways into an electrolyte com- posed of such condenser-molecules it finds the molecules with their axes distributed equally in all directions. It will pick out those already facing in the direction suitable for it — those requiring the least energy for its purpose, and those alone need be drawn. In Fig. 4 (a) two lines of induction, XX, YY, are represented as ready to move into an electrolyte, in which AB, AB represent the suitable molecules. The successive stages are represented by (6), (c), (d), (e), and the final result is that the highest B atom is delivered up to the + electrode, and the lowest A atom to the — electrode, while all the intermediate atoms in the chain change partners, merely forming molecules like the original ones, but with their axes reversed. If more lines come in they will find new molecules directed as they require, and soon also the collisions occurring in the liquid will distribute the axes of 18—2 276 MOLECULAR ELECTRICITY the molecules of the first chain into various directions, so that some of these, too, will be ready for later lines of induction. The coalescence and destruction of the parts of two tubes which run alongside each other in opposite directions, upon which the whole process •f Etectrode. ..«___ 1^ /s X —Electrode Fig. 4 (a). Fig. 4(6). depends, will naturally tend to take place, as less potential energy will be needed for the coalesced tubes than for the two side by side. The energy of the final configuration is the same as that of the initial configuration, except at the two ends, so that if XX and Y Y bring in enough energy to provide for the new configurations at the electrodes the process will continue. If XX MOLECULAR ELECTRICITY 277 and 77 bring in an excess of energy we may regard the change of partners as occurring with more or less of a rush, and the excess is converted into internal molecular energy, partly kinetic, partly potential; in fact, the chain y \ \ J > f It \ f i i V 1 < Fig. 4 (c). Y ) c Sr B Sr f Q Sr B r^-] V Fig. 4 {d). of recombined molecules is warmer than it was, and we have the Joule C^R effect. For simplicity we have considered the tubes XX, 77 as stretched through free space till they come to the chain of molecules which they are to rearrange. But it is easy to see how a tube may be handed on from 278 MOLECULAR ELECTRICITY y X -f- Electrode N' V B V X Fig. 4 (e). — Electrode Fig. 5 (a). X Fig. 5 (&). MOLECULAR ELECTRICITY 279 chain to chain without any free existence. In Fig. 5 (a) a tube XX already threads a chain of AB molecules and in (6), (c), (d) are shown the successive stages of its transfer to a neighbouring chain. The process is rendered more capable of representation by supposing each member of the second set of molecules lifted its own depth upwards. The derivation of each stage from the preceding is evident. This is doubtless the process which we must Fig. 5 (c). Fig. 5 id). suppose to occur in all cases, and the tubes must be regarded as threading such chains in the dielectric, even before they enter the electrolytic cell. Let us now consider what must happen at the electrodes. If the ions unite chemically with the electrodes, the positive and negative atoms find respectively negative and positive atoms ready to combine with them, and no difficulty is introduced at this point. But if the ions are set free, as are hydrogen and oxygen at platinum electrodes in dilute acid, we are brought 280 MOLECULAR ELECTRICITY face to face with the great, and, it is to be confessed, the yet unsolved difficulty in the hypothesis. Take the case of the hydrogen atoms released at the cathode, the atoms such as the lowest A in Fig. 4 (e). We may think of them as connected at first by tubes of induction going from them to platinum atoms on the surface of the cathode. But soon the gas bubbles up, and there is no manifestation of the charge all of one kind which we have ascribed to the separate hydrogen atoms. We may attempt several explanations. The common notion appears to be that the hydrogen gives up its charge to the platinum, and then rises up quite deprived of electricity. This might occur by the breaking away of the tubes from the hydrogen atoms and the with- drawal of the positive ends into the platinum, the positive charge moving across the separating space, like a disembodied soul, from its hydrogen habitation into the platinum. The platinum having two neutralising souls, becomes as merely material as the deserted hydrogen. But this is strongly against our experimental knowledge of charge, which is always, so far as we know, on matter, whereas here charge crosses over a separating space. We might suppose this got over by thinking of the hydrogen and platinum atoms as coming absolutely in contact, shortening the connecting tubes to nothing, and putting them out of existence. If the tubes are not re-created when the hydrogen rebounds the gas rises up without electricity. But, according to the current ideas of chemistry, the hydrogen issuing from the cell is not atomic, but molecular, and consists of paired atoms at least; that is to say, it is a chemical compound, differing from ordinary compounds it is true, in that the two members of each molecule are the same in kind, but the atoms are held together by chemical attraction, and satisfy each other's chemical affinity just as much as if they were different elements. If our hypothesis has any truth in it, this means that electric induction exists between the atoms and holds them together, a view supported, I think, by the phenomena of electric discharge in gases and the electrolysis which appears to take place in that discharge. To be consistent, then, we must suppose that the hydrogen, when it rises up, consists of pairs of + and — atoms, and for us the difficulty is to explain, not what has become of the hydrogen charge, but how half the atoms have succeeded in exactly reversing their charges, so that they are able to combine with the other half to form neutraHsing pairs. And, of course, we must extend our supposition of the constitution of an element as being made up of + and — atoms to the platinum also. When the lines of induction move into the platinum, the — atoms of the surface pairs are freed from their + partners, and are free to form pairs with the + hydrogen atoms. But when half the hydrogen reverses its charge half the platinum must do the same in order that it too may effect the neutral combination which we find when the electrolysis has ceased. Perhaps the best course is to say that if our main hypothesis of identity of chemical attraction and electric induction is real, then the reversal of charge, such as MOLECULAR ELECTRICITY 281 occurs on our supposition at the electrodes, is up to the present an unexplained, an ultimate fact. Without attempting explanation, but rather as a crude mode of picturing a process which would lead to the result, we might think of a + H and a — Pt as coming absolutely in contact, so that the tube-ends can shde round on the surface of the atoms and across the bridge of contact until + is on the Pt and — on the H. Possibly Fig. 6 will make this clearer. Fig. 6. In (b) two atoms have come in contact, in (c) the tubes are beginning to slide round, in (d) the ends have changed places, and in (e) the atoms have drawn apart with reversed charges. Having thus, or otherwise, reversed half the pairs, combination with the other half may ensue as in Fig. 7, the result being hydrogen molecules and platinum molecules each consisting of + and — pairs. At first sight it seems as though the reversal occurring in the spark-dis- charge of a Ley den jar or a Hertz vibrator would give us the key to the reversal pictured in Fig. 6. But further examination takes away this hope. The jar-reversal is never complete and exact such as we have to suppose the atomic reversal. Indeed, the induction may have any value, -f or — , less numerically than the initial value, as it gradually dies down to zero through the successive vibrations. We shall see later how some account may be given of the reversal in vibrators, an account which quite destroys any analogy with atomic reversal. Perhaps a better analogy is afforded by two vortex rings A and B when they are playing at leap-frog. The first widens out and lets the second go through it. Before the passage the liquid is streaming through A towards B. After the passage it is streaming from B towards A, so that if we think of induction as corresponding to direction of flow we have here a reversal. 282 MOLECULAR ELECTRICITY Pt Pt (a) (b) Pt But I suspect that this liquid- analogy is much more appropriate to a magnetic than to an electric hypothesis. Were it not for the magnetic difficulties involved there would be some temptation to attempt a theory in which the sign of charge is merely a state- ment of the kind of atom concerned, and that the tubes, Hke gravitation- tubes, have nothing but ahgnment and neither + nor — direction. But this would, I fear, be getting over one difficulty only by introducing much greater difficulties hereafter. We shall assume, then, that in some way or other reversal can take place in certain cases, and we may apply it at once to explain the apparently neutral condition of the ions in the electrolytic cell. We suppose that half the ionic atoms in Fig. 4 reverse or change charges with the electrode atoms, as in Fig. 6, and that then they go through the process of Fig. 7 with the other half, and so form + and — pairs, exhibiting no external electrification. A similar reversal will enable us to give an electrolytic account of metallic conduction. The great distinction between electrolytes and metals is that in the former there are opposite atomic processions, while in the latter there is apparently no motion in either direction. Electrolysis with- out procession means, as we have just seen with the electrodes, where the procession stops, re- versal of charge. Suppose, then, that such reversal is possible with every metallic molecule. Imagine a copper wire carrying a current to be made up of molecules, as represented in Fig. 8 {a), and let tubes of induction XX, YY be just movmg into the wire. Before the current is established the molecular axes are evenly distributed in all directions, but the tubes entering in select those most suitable, as shown in {a), turning the configuration to that shown in (6) ; if now reversal takes place we get (c). Let two more tubes come in, and we get (d), a reversal of which gives the same arrangement as (a) as far as the copper is concerned, but the four entering tubes of induction have entirely disappeared. We must now attempt to give some account of the action at the source of a voltaic current. As we simplified the electrolyte so we shall simplify the active liquid of the voltaic cell by imagining that we have merely sulphuric acid, that is, we shall neglect the solvent, water. We shall take as the two metals zinc and copper. We know that the result of putting the copper and zinc in the acid is that (c) Fig. 7. MOLECULAR ELECTRICITY 283 the air above the acid tends to become the seat of tubes of induction running from copper to zinc. This will be neutralised by the oxidising tendency of the air on the zinc— at any rate that is the 'chemical theory' of voltaic action. To eliminate this action of the air we shall suppose the zinc to have Fig. 8 (a). -Cu. ■Cu^ . Cu, •Cu, .Cu, ii Ou. -Cu. .Cu^ -Cu. -Cu, T Fig. 8 (6). U OUc -Cug .Cu^ Cuj Cu. Cu, ■ Cu, Cu, Cu, -Cu, Fig. 8 (c). Fig. 8 id). a copper terminal in the air, as it always has in practice. Experiment shows that these tubes of induction always run from copper terminal to zinc terminal with a fall of potential which may be as much, say, as a volt and a half. To represent the establishment of this induction let us imagine the cell initially to consist of such molecules as are shown in Fig. 9 (a), where we represent the 284 MOLECULAR ELECTRICITY acid molecules as consisting of + Hg paired with - SO4, and the copper and zinc plates as ± Cu and ± Zn. Of course, the molecules in each substance are evenly distributed in all directions, but only those are shown in the figure which are suitably directed for the action which is going to occur. The first stage is represented in Fig. 9 (6), where change of partners has occurred between metal and acid both with copper and zinc. The HgZn molecules do not reverse, for it is an experimental fact that the hydrogen does not rise up where it is first turned out. The reversal may perhaps be prevented by the presence of the ZnS04 Cu- Cu j\ /( ■Cu • Cu Ou- Cn- SO^ >S Ji Zn — H SO4 > 1 d k , , f —< — < r- H 7n- *^ SO4 H2 SO4 Hg Fig. 9 (a). •Cu ■Cu -Cii Cu Zn Z-n SO^ SO4 Hj SO4 Hj Fig. 9 (&). SO4 Cu Cu molecules, but it is more probably due to the electrical energy put into the HgZn. For the ZnS04 contains much less energy than the H2SO4 it replaces ; so that the HgZn contains much more than the ZnZn it replaces. The electrical energy put in makes the atoms separate too widely, we may imagine, to allow of the contact needed for reversal. Probably at first there is no reversal in the HgCu molecules at the other plate, and for similar reasons. The electrical energy in the HgZn and HgCu molecules will imply that the acid is at a higher potential than the metals, and tubes of induction will spread out from the Hg atoms. Since the HaZn molecules .contain the most energy, we may represent their tubes as spreading rather than those of the MOLECULAR ELECTRICITY 285 HgCu, and in Fig. 9 (c) we have the tubes shown as going out towards the copper and then doubling back again. In (d) the tubes have entered into the neighbouring molecules by coalescence with oppositely-directed tubes, and we are left with two tubes running from the + to the — terminal with a ZnS04 molecule and an HgCu molecule, the CUSO4 molecule having been dissociated in the process. The molecules of acid have changed partners, but still have the same constitution. If the action is not continuous, but merely goes on till the terminals are charged, we must suppose that HgZn and HaCu pairs are left against each Zn SO4 Cu 8O4 Hj 8O4 Hg Fig. 9 (c). Ou Cu „J_L M M M Zn 80^ -Cu -Cu Cu Fig. 9 (d). plate with an average fall from H to metal of, say, 1| volts. The pairs against the copper plate are no doubt the agents in polarisation. The reverse current which we get on replacing the zinc by a fresh copper is to be set down to the straying-out of the tubes of induction of these HgCu pairs. If the action is made continuous by connecting the terminals with a wire, the hydrogen rises up from the copper plate, and we must suppose that half the HgCu pairs have reversed and have then changed partners with the other half. Perhaps this reversal is rendered possible by the resolving of the CUSO4 molecules, perhaps by the outlet provided for the energy of the HgCu pairs in the external circuit. 286 MOLECULAB ELECTRICITY In a very similar way we may account for the contact- difEerence of potential of copper and zinc in air. We know that if the two metals are brought into contact a fall of potential occurs from the air near the zinc to the air near the copper ; that is, electric induction passes from the neighbourhood of one to the neighbourhood of the other. Both metals can be oxidised, but zinc by far the more readily. Let us suppose the oxygen molecules in the air to be made up of pairs of opposite atoms, and that before contact we have a state of affairs, represented by Fig. 10 {a), developing into (6) by the actions at the metals. On each metal we have the normal oxides ZnO, CuO, where the positive atom comes first, and the unstable molecules OZn and OCu. If the tubes of induction of these unstable molecules stray out they have to double Zn- Zn- U V Zn- Zn- qO _ o o ""^~ Q . . ' 1 ' ' O A-« /■H ■ — < — Oo o Fig. 10(a). Zn- .1 >i r V -Cu -CU -Cu -Cu -Cu -Cu Zn Zn O < o *: Cu Cu o o Fig. 10 (&). back on themselves, so that as long as there is no contact there are equal numbers in the two directions passing through the air, and no fall of potential except from the surface-layer of the air to the surface-layer of the metal close to it. This fall will be greater at the zinc than at the copper, since there is presumably more energy given up at the zinc surface. Probably reversal is prevented either by the presence of the normal oxides or by the want of outlet for the energy. If, however, it does occur in some of the unstable molecules there will be re-pairing and the formation of new metal molecules and new oxygen molecules, with the net result that each metal is left slightly oxidised. But let us make the two metals touch as at J in Fig. 11 {a). If the tubes of induction of the OZn molecule now move out Uke those of the HgZn molecule in Fig. 9 (c), the upper returning part can enter into the continuous metal bridge and there be dissipated; while the lower outgoing part will MOLECULAR ELECTRICITY 287 thread the oxygen molecules, as shown in Fig. 11 (b), and decompose the CuO molecule already found. We shall as a net result have ZnO on the surface of the zinc, a fall of potential from on the zinc surface to on the copper surface, and the unstable OCu, which possibly reverses and re-pairs, leaving the copper surface unacted on. If it remains then the copper is polarised, and if the zinc were suddenly removed and replaced by a new copper it would appear that the old copper should show a fall of potential towards the new. Zn Zo Zn Zn Zn Zn Oy Co Cu Cu n >i < — — < — — < — — e O O GO < 1 t < — I I < 1 I ^ > 1 I — < 1 I < — I I ► o o O O O o > — — ^ — — < — — )> Cii -Cu Cu Cu Fig. 11 (a) o o o o Fig. 11 (b). Zn Cu Fig. 11 (c). But any such fall would probably be disguised by the induction ending in the atoms next to the copper. We have probably gone too far in this account in supposing that all the induction of the OZn molecules goes out in this way. If, for example, we suppose that only half goes out, we get a chain extending from Zn to Cu, as in Fig. 11 (c). Probably some such supposition must be made in order to explain how the + atoms remain at the zinc surface and the — atoms at the copper surface after the break of contact. That they do remain is shown by the -i- electrification of the zinc and the 288 MOLECULAR ELECTRICITY — of the copper when tested by an electrometer. But our account must be regarded as a first attempt and not as a complete explanation. m L i ' M i f . •^ i t. < ' ' k , . . ^ r Fig. 12(a). \< .1 • Fig. 12(c) ■!■' - - w 1 ' y 't — u !■ , ,, Fig. 12(e). Fig. 12(6). V ^^ >^ V V V u >r >r Fig. 12 (cZ). Leaving electrolysis, let us consider how an ordinary insulator may be affected by induction, and how we may represent the condition of affairs leading up to spark-discharge. We shall thus get some useful ideas which will supplement our account of electrolysis and make it somewhat easier to understand the true nature of the process. MOLECULAR ELECTRICITY 289 We shall suppose that we are deaUng with a dielectric consisting of paired + and — atoms, and we shall suppose, further, that these molecules are initially self-contained — that is to say, that their tubes do not stray out to surrounding molecules. This is, no doubt, a simplification not existing in nature; but, as with electrolysis, so here also we must be content to begin with an abstract case. When there is no apparent electrification in the system the axes of the molecules will be equally distributed in all directions. But if electrification is communicated to a pair of conductors bounding the dielectric, tubes of induction move into the dielectric, and, selecting the suitably arranged molecules, connect these in chains. It will be convenient now to suppose at least four tubes of induction to pass from atom to atom in a molecule. Let Fig. 12 (a) represent a number of molecules ready for a tube of induction to affect them. In (6) it has moved in and formed the molecules into a chain stretching right through the dielectric. If we suppose another tube to move in, as in (c), we get a condition of instability, for now we are just half-way to a change of partners all along the line. A third tube will give us (d), a configuration with the same amount of energy as (b), since the molecules in the two cases are similar, the axes only being reversed. Hence in passing from (c) to (d) energy is given up, another way of saying that (c) is unstable. A fourth line entering will change {d) to (e), where the change of partners is complete, and where electrolysis has occurred. We may usefully follow out the process by the aid of a diagram representing the relation between energy put in and induction. Beginning with an ordinary condenser, let distances along OX (Fig. 13) represent induction put in per unit area cross-section, distances along Y difference of potential between the end- plates. At first we have V = r^ > p. c. w. 19 290 MOLECULAR ELECTRICITY where V is the potential difference, d the thickness, a the surface- density, and K the specific inductive capacity. But since the induction D = cr. If then K is constant, the relation between F and D is represented by a straight line, which makes with OX an angle 6 given by 4:77(Z tan c/ = -^ . The energy stored per unit cross-section is equal to -^, or equal to the area from the origin up to the ordinate V, and bounded by the hne representing the relation between F and D. We may indeed regard the diagram as showing the relation between induction and energy stored, and this is probably a better point of view, since we cannot attach much idea to potential in the later parts of the process now to be considered. From this point of view the abscissa is the induction through unit area and the ordinate is the energy added per unit addition of induction. All experiments hitherto made appear to show that K is practically constant so long as the medium can continue to store up energy, though the Kerr effect shows that K does alter slightly as D increases, apparently sometimes increasing and sometimes diminishing, or, perhaps, always in- creasing, but sometimes most along the lines of induction, at other times most at right angles to them. Sooner or later, however, a point of instability is reached — sooner in gases, later in liquids and solids ; and discharge occurs along one track and all the energy is dissipated. We can see how the curve in Fig. 13 must run in order to represent this instabiHty. Making the unit area small enough to represent the cross-section of a molecule, the curve expresses the relation between induction going right through a chain of molecules from plate to plate and energy put in. At first, while only a small part of the induction is continuous, as in Fig. 12 (6), we know that the energy stored is proportional to the square of the induction, and the curve is, as we have seen, a straight line. But as more induction becomes continuous from plate to plate, and as the atoms are pulled in both ways, the force resisting separation does not go on increasing so rapidly, and the curve falls below a straight line. At some point on the way to instability and subsequent change of partners the force will reach a maximum, and after that the curve will turn down, successive equal additions of induction requiring diminishing additions of energy. At last, when the fines of induction from each atom run half one way, half the other, as in Fig. 12 (c), the energy put in is a maximum, and the curve crosses the X axis, as at A (Fig. 13). The configuration is now unstable. MOLECULAR ELECTRICITY 291 and will of itself pass through (d) and (e), absorbing two more positive hnes or extruding two negative hnes till we arrive at (e), represented by the point B in Fig. 13, when the condition is again stable, like that at the beginning. The energy put in between and A is given out again between A and B in part, no doubt, as the Hght, heat, and so on of the discharge. The curve OAB may be termed, perhaps, a molecular characteristic. It should be noted that the passage from (c) to (d) (Fig. 12), or past A (Fig. 13), may be effected D D Fig. 14(a). Fig. 14(6). Fig. 14 (c). M V ^' " A V ^r Fig. 14 id). either by absorbing more positive tubes, or by sending out negative tubes. If the passage through the position A occurs with a rush, then we may regard the chain as sending out negative tubes by some such process as is illustrated in Fig. 14, (a) to (d). Though the instability of a single chain will not be reached till its condition is represented by A (Fig. 13), the instability for a great number of parallel chains is reached as soon as they are all at or near the highest point of the characteristic. In the ascending part of the curve all the parallel chains will tend to have the same amount of induction through 19—2 292 MOLECULAR ELECTRICITY them, for if any one has more than another, energy will be yielded up by a redistribution between them. Thus let one chain have induction OM (Fig. 15), another near it induction ON, the energies stored being OMP, ONQ. If L bisect MN the second chain may give up induction LN = LM to the first, and at the same time there will be a yield of energy equal to the difference between the areas PL and QL. Probably the induction is really distributed about an average, some of the chains having more, others less, for no doubt the condition is kinetic, and when there appears to be equilibrium it is not static but ' mobile.' Fig. 16. If the average condition is represented by a point far up the slope and near the crest, and if any one of the chains gets past the crest, as far below it on the other side as the average is below on the first slope, then this advanced chain will at once receive induction from the others, and continue to move down the second slope towards the point of instabihty. Thus, let there be n chains in all. Let OH A (Fig. 16) be the characteristic of that with the greatest induction, OKA' the sum of the other n — 1 characteristics. If the first chain is at P while the others are at p on the same level, by a transfer of induction mn = MN from the general body in the first chain, there is a yield of energy, since the area of pmnq is greater than the area of PMNQ, for the two slips are of the same breadth, but the slope of PQ is steeper than that of fq. The transfer will therefore take place, and the chain moves from P to Q. At Q, a fortiori, a new transfer will take place, and so on, and the chain will move towards the position of instability. This will all probably MOLECULAR ELECTRICITY 293 occur even when the average is far below the crest if the induction is widely distributed about that average, for as soon as one chain gets over the crest it will probably find others between it and the average ready to hand on their induction and energy to it, and send it down the second slope. As, then, the general average rises towards the crest, the most advanced chains, as soon -as they pass the crest, tend to discharge the rest, and there will at a certain point be a rush in sideways on to these advanced tubes. They will move down to the condition of instability represented by A on the diagram. They can of themselves move past this point, extruding negative tubes, till they arrive at the point B, where the change of partners is complete. It appears at least probable that this yield of negative tubes gives the opposite charging of the medium which takes place in the second quarter- period of the oscillation accompanying rapid discharge, now made so familiar by the work of Hertz. We may, perhaps, think of the process somewhat as follows: If the inrush of positive tubes during the first stage is rapid they will concentrate on the chains in the neighbourhood of that which began the breakdown, and carry them past the unstable point. Then will begin the outrush of negative tubes, and when this is complete the central chains will be discharged and in the condition of Fig. 12 (e), while the surrounding medium will contain negative induction. There will be now an inrush of negative induction into the locus of the first discharge, for not only is this free from induction, but also its molecules are suitably arranged to take up negative tubes, and during the second half-period of swing from negative to positive there will be a second change of partners along the same line. And so on with the successive alternations of charge, and there is a tendency, evidently, to keep the same line of discharge. Each change will give atoms at the two end-plates, which will combine either with each other or with the electrodes, or perhaps be taken again into the chains in the following changes. In the sudden changes of partners some of the energy goes to atomic vibration, and we have evidence of this in the atomic radiation which we call spark. There is also energy of translation of the atoms from one partner to another, which appears as heat in the molecules. This heat possibly produces the sound of the spark through the sudden expansion. We have then dissipation of energy as well as the radiation out to space in the Hertzian waves, and the two gradually reduce the electrical energy of the system. We can see, too, how the amplitude of charge may lessen in the process. For if the negative tubes begin to move out at the middle of each oscillation before the positive tubes have all moved in, or vice versa, then the first of the issuing kind will destroy the last of the incoming kind, and the final charge will be diminished by the amount of this overlap. In the extreme case of slow discharge, as through a wet thread, the issuing tubes are neutralised as they come out by the incoming ones, and the motion is dead-beat. 294 MOLECULAR ELECTRICITY These negative tubes turned out may also supply the negative tubes required in the theory of Prof. J. J. Thomson (Recent Researches, Chapter i). Now let us take a conducting dielectric such as water. . As guiding us to an account of what goes on we have the facts (1) that, however small the E.M.F., conduction and presumably electrolysis take place, and no finite difference of potential can be maintained between the electrodes unless we continually supply fresh energy, and (2) that a solute such as sulphuric acid, which appears to combine with the solvent in some way, enormously increases the conductivity. From (1) we gather that some of the molecules must be just ready for change of partners, and from (2) we may at least guess that molecular groups are formed much more complex than the atomic pairs we have hitherto dealt with. We can see how such molecular groups might arise by the straying out and coalescence of induction-tubes of neighbouring molecules, and the consequent formation of new connections. Thus, if two OC '^ (fl) a {(') Fig. 17. Fi2. 18. pairs come together, as in Fig. 17 (a), we may have them connected into a single group, such as (6), and any number of molecules may be brought into circuit in the same way, so that we may have a parallel arrangement as in Fig. 18. Or if the coalescence occurs by the approach of two or more groups, such as Fig. 17 (6), we may have a series arrangement as in Fig. 19, the kind which we shall suppose to be effective in conduction. We must further suppose that small quantities of acids or salts in solution enormously increase the number of complex molecules. Some of these groups would appear to be more energetic than the initial pairs. Probably on that account they are continually breaking up and re-forming, the energy of translation being no doubt converted at each collision and reformation into energy of electrical separation. There are no doubt all degrees of connection from those of Fig. 17 (h) and Fig. 19 (where the result may be obtained practically by adding one closed ring-tube of induction to a series of pair-molecules) to the case where the tubes from each atom go half one way and half the other. MOLECULAR ELECTRICITY 296 But I imagine that at any given instant only a small fraction of the molecules are thus connected into groups or circuits. If a tube of induction running from above downwards moves sideways into such a group as that in Fig. 19 or Fig. 20 (a), it finds the right-hand side made ready for it, and we may possibly have in succession Fig. 20 (a), (b) and (c), where we suppose that the left-hand side splits up into pairs, while the right- hand side remains threaded on the incoming hne. When a liquid contains many such groups, some of the chains of molecules are almost ready-made, and the chains are very easily completed from plate to plate, since the entering tubes have only to furnish a link, as it were, here and there. The ' electric elasticity' \t yr yr >r V y Fig. 19. Fig. 20(a). >f >r 1, >r V ^' ;. M '< >r \f >r i^ A M >f V >' ^f >k i^ ik Fig. 20 (&). Fig. 20 (c). IjK will be a sort of average of the elasticities or of the difficulties of forming chains in all the various kinds of molecules present. To give a numerical illustration, let us have two condenser-plates in a liquid in which we suppose that in the simply paired molecules, such as those on the left in Fig. 20 (c), the value of K is 1-77. Let us suppose also that these occupy 99 per cent, of the paths from plate to plate, while the other 1 per cent, is occupied by very- much-connected molecules with K equal to 7500. Then if the capacity of the condenser in air were 0, it would be with this arrangement of molecular paths 99 X 1-77C + 7500C 100 = 76-76C, or the resultant specific inductive capacity is 76-76. 296 MOLECULAR ELECTRICITY But this is only true in the mass. If we have very minute electromagnetic waves going through the liquid they will, for 99 per cent, of the molecules, use K= 1-77. For a very large fraction of the remaining 1 per cent, they will probably also use this K, for they are not concerned with the group as a whole, but only with the individual members to which they add or from which they subtract small quantities of induction ; and as long as the points repre- senting these members of groups are on the straight part of the characteristic their K for waves is still 1-77. We may therefore expect the value for small waves generally to be very little more than 1-77. This appears to indicate a possible explanation of the high inductive capacity of such substances as water and alcohol. Accompanying this high value there is generally conductivity and no doubt electrolysis. If we suppose that some of the groups are already so far towards decomposition that half the tubes from each atom run one way and half the other, tubes entering the substance will concentrate on such groups, for a further addition of induction will yield energy instead of requiring it, and the point of instability being passed electrolysis will occur. Perhaps even before the tubes are thus evenly divided, tubes entering the substance may prefer to pass through the groups rather than through the simply paired molecules, for while more energy may be stored in the one half of such a group as that represented in Fig. 20, when a new tube enters in it, less may be required in the discarded half, and so, on the whole, energy may be given up. But it does not seem possible to give any satisfactory account of the process in our ignorance of the real constitution of the complex molecules. All we can say is that some such process probably occurs, inasmuch as conduction does occur even with the smallest external e.m.f. We may perhaps suppose that in metallic conduction we have a similar process. If the metallic molecular structure is very complicated, with many groups having unstable, or nearly unstable, construction, then tubes of induction entering a metal will select such groups in preference to the more simple stable molecules and electrolyse them. To account for the absence of transfer of atoms along the line of current we must, I think, introduce the supposition of change of charge as already explained on p. 282, and to account for the continuance of conduction we must suppose that there is a continuous formation of new groups as the old ones are broken up. Perhaps we have here some key to the rise of resistance with rise of temperature. As the molecules become more energetic we may expect that the groups will be more broken up by the motions of vibration and translation, and that the number of unstable groups is diminished and their rate of formation is decreased. Hence the entering tubes will find fewer groups ready for them, and if the external e.m.f. remains constant, the rate at which the tubes are dissipated will be decreased. The difficulty of such explanation consists in MOLECULAR ELECTRICITY 297 understanding its inapplicability to electrolytes. Perhaps, too, we have here a hint as to the superior conductivity for heat of metals. If a metal consists largely of many-atomed groups entangled together, and continually breaking up and re-forming, energy supplied to one molecule will, we may imagine, be more readily transferred to its neighbours than if each molecule is self- contained and permanent. This will certainly be the case if energy given to an atom in a molecule is more rapidly transferred to its fellows in the same molecule than to atoms at the same distance in the surrounding molecules. To use an illustration which may put the suggestion in a clearer way, news will be transmitted through a population dwelling in villages and towns much more rapidly than through an agricultural population of the same average density scattered over a country in isolated homesteads. The theory which I have been trying to set forth may be regarded as a theory of the conservation of induction-tubes, and of their beginning and ending on atoms. That the atoms always have charges on them, and there- fore have tubes proceeding to or from them, appears to be generally held as necessary if we accept the electromagnetic theory of light. If the molecules give rise to waves of electric induction they must necessarily be electric systems, and their parts must almost certainly be bound together by electric forces. Whether it is possible for an induction-tube to exist without atoms is at present merely a matter of speculation. At present we know of no such thing. If such a tube exist, it can only be as a closed ring, like a closed ring-tube of magnetic induction round a current, for an unclosed tube would have opposite charges at its ends in free space, and charges not on matter are so entirely outside experience that we cannot accept their existence. The weight of evidence appears to me rather against the view of matter-free induction, and though at first we might be inclined to think that the passage of light-waves across interstellar space implied such induction, yet even in this case we have possibly quite enough matter to supply atomic ends for the tubes to attach themselves to. If we accept the electric discharge theory of comets' tails we apparently assume the existence of enough matter to carry electric induction from a cometic nucleus outwards. I suppose that the theory implies that the nucleus is charged in one way, say positively, and that it has become separated from the matter bearing the negative, which remains far out in space. The sun is itself to be regarded as charged with the same sign as the nucleus, while the corresponding solar negative is also somewhere in space. When the induction of the comet is added to that of the sun the strain is sufficient to break down the feeble insulation of inter- planetary space, and a discharge results straight out, or nearly straight out 'from the nucleus, and this discharge is through the interplanetary matter. But though this appears to be the only reasonable account of the discharge- theory, after all it is bringing Httle more than a speculation to bolster up 298 MOLECULAR ELECTRICITY another speculation, viz., the existence of matter sufficient to carry waves of induction in the interatomic form wherever light- waves travel. At first sight this theory of molecular electricity appears to be very different from the chemical dissociation-theory now generally held ; but if the dissociated atoms of that theory have charges, they have also tubes of induction proceeding from the charges. When the tubes are taken into account they must, I believe, lead to some such hypothesis as that of which I have attempted an inadequate and imperfect explanation. I am only too deeply conscious of the difficulties unsurmounted. But in working at the subject I have felt all through that, since so much is nearly but not quite explained, there must be hope of progress on these or similar Hues if we can only supply some as yet unrecognised idea. Perhaps the very imperfections of my account may stimulate some reader to take up the subject afresh from some better point of view, and, with new ideas, achieve success. PAET III. WAVE PROPAGATION— KADIATION— PRESSURE OF LIGHT— AND RELATED SUBJECTS. 19. NOTE ON AN ELEMENTARY METHOD OF CALCULATING THE VELOCITY OF PROPAGATION OF WAVES OF LONGITU- DINAL AND TRANSVERSE DISTURBANCES BY THE RATE OF TRANSFER OF ENERGY. [Birmingham Phil. Soc. Proc. 4, (1885), pp. 55-60.] [Bead Nov. 8, 1883.] Waves of Longitudinal Disturbance. A wave of sound may be considered as energy of a particular type, partly potential and partly kinetic, which is being passed on from point to point through the medium, so that the energy which is at any moment occupying a particular portion of space will have passed in a second later to a distance equal to the velocity of sound. Of the two energies the potential is due to the strain of the medium, and, when in this strained condition, each part of the medium exerts force on the neighbouring parts. But it also has kinetic energy, that is, the part considered is in general in motion and there is therefore motion of the point of application of the force which it exerts on the contiguous parts through the existence of its potential energy ; that is, it does work and passes on energy to the contiguous parts. If we consider for instance a series of plane waves of sound which move on unchanged, the work done in any small time t at any plane perpendicular to the direction in which the sound is moving must be equal to the sound-energy contained in the space immedi- ately behind the plane through which the sound will travel in the time t. This gives us one relation between the various quantities, and we obtain another from the consideration that any condition as to velocity and dis- placement which is now at a particular point will have travelled on unchanged 300 VELOCITY OF PROPAGATION OF WAVES OF LONGITUDINAL AND in the time t to a distance Ut where TJ is the velocity of sound. From these two relations we can at once find the velocity TJ*. Let BA be the direction in which the sound is travelling, and let AP be the trace of a plane perpendicular to AB. Draw a curve CPQD whose height above each point of AB shall represent the displacement of the particle at that point (these displacements are actually of course along AB). Thus AP represents the displacement at A along AC, and MQ the displacement at M along MA. Consider a small volume V with unit area on the plane through AP as base and height AM = V. The volume of the medium which had height F before the disturbance reached A will now be compressed; for the end A has moved forward a distance AP, while the end M has moved forward a greater distance MQ. The compression is therefore the difference between these, viz.; QR = v, say. But if the sound takes a time t to travel over MA, p Q,__ 1 — "^'^ n ^ ■^ R ^^^^^-\ c A M DB after that time the displacement of A will be equal to the present displacement of M. Or if u be the actual velocity of the particle at A, Qn = v= ut (1) Now, considering the energies, we have the kinetic energy of the volume V = p -^, where p is the density of the medium. The potential energy equals the work done in compressing. If P is the original pressure, and P + f the pressure in the compressed state, P -f ^ is the average pressure during the compression, and the distance through which this has moved is numerically equal to the diminution of volume v. Then the potential energy is (p.g. * This method of treating the subject of wave- propagation is given by Lord Rayleigh in a note in vol. 9, no. 125, of the Proceedings of the London Mathematical Society (republished at the end of vol. 2 of his Theory of Sound). This paper is merely an application of his method to two particular cases. TRANSVERSE DISTURBANCES BY RATE OF TRANSFER OF ENERGY 301 The work done in t sees, across unit area at A is equal to the pressure exerted by the medium to the right on the medium to the left, multiplied by the distance which the particle has moved in the time t, or (P + p) ut. We may equate this to the sum of the two energies, potential and kinetic, contained in 7, for this energy is passed across the plane in the time t. Then (P + ^)^,^=(p + g^ + pIJ.^ (2) But ut = -y by equation (1) ; or ^ = P t^ 2^2 Vv But V = AM = (distance travelled by the sound in t) = Ut; Vv V V^ V ^^„ p V Vv V But — - , or — , that is the ratio of a small increase of pressure to the V change per unit of volume thereby produced, is the elasticity of the medium. We therefore obtain jj^ elasticity density ' or velocity = Velasticity -f- density. Waves of Transverse Disturbance. The velocity of propagation of plane waves in which the disturbance is in the plane of the wave can also be easily found by this method. Since the displacements are all perpendicular to the direction of propagation, the force acting across a plane perpendicular to this direction will be entirely tangential and the rigidity of the medium will alone be brought into play. As before, let the waves be travelling in the direction BA and let the curve CPQD represent the displacement. The curve may now represent the actual displacements since they are perpendicular to AB. If, as before, u is the velocity of the particle displaced from A, andt the time the wave takes to travel over the small distance MA, then the displace- ment AP becomes equal to MQ in t, or QR = ut. 302 VELOCITY OF PROPAGATION OF WAVES OF LONGITUDII^AL AND Now consider the energies in the volume V, with unit area on the plane through AP and height AM numerically equal to V. If p is the density, the kinetic energy is p —^ . The potential energy is equal to the work done in bringing the medium from its normal state into its present state of shear — the angle of shear at A being QPR, and since MA is very small this angle is measured by QR QR ut — — or — — = — PR V V If G be the modulus of rigidity, the tangential force per unit area is ut G X angle of shear = G ^. Now the average value of this force in bringing the shear to its present value is half this, and the distance of displacement of the force is QR. Then the potential energy in the volume V is The work done in t sees, across unit area at A is equal to the tangential force at A multiplied by the distance through which the particle at A moves in t ; lit u^t^ or ^ V ■ ^^^ "" ^ 'v • Equating this to the sum of the two energies, potential and kinetic, con- tained in V we have or Gj-pV. But V = AM = (distance through which the wave travels in t) = Ut; G V^ P t' or U ■-= Vmodulus of rigidity h- density. It follows at once from the above that the energy in any part of the medium is equally divided between the two forms kinetic and potential, for the equation ~r ^ 2T "^ ^ 2 gives G ^y ^^'2 ' or potential energy in F =^ kinetic energy in F. TRANSVERSE DISTURBANCES BY RATE OF TRANSFER OF ENERGY 303 This result also holds for waves of longitudinal disturbance if the initial pressure is zero. That is if we may put P = in the equation (P + V)v=(^P + l)v + p^^ for the potential energy is then ^, and the above equation gives us pv Vv^ or potential energy in F = kinetic energy in V. 20. RADIATION IN THE SOLAR SYSTEM: ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES. • [Phil. Trans. A, 202, 1903, pp. 525-552.] [Received June 16. Read June 18, 1903.] PART I. Temperatuee. When a surface is a full radiator and absorber* its temperature can be determined at once by the fourth-power law if we know the rate at which it is radiating energy. If it is radiating what it receives from the sun, then a knowledge of the solar constant enables us to find the temperature. We can thus make estimates of the highest temperature which a surface can reach when it is only receiving heat from the sun. We can also make more or less approximate estimates of the temperatures of the planetary surfaces by assuming conditions under which the radiation takes place f, and we can determine, fairly exactly, the temperatures of very small bodies in inter- planetary space. These determinations require a knowledge of the constant of radiation and of either the solar constant or the effective temperature of the sun, either of which, as is well known, can be found from the other by means of the radiation-constant. It will be convenient to give here the values of these quantities before proceeding to apply them to our special problems. * A surface which absorbs, and therefore emits, every kind of radiation is usually described as 'black,' a description which is obviously bad when the surface is luminous. It is much better described as 'a full absorber' or 'a full radiator.' t This was pointed out by W. Wien in his report on 'Les Lois Theoriques du Rayonnement ' {Congres International de Physique, vol. 2, p. 30). He remarks that Stefan's law enables us to calculate the temperatures of celestial bodies which receive their light from the sun, by equating the energy which they radiate to the energy which they receive from the sun, and states that for the earth we obtain nearly the mean temperature, using the reflecting power of Mars, while the temperature of Neptune should be below - 200° C. RADIATION IN THE SOLAR SYSTEM 305 The Constant of Radiation. If R is the energy radiated per second per square centimetre by a full radiator at temperature 6° A (where A stands for the absolute scale), the fourth-power law states that R = ad\ where a is the constant of radiation. According to Kurlbaum* the constant is CT = 5-32 X 10-5 erg/cm. 2 sec. deg.*. The Solar Constant. The solar constant is usually expressed as a number of calories received per minute by a square centimetre held normal to the sun's rays at the distance of the earth. The determinations by different observers differ so widely that it is not necessary for our present purpose to consider whether the constant really exists or whether there are small periodic variations from constancy. Angstrom estimated the value as 4 calories per square centimetre per minute, and this value is adopted by Crova as very probable I . When converted to ergs per second this gives Sa = 0-28 X 10' ergs/cm.2 sec, where the suffix denotes that it is Angstrom's value. LangleyJ assumed that the atmosphere transmits about 59 per cent, of the energy from a zenith sun, and from his measurement of the heat reaching the earth's surface he estimated the value of the constant at 3 cal./cm.^ min. This gives Si = 0-21 X 107 ergs/cm.2 sec. Rosetti§ assumed a transmission of 78 per cent, from the zenith sun, but Wilson and Gray|| consider that 71 per cent, represents Rosetti's numbers better than 78 per cent. If in Langley's value we replace 59 per cent, by 71 per cent., we get 2-5 cal./cm.^ min. This gives Sr = 0-175 X 107 ergs/cm.2 sec. * Wied. Ann. vol. 65, 1898, p. 748. f Congres International de Physique, vol. 3, p. 453. J Phil. Mag. vol. 15, 1883, p. 153, and Researches on Solar Heat. § PhU. Mag. vol. 8, 1879, p. 547. II Phil. Trans. A, 1894, p. 383. p. c. w. 20 306 RADIATION IN THE SOLAR SYSTEM: The Radiation from the Sun's Surface. If s is the radius of the sun's surface, R the radiation per square centi- metre, then the total rate of emission is 4:7ts^R. This passing through the sphere of radius r, at the distance of the earth and with surface 4^r^, gives where S is the solar constant. Hence R = p= (ifx" lO^' ^ = ^^'''^^^- Corresponding to the three values of S just given we have three values of R, viz., Ra = 1-29 X 10" ; Ri = 0-945 x lO^i ; R^ = 0-805 x lO^i. The Effective Temperature of the Sun. If we equate the sun's radiation to g6^, where g is the radiation-constant, we get 6, the 'effective temperature' of the sun, that is the temperature of a full radiator which is emitting energy at the same rate. Thus 5-32 X 10-5 ^^4 _ 1.29 x lO^S whence 0^ = 7000° A approximately. Similarly 6', = 6500° A ; 9, =--- 6200° A. Wilson* made a direct comparison of the radiation from the sun with that from a full radiator at known temperature. Assuming a zenith trans- mission of 71 per cent., he obtained 5773° A as the effective solar temperature. If we put 46,0006' = 5-32 X 10-^ x 5773^, we get S^ 0-128 x 10^ This is no doubt too low a value. Either then Wilson's zenith transmission was less than 71 per cent, or Kurlbaum's constant is too small. The low value is probably to be accounted for chiefly by the first supposition. Wilson points out that if x is the true value of the transmission, his value of the temperature is to be multiplied by (lljx)^. If we take 9^ = 6200°^ as the true value, then x will be given by This low value is not necessarily inconsistent with the much higher value 71 per cent, used above in finding Rosetti's solar constant, for no doubt the transmission varies widely with time and place, and we have no reason to * Roy. Soc. Proc. vol. 69, 1901-2, p. 312. ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 307 I assume that 1-77 calories per minute, obtained by Langley, would have been received from the zenith at the time and in the place where Wilson was making his determination. The Effective Temperature of Space. In determining the steady temperature of any body as conditioned by the radiation received from the sun, we have to consider whether it is necessary to take into account the radiation from the rest of the sky. If it receives S from the sun, p from the rest of the sky, and if its own radiation is R, then in the steady state R = S-\-p or R-p = S. It behaves therefore as if it were receiving S from the sun, but as if it were placed in a fully radiating enclosure of such temperature that the radiation is p. This temperature is the 'effective temperature of space.' The temperature may perhaps be more definitely described as that of a small full absorber placed at a distance from any planet and screened from the sun. Various well-known attempts have been made to estimate this temperature, but the data are very uncertain. The fourth-power law however shows that it is not very much above the absolute zero, if we can assume that the quality of starlight is not very different from that of sunlight. According to Hermite* starlight is one-tenth full moonlight. Full moonlight is variously estimated in terms of full sunlight. Langley f takes it as 4QoVoo- These two values combined give sunlight as 4 x 10^ starlight. But starlight comes from the whole hemisphere, while the sun only occupies a small part of it. In comparing temperatures we have to use the brightness of sunlight as if the whole hemisphere were paved with suns. If B is the illumination of a surface at 0, Fig. 1, lighted by the sun in the zenith at S, and if TTS^ is the area of the sun's diametral plane, then B/tts^ is the illumination at due to each square centimetre. If the hemisphere were all of the same brightness as the sun, the illumina- tion at due to the ring of sky between 6 and 6 -\- dd would be ^27rf2sin6>cos^^^, 7TS^ where r is the distance of the sun. Integrating from ^ = to ^ = 7r/2, we have Total illumination = Br^js^ = 46,000 B. * U Astronomie, vol. 5, p. 406. t 'First Memoir on the Temperature of the Surface of the Moon.' National Academy of Sciences i Memoirs, vol. 3, 1884. 20—2 308 RADIATION IN THE SOLAR SYSTEM I The illumination from a hemisphere paved with suns is therefore 46,000 X 4 X 10^ = 1-84 X lO^^ times that from the stellar sky. If we assume that the quality of the radiation is the same in both cases, that is, if we assume that the energy is proportional to the light-part of the spectrum, we have by the fourth-power law ^ ^ . „ effective temperature of sun Enective temperature oi space = i (0-184 X 10i2)t effective temperature of sun f ^ 655 As the temperature of the sun probably lies between 6000°^ and 7000^^, this gives Effective temperature of space = 10° A. If, then, a body is raised by the sun to even such a small multiple of 10° as, say, 60°, the fourth-power law of radiation implies that it is giving out, and therefore receiving from the sun, more than a thousand times as much energy as it is receiving from the sky. The sky-radiation may therefore be left out of the account when we are dealing with approximate estimates and not with exact results, and bodies in the solar system may be regarded as being situated in a zero enclosure except in so far as they receive radiation from the sun. Temperature of a Planet under Certain Assumed Conditions when placed at a Distance from the Sun equal to that of the Earth. The real earth presents a problem of complexity far too great to deal with. I shall therefore consider an ideal earth for which certain conditions hold, more or less approximating to reality, and determine the temperature of its surface on the assumption that it receives heat from the sun only. Let us suppose : 1. That the planet is rotating about an axis perpendicular to the plane of its orbit, which is circular. This will give us too high a temperature at the equator, and the absolute zero, which is too low, at the poles. The mean, however, over the planet will probably be not much affected by the supposition. 2. That the effect of the atmosphere is to keep the temperature in any given latitude the same, day and night. This is not a great departure from reality. On the sea, which is more than two-thirds of the earth's surface, the daily range is very small, of the order of 1° or 2° C, while even on the land it is, in extreme cases, not more than 15° C, which is not a large fraction of the absolute temperature. ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 309 3. That the surface and the atmosphere over it at any one point have one effective temperature as a full radiator. This is no doubt a departure from reality. How wide a departure we have no present means of estimating. 4. That there is no convection of heat from one latitude to another. This is a very wide departure from reality. But, as we shall see below, the mean temperature of the planet is very little affected by convection, even if we assume that it is so extensive as to make the surface of uniform temperature. 5. That the reflection at each point is -fjj of the radiation received. This is probably of the order of the actual reflection from the earth. According to Langley* the moon reflects about J of the radiation received. The earth certainly reflects less. The temperatures determined hereafter are proportional to the 4th root of the coefficient of absorption. Even if this coefficient is as low as 0-9 its 4th root is 0-974. Hence if the actual value is anywhere between 0-9 and 1, the assumed value of 0-9 will not make an error of more than 2J per cent, in the value of the temperature. 6. That the planet ultimately radiates out all the heat received from the sun, no more and no less. This again is very near the condition of the real earth, which, on the whole, radiates out rather more than it receives — perhaps on the average a calorie per square centimetre in three days. r cos \ d\ T"" / \r dA Fig. 2. Making these six suppositions, let us calculate the temperature of various parts of this ideal planet. Consider a band between latitudes A and A + dX. The area receiving heat from the sun at any instant, if projected normally to the stream of solar radiation, is (Fig. 2) 2r cos XrdXcosX= 2r^ cos^ XdX, where r is the radius of the planet. * 'Third Memoir on the Temperature of the Moon.'. National Academy of Sciemes, Memoirs vol. 4, part 2, p. 197. 310 BADIATION IN THE SOLAR SYSTEM I If S is the solar constant, this band is absorbing, with coefficient 0-9, 0-9^ X 2r^cos^XdX. But the band all round the globe is radiating equally, according to the second supposition, and the radiating area is 27rr cos A . rc^A = 27Tr^ cos A^^A. Hence the radiation emitted per square centimetre per sec. is 0'9S . 2r^ cos^ XdX _ 0-98 cos A 27rr^ cos XdX tt ' If the effective temperature in this latitude is 6^, we have 0-9iS cos A 5-32 X 10-^ 6>A^ 77 or / 0-9 X 10^S \i [ 5-3277 ; cos* A. If we put A = 0, we get the equatorial temperature corresponding to each of the different values of S given above, viz. : Equatorial 6^ = 350° A approximately. e^ - 325° A „ e, ^ 312° A The temperature in latitude A is 6^ = equatorial temperature x cos* A. Thus, in latitude 45°, it is 0-917 x equatorial temperature. The average temperature over the globe is J r2 f 1 277/-2 cos A 6e cos* XdX, 'J where 6j^ is the equatorial temperature. The average temperature, then, is httle more than 1 per cent, above the temperature in latitude 45°. If we use the three values of 6^ just given, we have Average 6^ = 325° A approximately. „ 01 = 302° A d, = 290° A Our fourth supposition was that there is no convection by wind or water from one latitude to another. Let us now go to the other extreme and ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 311 suppose that the convection is so great that the temperature is practically uniform all over the globe. We then have a receiving surface virtually irr^, and a radiating surface iTrr^. Then we get the radiation emitted per square centimetre 47rr2 ~ 40' and if 6 is the temperature required for this, 5-32x10-5^4=^; whence Uniform d^ = 330° A approximately, 01 = 307° A Or = 293° A values not more than 5° above those obtained for the average on the supposition of no convection. Comparing these results with the temperature of the real earth, it is seen at once that they are of the same order. The average temperature of the earth's surface is usually estimated at about 60° F., say 289° A. The temperature of the atmosphere is on the whole decidedly lower than that of the surface below it. We should therefore conclude that the earth's effective temperature is somewhat below 289° A. Again, the earth and the atmosphere, taken as one surface, do not con- stitute a full absorber, but are to some extent selective. Hence we should expect the earth to be, if anything, of a higher temperature than a full absorber and radiator under the same conditions. For both these reasons, then, the ideal planet might be expected to have a temperature below rather than above 289° A. The lowest estimate obtained above is therefore probably nearest to the truth, and it would appear that even that is somewhat too high. This tends to show that, if we accept Kurlbaum's value of the radiation-constant, we cannot put the solar constant so high as 3 or 4, but must accept a value much nearer to that which I have called Rosetti's value, viz., 2-5. In what follows I shall therefore take Rosetti's value and the resulting value of the solar temperature, viz., 6200° A. The calculation made above may be turned the other way round, and may be used for a Determination of the Effective Temperature of the Sun from the Average Temperature of the Earth. Assuming that the real earth may be replaced by the ideal planet already considered, the radiation per square centimetre from the equatorial band is . But the radiation per square centimetre from the sun's surface is 312 RADIATION IN THE SOLAR SYSTEM: 46,000/S'. If then B^ is the earth's equatorial temperature, and 6^ is the solar temperature, — ^ : 46,000>S = Se^ : Os^ 7T whence 0^ = ^sl^^- The average temperature of the earth is 0-93 of the equatorial temperature. If this average is ^^ , then e^ = es/21-5. If we take the temperature of the real earth as 289° A, and as being equal to that of the ideal, d^ = 21-5 X 289° = 6200° A approximately. Upper Limit to the Temperature of a Fully Radiating Surface exposed normally to Solar Radiation at the Distance of the Earth from the Sun. The highest temperature which a full radiator can attain is that for which its radiation is equal to the energy received. This will only hold when no appreciable quantity of heat is conducted inwards from the surface. To obtain the upper limit in the case under consideration, we have to equate the radiation to the solar constant, which we shall now take as Sr = 0-175 X 10^. Then 5-32 X 10-5^4 = 0-175 X 10^ whence 6 = 426° A. If the surface reflects some of the radiation and absorbs a fraction x of that falling on it, then the effective temperature is x^ X 426° A. The Limiting Temperature of the Surface of the Moon. We may apply this result to find an upper limit to the temperature of the moon's surface. This upper limit can only be attained when it is sending out radiation as rapidly as it receives it, and is therefore conducting no appreciable quantity inwards. 117 r, n . 1 T 1 ? ^- . /7 •. X i- reflected radiation 1 We shall take Langley s estimate (loc. cit.) of — ^——-^ :rr~-. — = -— - . ^ -^ ' emitted radiation 6-7 This is represented nearly enough by ;z; = | . The upper limit of temperature of the surface exposed to a zenith sun is, therefore, ^ = 426 X (I)* = 426 X 0-967 - 412° A. This, then, is the upper Hmit to the temperature of the hottest part of an airless moon. ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 313 For a surface with normal at angle A with the line to the sun, ^x = 412 cos* A. If we take this as the law of temperature of the side of the moon exposed to the sun, we can find the effective temperature of the full moon as seen from the earth, i.e., the uniform temperature of a flat disc of radius equal to that of the moon, sending to us the same total radiation. If Ndo) is the normal stream of radiation from 1 cm.^ of surface of the moon immediately under the sun sent out through a cone of angle doj, that sent out in direction A to the normal is iVcosA<?a;. But 1 cm.^ on the moon's surface, with normal inclined at A to the sun's rays, only receives cos A of the radiation received by the surface immediately under the sun. It therefore sends in the direction of the earth, also at A to the normal, onlv iVcos^A^o). Hence the total radiation to the earth, obtained by inte- grating, is / 2 N cos^ A . 27rm2 sin XdX ^^ where m is the radius of the moon and r is its distance from the earth. Let Nj^ be the normal stream from the equivalent flat disc, then Trm^Nn _ 277 m- ^ 2 whence Nz) = ^N. o The effective temperature of the flat disc is therefore Vf that of the surface immediately under the sun at the same distance from it. Then the effective average = 412 x v^f = 412 x 0-9 = 371° A. The upper limit, then, to the average effective temperature of the moon's disc is just below that of boiling water. This is very considerably above Langley's estimate, that the surface of the full moon is a few degrees above the freezing-point. There can be no doubt that a very appreciable amount of heat is conducted inwards. The observations during eclipses by Langley* and by Boeddicker show that some heat is still received from the moon's surface when it has entered the full shadow, and that it takes time after the eclipse has passed to establish a steady temperature again. It might be possible to make some rough estimate of the amount conducted inwards from the Fourier equation, but the problem is not an easy one. Perhaps we get the best estimate by comparing the actual temperature with that found above. * 'Third Memoir,' p. 159. 314 RADIATION IN THE SOLAR SYSTEM: If the actual temperature is taken as about^ | of the upper limit, say 297°^, then the radiation outwards is of the order (|)* = 0-41 of that where no con- duction exists. Then nearly f of the heat is probably conducted inwards. If the moon always turned the same face to the sun instead of to the earth, the upper limit would be approached. Temperature of a Spherical Absorbing Solid Body of the Order 1 cm. in diameter at the Distance of the Earth from the Sun. The calculation of the temperature of such a body is interesting for two reasons. Firstly, the body will be at nearly the same temperature through- out, and secondly, as we shall show in the second part of this paper, the mutual repulsion of two such bodies, due to the pressure of their radiation, is of the same order as their gravitative attraction. If the radius of the body is a, its effective receiving area is 7ra^, and it receives TTa^S ergs/sec. Its radiating surface is iira'^, and therefore its average radiation per square centimetre per sec. in the steady state is Tra^S/iTTa^ = iS. If we take S=2'6 cal./cm.^ min. or 0-04 cal./cm.^sec, and if the conductivity is of the order of that of terrestrial rock lying, say, between 0-01 and 0-001, it is evident that a difference of temperature of only a few degrees between the receiving and the dark surfaces will convey heat sufficient to supply radiation, 0-01 cal./cm.^ sec, equal to the average. Thus, if the conductivity is 0-001 and the diameter is 1 cm., a difference of temperature of 10° suffices. We may therefore take the temperature of the surface as approximately uniform when the steady state is reached. Let the temperature be 6, and let the solar temperature be Og. Then we have and If ^e = This will be the temperature of fully absorbing bodies of diameter less than 1 cm., so long as they are not too small to absorb the radiation falling on them. : 0s' = 1 : 46,000;S e- 20-7' 6200° A, 300°. 4 approximately ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 315 Variation of Temperature with Distance from the Sun. Since the radiation received varies inversely as the square of the distance from the sun, that given out varies in the same ratio. The temperature of the radiating surface varies therefore as the fourth-root of the inverse square, that is inversely as the square-root of the distance. This enables us to deduce at once the temperatures of the various surfaces and bodies which we have considered, if placed at the distances of different planets as well as at the distance of the earth. We have merely to multiply i-u ^J. 1,'j.v. J. £ 1 T_ /earth's distance the results hitherto found by </ -^ , ... . V planet s distance The following table contains the values of the temperatures at selected distances, all on the absolute scale : Table of Temperatures of Surfaces at Different Distances from the Sun. All on the Absolute Scale. I II III IV V VI VII VIII IX Equatorial Average tem- Upper limit of a Average tem- perature of Tem- peiature Tem- perature At the Distance, Square- tempera- surface four- of distance of Earth's root of ture perature of ideal planet reflecting fifths small the planet distance = 1 (distance)-i of ideal one-eighth that of absorb- planet under zenith sun lent disc equiva- lent disc ing sphere Mercury 0-3871 1-61 502 467 664 598 478 483 Venus 0-7233 1-18 368 342 486 438 350 358 Earth 1-0000 1-00 312 290 412 371 297 300 Mars 1-5237 0-81 253 235 337 300 240 243 Neptune 30-0544 0-18 56 52 74 67 53 54 We have omitted the larger planets except Neptune, since in all probability they radiate heat of their own in considerable proportion. Neptune is inserted merely to show how low temperatures would be at his distance if there were no supply of internal heat. The results given in the table may not be exactly applicable to any of the planets, but they at least indicate the order of temperature which probably prevails. If, for instance, Mars is to be regarded as having an atmosphere with-, regulating properties like our own, his equatorial temperature (Column IV) is probably far below the temperature of freezing water, and his average temperature (Column V) must be not very different from that of freezing mercury. If, on the other hand, we suppose that his atmosphere has no regulating power, we get the upper limits not very different from those in 316 RADIATION IN THE SOLAR SYSTEM: Columns VI and VII. These are the Hmits for the bright side, and they imply nearly absolute zero on the dark side. If we regard Mars as resembling our moon, and take the moon's effective average temperature as 297° A, the corresponding temperature for Mars is 240° A, and the highest temperature is I X 337 = 270°. But the surface of Mars has probably a higher coefficient of absorption than the surface of the moon — it certainly has for light — so that we may put his effective average temperature on this supposition some few degrees above 240° A , and his equatorial temperature some degrees higher still. It appears exceedingly probable, then, that whether we regard Mars as like the earth, or, going to the other extreme, as like the moon, the temperature of his surface is everywhere below the freezing-point of water. The only escape from this conclusion that I can see is by way of a supposition that an appreciable amount of heat is issuing from beneath his surface. We cannot draw any definite conclusions as to the temperatures of Mercury and Venus till we know whether they have atmospheres and whether they rotate on their own axes. If we make both these suppositions and further suppose that their conditions approximate to those (given in Columns IV and V) of the ideal planet at their distances, then they may well be surrounded by hot clouds, as is sometimes supposed, entirely screening their solid bodies from us. If, on the other hand, their atmospheres are ineffective as regulators and if they always present the same face to the sun, the hottest part of Mercury is probably not far from 650° A, and that of Venus not far from 500° A. If a comet consist of small solid particles of diameter of the order 1 cm. or less, then the temperatures of these particles are given in Column IX. At one-quarter of the earth's distance, say 23 million miles from the sun, the temperature is 600° A, about the melting-point of lead. At one-twenty-fifth, say 3| million miles, it will be about 1500° A, say the melting-point of cast- iron. Nearer than this the temperature no doubt increases rapidly, but the law of temperature, deduced from the inverse-square law for the radiation received, requires amendment, as that law was based on the supposition that a hemisphere only is lighted by the sun, and that the whole of his disc is visible from every part of that hemisphere. Both of these suppositions cease to hold when the distance from the sun is only a small multiple of his radius. PART 11. Radiation-Pressures. The pressure of radiation against a surface on which it falls, first deduced by Maxwell from the Electromagnetic Theory of Light, is now established on an experimental basis by the work of Lebedew, confirmed by that of Nichols and Hull. ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 317 Though this pressure was first deduced as a consequence of the Electro- magnetic Theory, Bartoli showed, independently, that a pressure must exist without any theory as to the nature of light beyond a supposition which may perhaps be put in the form that a surface can move through the ether, doing work on the radiation alone and not on the ether in which the radiation exists. Professor Larmor* has given a proof of this pressure and has shown that it has the value assigned to it by Maxwell, viz., that it is numerically equal to the energy- density in the incident wave, whatever may be the nature of the waves, so long as their energy-density for given amplitude is inversely as the square of the wave-length. We may, in fact, regard a pencil of radiation as a stream of momentum, the direction of the momentum being the axis of the pencil. If E is the energy-density of the pencil, U its velocity, the momentum-density may be regarded as EjU. If the stream of "radiation is being emitted by a surface, the surface is losing the momentum carried out with the issuing stream, and is so being pressed backwards. If the stream is being absorbed by the surface, then it is gaining the momentum and is still being pressed backwards, the forces being in the line of propagation. As the expressions for the radiation-pressure in various cases are probably not very well known, it may be convenient to state them here for use in what follows. Values of Radiation-Pressure in Different Cases. If 1 cm.'-^ of a full radiator is emitting energy R per second, and if N dw is the energy it is emitting through a cone dco, with axis along the normal, then in direction d its projection is cos d, and it is emitting iVcos ddoj through a cone do). Putting dcxi = 2tt sin Odd, and integrating over the hemisphere, we have R= \ iV cos ^ . 277 sin (^(9 = ttN. J If we draw a hemisphere, radius r, round the source as centre, the energy falling on area r^doj is iVcos ddoj per second, and, since the velocity is U cm. per second, the energy-density just outside the surface on which it falls is iVcos diUr^, and this is the rate at which the momentum is being received, that is, it IS the normal pressure. The total force on area r^dcj is iVcos Odw/U. This is the momentum sent out per second by the radiating square centimetre through the pencil with angle dw, in the direction 6, and is therefore the force on the square centimetre due to that pencil. Resolving along the normal and in the surface we have Normal pressure = N cos^ ddco/U, Tangential stress = iV cos ^ sin Odco/U. * Brit. Assoc. Report, 1900; Encyc. Brit. vol. 32, Art. 'Radiation.' 318 RADIATION IN THE SOLAR SYSTEM I Putting d(x) = 277 sin ddd and integrating over the hemisphere, we get IT Total normal pressure = ( [N cos^ d . Iir sin d dOjU) - ^ttNJW = 2i?/3C7. J Total tangential stress = 0, since the radiation is symmetrical about the normal. If the surface is receiving radiation, let us suppose that the stream is a parallel pencil S ergs per second per square centimetre held normal to the stream, and that it is inclined afc an angle 6 to the normal to the receiving surface. The momentum received per second is S cos djTJ. This produces Normal pressure ^ S cos^ djU , Tangential stress =^ S cos d sin djU. If the stream is entirely absorbed both these forces exist. If the stream is entirely reflected, the reflected pencil exerts an equal normal force and an equal and opposite tangential force, and we have only normal pressure of amount 2S cos^ d/U. If only a fraction /x is reflected, the incident and reflected streams will give Normal pressure = (1 + /jl) S cos^ O/U, Tangential stress = (1 — /jl) S cos 6 sin d/U. To the normal pressure must be added the pressure due to the radiation emitted from the surface. Radiation- Pressure in Full Sunlight. If a full absorber is exposed normally to the solar radiation at the distance of the earth the pressure on it is S/U, or - -^ .w.f.-~ = 5-8 x 10~^ ^ ' 3 X 10^^ dyne/cm.^. The Radiation-Pressures between Small Bodies. Comparison ivith their mutual Gravitation. It is well known that the radiation-force on a small body, exposed to solar radiation, does not decrease so rapidly as gravitative pull on the body when its size decreases. If the body is a sphere of radius a and density p, and with a fully absorbing surface, and if it is so small that it is practically at one temperature all through, it is receiving a stream of momentum • ira^SjU directed from the sun. Its own radiation outwards being equal in all directions has zero resultant pressure. ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 319 The gravitative acceleration towards the sun at the distance of the earth is about 0-59 cm./sec.^. Then we have Radiation-pressure. tto^S Gravitation-pull U x ^ira^p x 0-59 ' The two will be equal when a -- S Up X 0-59' If we put /o=l; /S = 0-175 X 107; U = ^xlO^^; we get a = 74 X 10-6. This is the well-known result that a body of diameter about two wave- lengths of red light would be equally attracted and repelled if we could assume that a surface so small still continued to absorb. But, of course, when we are getting to dimensions comparable with a wave-length that assumption can no longer be made. It is not, I think, equally well recognised that if the radiating body is diminished in size, the radiation-pressure due to it also decreases less rapidly than the gravitative pull which it exerts. For the radiation decreases as the square of the radius of the emitting body and its gravitative pull as the cube. We can easily compare the forces due to radiation and gravitation between two bodies, if for simplicity we assume that their distance apart is very great compared with the radius of either. Fig. 3. Let A, B, Fig. 3, be two spheres with full radiating surfaces. Let their radii be a, b, and let their centres o, o' be d apart. If this distance is great compared with a and 6, each may be regarded as receiving a parallel stream from the other. Let A send out a normal stream N dco per square centimetre through cone do), while B sends out N' dco. B receives the stream of cross-section nb'^ or the angle of the cone is nb^/d'^, and it issues virtually from area na^, for at B, A will appear as a uniformly bright flat disc. 320 RADIATION IN THE SOLAR SYSTEM: Then the total force on B is where R — ttN. The force on A due to B is ira^hm' jUd^, where E = ttN\ These are not equal unless R = R' , i.e., unless the two bodies have the same temperature, an illustration of the fact that equality of action and reaction does not hold between the radiating and receiving bodies alone. They no longer constitute the whole of the momentum-system. The ether, or whatever we term the light-bearing medium, is material, and takes its part in the momentum-relations of the system. If the surfaces are partially or totally reflecting, the forces are easily obtained. Thus if one is totally reflecting, it can be shown that the force is only half as great as when it is fully absorbing. But it will be sufficient to confine ourselves to the case of complete absorption, followed by radiation of the absorbed heat equally in all directions from all parts of the surface. More general assumptions do not alter the order of the forces found. If G is the constant of gravitation = 6-67 x 10"^, and if p, p' are the densities of A and B, the gravitation-pull, P, is 6^ ^ ar}2^^^ • Radiation-push F _ 97Ta^b^R Gravitation-puff P WGUn^a^b^pp" F 9R P WGUTTabpp" lia = b; p=p'; R^ 5-32 x 10-^^^ we have 0-69 X 10-4^2 /p ^ or If we suppose the two bodies to have the temperature of the sun, say 6200°^, and its density, say 1-375, then F = P, when a =1930 cm. or 19-3 metres f. Of course two globes of this size would soon cool far below the temperature of the sun, even if for an instant they could be raised up to it. If we suppose 6 = 300° A — the approximate temperature of small bodies at the distance of the earth from the sun — and if we take /o = 1, then F = P when a = 6-2 cm. I * [The original has 2-18, instead of 0-69, which is a shp due to the omission of ^l^- "^^^ density of the sun is also wrongly taken as 0-25 instead of 1-375 (see Art. 65, p. 709). This necessitates some corrections in the succeeding part of the paper, which have in all cases been marked with a f.' Ed.] t [See note above. Ed.] ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 321 Thus two globes of water — probably nearly full absorbers at 300° A — will at that temperature neither attract nor repel each other if their radii are about 6 cm.f If the density of the spheres is 11, about that so often used for masses in the Cavendish experiment, F = P when a = 0-564 cm.t This does not throw any doubt on the results of Cavendish experiments, for it only holds when the radiators are in an enclosure of very low absolute temperature. In all Cavendish experiments the greatest care is taken to make the attracted body and its enclosure of one uniform temperature. The really interesting case is that of two small meteorites, in interplanetary space. To judge from the specimens which succeed in penetrating the earth's atmosphere they are very dense. Let us suppose them to have density 5-5 — that of the earth — and temperature 300° A, that which they will have at the earth's distance. Then F = P when a = 1-13 cm.f If the radii of the bodies are less than the values found for equality of F and P in the different cases, the net effect is repulsion. The ratio of -F to P is inversely as the square of the radius, so that, as the radii are decreased from the values giving F = P, the radiation -repulsion soon becomes enormously greater than the gravitation-pull, and the latter may be neglected in comparison. Thus for two drops of water at 300° A in a zero enclosure, with radii 0-001 cm., the pressure is nearly 40,000,000 times the pullf. It is not, however, that the radiation-force is great, or even its acceleration. The force becomes exceedingly minute, but the gravitation much more minute. Thus consider two drops of water at 300° A placed in a zero enclosure at a distance d = 10a apart. Our assumption of parallel radiation from one to the other is now only a rough approximation, but the result will be of the right order. The radiation-push is Tra^R/Ud^, and the acceleration is 3aR/4:Ud^ =" TTv? ^ ~ approximately. This only becomes considerable when the drops approach molecular dimensions, and long before this they cease to absorb fully the stream of momentum falling on them. Still, even molecules are selective absorbers, and absorb especially each other's radiations. And we may expect that if two gas-molecules collide and set each other radiating much more violently than before, they will be practically in an enclosure of much lower temperature than their own, and their mutual radiation may result in very rapid repulsion — repulsion of the order of the fourth power of the temperature reached. t [See note on p. 320. Ed.] P. c. w. 21 322 RADIATION IN THE SOLAR SYSTEM: Radiation-Pressure between Small Bodies at Different Distances from the Sun. We have seen above, that if two small spheres of density 5-5 are at the distance of the earth from the sun, their gravitation will be balanced by their radiation-pressure when the radius of each is 1-13 cm.f Now the balancing radius is proportional to the square of the temperature, that is, inversely proportional to the distance, since the temperature (Part I) is inversely as the square-root of the distance. Thus, at the distance of Mercury, the radii would be about 3 cm.f ; a million miles from the sun's surface they would be about 100 cm.j ; out at Neptune they would be about 0-4 mm.| We see then that the mutual action between small bodies of density that of the earth, will, at different distances, change sign for different sizes of body, ranging from something of the order of 2 metres diameter")* near the sun to the order of 1 mm. diameter f at the distance of Neptune. A ring of small planets, each of radius 1-13 cm.f, and density 5-5, would move round the sun at the distance of the earth without net mutual attraction or repulsion, and each might be regarded as moving independently of the rest. It appears possible that if Saturn is hot enough, considerations of this kind may apply to his rings. The repulsion between small colliding bodies, even if not heated by the sun, must lead to some delay in their final aggregation. This is obvious when there are only two small bodies, and their temperature is very con- siderably raised by the collision. But there is also delay if instead of a single pair we suppose two swarms to collide. Near the boundary of the colliding region, a body will experience radiation-pressure chiefly on one side, and will tend to be driven out of the system. Of course, if the swarms are so dense that a member near the outside cannot see through the rest, this effect will be less. A body in front of another entirely screens its radiation, but the gravitation is not screened. Hence, a body near the boundary of a densely-packed region of collision may be repelled only by the colliding bodies just round it, while it will be attracted by all ; or, to put the same idea in another way, a body in a spherical swarm of uniform temperature will only be pulled equally in all directions at the centre of the swarm, but it will be equally repelled in all directions as soon as it is sufficiently deep to be surrounded by its fellows wherever, so to speak, it looks. Inequality of Action and Reaction between Two Mutually Radiating Bodies. We have seen that two distant spheres push each other with forces TTa^b^R/Ud^ and -na^b'^R' jU d^, and that these, though opposite, are not equal unless R = R' . It would be easy to imagine cases in which the forces were not even opposite or in the same directions. At first sight, then, it would appear that t [See footnote, p. 320. Ed.] ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 323 we have two bodies acting upon each other with unequal forces, but of course this statement is inexact. The bodies do not act upon each other at all; each sends out a stream of momentum into the medium surrounding it. Some of this momentum is ultimately intercepted by the other, and in its passage the momentum belongs neither to one body nor to the other. If we assume that the momentum is conserved, and of course everything in the methods of this paper depends on that assumption, the action on one of the bodies is equal and opposite to the reaction on the light-bearing medium contiguous to it. There is no failure of the law of action and reaction, but an extension of our idea of matter to include the medium. There should be no difficulty in this extension ; indeed, we have made it long ago in endowing the medium with energy-carrying properties. Whether the momentum in the medium is in the form of mass m moving with velocity v in the direction of propagation is perhaps open to doubt. We may, perhaps, have different forms of momentum just as we may have different forms of energy, and possibly we ought not to separate the momentum in radiation into the factors m and v, but keep it for the present as one quantity M. An interesting example of inequality of the radiation-forces on two mutually radiating bodies is afforded by two equal spheres, for which, at a given temperature, the radiation-push F balances the gravitation-pull P. Raise* one in temperature so that the push on the other becomes F\ Lower the other so that the push on the first becomes F", but adjust so that r + F" = 2F=: 2P, then P-F" = F' -P. There will then be equal accelerations of the two in the same, not in opposite directions, and a chase will begin in the line joining the centres, the hotter chasing the colder. If the two temperatures could be maintained, the velocity would go on increasing; but the increase would not be indefinitely great, inasmuch as a Doppler effect would come into play. Each sphere moving forward would crowd up against the radiation it emitted in front, and open out from the radiation it emitted backwards. This would increase the front and decrease the back pressure, and ultimately the excess of front pressure would balance the accelerating force due to mutual radiation. Let us examine the effect of motion of a radiating surface on the pressure of its radiation against it. Application of Doppler^s Principle to the Radiation- Pressure against a Moving Surface. If a unit area A, Fig. 4, is moving with velocity u in any direction AB, making angle with its normal AN, the effect on the energy-density in the stream of radiation issuing in any direction AP is two-fold. If the motion is such as to shorten AP, the waves and their energy are crowded up into 21—2 324 RADIATION IN THE SOLAR SYSTEM: less space, and if such as to lengthen AP, they are opened out. At the same time, in the one case A is doing work against the radiation-pressure and in the other is having work done on it. We shall assume, as in the thermo- dynamic theory of radiation, that this work adds to or subtracts from the energy of radiation. Both effects, (1) the crowding, and (2) the work done, or the reverse of each, combine to alter the energy and therefore the radiation- pressure. We have no data by which we can determine whether the motion alters the rate at which the surface is emitting radiation, but it appears worth while to trace consequences on the assumption that the radiation goes on as if the surface were at rest*, but that it is crowded up into less space or spread over more, and that we can superpose on this the energy given out to, or taken from, the stream by the work done by, or on, the moving surface by the radiation- pressure. This work can evidently be calculated to the first order of approximation by supposing the pressure equal to its value when the surface is at rest. Let us draw from A as centre a sphere of radius U, equal to the velocity of radiation. The energy which, in a system at rest, would be radiated into a cone with A as vertex, length U, and solid angle do), in the direction AP making an angle x with the direction of motion AB, will now be crowded up into a cone of length U — u cos x, since u cos x is the velocity of A in the direction AP. We shall suppose that u/U is very small. Hence the energy-density in the cone is increased in the ratio U + u cos x ucosx Considering now the effect of the work done, the force on A due to the stream in dw is N cos ddoj/U, and the work done in one second is {N cos ddco/U) X u cos X- When A is at rest the energy in this cone is N cos ddo). U or by the factor * Added August 20, 1903. Since the above was written Professor Larmor has pointed out to me that the results obtained in the text from this assumption, along with the hypothesis of crowding of the radiation and its increase by an amount equivalent to the work of the radiation- pressure, can be justified by an argument based on the following considerations. A perfect reflector moving with uniform speed in an enclosure, itself also moving at that speed, and so in a steady state, must send back as much radiation of every kind as a full radiator in its place. Now the electrodynamics of perfect reflection are known ; hence the effect of motion of a full radiator on the amount of its radiation can be determined. The result is equivalent to the statement that the amplitudes of the excursions of the optical vibrators are the same at the same temperature whether the source to which they belong is moving or not. ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 325 When A is movijig it is increased to T,, ^ , N cos ^c?ct> iv cos Udoj H ^ u cos ;)^, that is N cos 6>^a; (l + ^i^^ . Thus the effect of the work done is equal to that of the crowding, and the energy-density on the whole is increased in the ratio 2i^cosx.-. "^ U The pressure is increased in the ratio of the energy-density*. Then the force on A due to the radiation through dco is increased from Ncosddco , N cosddco /^ 2w cos v i to == 1 + ^ U U \ ' u If we resolve this along the normal to the surface A and integrate over the hemisphere we obtain the total normal pressure. As we only want to know the change in pressure P we may neglect the first term which gives the pressure on A at rest, and we have ^ [N cos^ 9 2u cos y , If cf) is the angle between the normal planes through B and P we have cos X = cos 6 cos e/f + sin ^ sin ijj cos cj). Putting dco = sin Odddcf), we get P = r2 c'i-^2Nu I TtT ^^^^ ^ ^^^ ^ (^^^ ^ ^^^ 'A + si^ ^ ^^^^ 'A COS (f)) ddd(f) ttNu cos i/r Ru cos j/f The change in the tangential stress, T, is evidently in the direction AC, that of the component of u in the plane of A. We may therefore resolve each element of tangential stress in the direction AC. Omitting the first term again, since in this case it disappears on inte- gration, the element due to dco in the direction AP will contribute N cos 6 sin 6 cos 2u cos x U ' U and integrating over the hemisphere we have dco, T = \ -jj^ cos 6 sin^ 6 cos cf) (cos 6 cosip + sin 6 sin ip cos ^) (?^ defy ttNu sin j/f i??i sin iff ^ 2JP '~^ ~~2W~' * [See note, p. 330. Ed.] 326 RADIATION IN THE SOLAR SYSTEM: Force on a Sphere moving with Velocity 'u' in a Given Direction. If a sphere, radius a, is moving with velocity u, we may from symmetry resolve the forces on each element in the direction of motion. The resolutes will be P cos ifj and T sin ifj. Evidently it is sufficient to integrate over the front hemisphere and then double the result. We have the 2 -r. , T T^ ^ r /^^ cos^ w , Ku sm2 e/f\ ^ . , , , Ketardmg _borce = -^ I fj^ \ i^jj2 j ^tt"^ sm i/jdifj va 3 U2 It is noteworthy that one half of this is due to the normal, the other half to the tangential stresses. If the sphere has density p the acceleration is obtained by dividing by iTTCi^p, then du/dt = - IRujU^pa. Effect on Rotation. If the sphere radius a is rotating with angular velocity co, then any element of the surface, A from the equator, is moving with hnear velocity aa> cos A in its own plane. This does not affect the normal pressure, but it introduces a tangential stress opposing the motion Rii/2U^ = RacocosXI2U\ Taking moments round the axis and integrating over the sphere, we obtain a couple TT ina^p . ^a^ , = ^_^- 27ra^ cos'^ A dX, ^ "" dt 2U^ j _^ 2 whence dco/dt = — qRco/U^pa. The rate of diminution of oj is therefore of the same order as that of v. To obtain an idea of the magnitude of the retardation of a moving sphere, let us suppose that one is moving through a stationary medium. Let its radius he a = 1 cm., its density p = 5-5, its temperature 300° A. Idu 2 X 5-32 X 10-5 X 300* Then u dt 9 X 1020 X 5.5 = 1-75 X 10-16. This will begin to affect the velocity by the order of 1 in 10,000 in, say, 10^2 seconds, or taking the year as 3-15 x 10^ seconds, in about 30,000 years. ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 327 The effect is inversely as the radius, so that a dust-particle 0-001 cm. radius will be equally affected in 30 years. The effect is as the fourth power of the temperature, so that with rising temperature it becomes rapidly more serious. Equation to the Orbit of a Small Spherical Absorbing Particle Moving in a Stationary Medium Round the Sun. It is evident from the above result, that the effect of motion on radiation- pressure may be very considerable in the case of a small absorbing particle moving round the sun. We shall take the particle as spherical, of radius a and distance r from the sun. We shall suppose the radius so small that the particle is of one tempera- ture throughout, the temperature due to the solar radiation which it receives, but that it is still so large as to be attracted much more than it is repelled by the sun. Both attraction and repulsion are inversely as the square of the distance, so that we shall have a central force which we may put as producing acceleration A/r^, where A is constant. We know that at the distance of the earth, putting r = b, A/b^ = 0-59 cm. /sec. ^ say 0-6 cm./sec.^. Then A = O-Qb^. The force acting against the motion produces retardation — 2RulU^pa. If S is the solar constant at the distance 6, its value at distance r is Sb^rK Putting iTram = ira^Sb^/r^, R = (S/i) {b^lr% then the acceleration in the line of motion is _ Sb"^ u__Ts 2lPpa ' r2 ~ /•2 ^ where T = Sb^l2U^pa, and -s- is now written for the velocity u. The accelerations along and perpendicular to the radius-vector give the equations '-"•=-^s <■' ls"''>-W^^ ■ ; <^> From (2) we get | {r^O) = - ^^^ whence r^O = C - TO, (3) where C is the constant of integration. 328 RADIATION IN THE SOLAR SYSTEM: If ^ is when t = 0, then C is the initial value of rW. Further, as 6 increases rW decreases and is when d = C/T. This gives a limit to the angle described. Equation (1) may be written r-re^=-i-^ (4) Putting u for r~^ we have dr X 1 du A ,„ „^, du ^ = -^ = -^^i^ = -(^-^^)i f-^^-^^^)' T{C-Te)n^^g-{C^Teru^^^ from (3). Substituting in (4) de^ ' (C - TOY' This can probably only be integrated by approximation. We can see the effect on the motion at the beginning by putting d'^u A / 2T 4- ^^ -= ^ 1 - ^. dd^ ' 02 {■-¥•). since TjC is small if we begin at the distance of the earth and with a particle having the velocity of the earth. An integral of this is The complementary function will be periodic and may be omitted. To the order of approximation adopted Then initially rjr = - (ITjC) 6. In applying these results, we may note that T = Sb^l2U^pa is constant for all distances, and that b, the earth's distance, is 493 U. Inserting the value of the solar constant, 0-175 x 10'^, and taking p ^ 5-5, we get T - 3-9 X 1010 . a-i. C will depend on the initial conditions. Assuming that the body considered is initially moving in a circle, then, at the beginning re^^~ or ^= /^= /^, since at r = 6 the acceleration to the centre is 0-6. Then C =^ r^O = V6^¥r. ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 329 Substituting these values in r/r we have r__ 7-8 X IQio This gives only the initial value of - and cannot be taken to hold for a time which will make T^d^/C^ appreciable. But by (3) we see that r = if 6 = C/T, so that CI27tT is a superior limit to the number of devolutions, even if we suppose the way clear right up to the centre. Putting the numerical values we get C/27rT= 46-5A*. Suppose, for example, that r = b = 493 x 3 x 10^^ ; a = 1, then r/r = - 3-5 X 10-16. If we multiply by 3-15 x 10^, the seconds in a year, we obtain (r/r) X 3-15 x 10^ = M x IQ-^. This implies that a sphere 1 cm. radius and density 5-5, starting with the velocity of the earth, and at its distance from the sun, will move inwards 10,0 00 ^^ ^^^ distance in about 10,000 years. It cannot in all make so many as 46-5 X 6^ = 1-79 x 10^ revolutions. If we put a = 0-001 cm., since the effects are inversely as a, then its distance will decrease by about 1 in 10,000 in 10 years, and it cannot make in all so many as 1-79 x 10^ revolutions. If instead of starting from the distance of the earth, the particle starts from, say, 0-1 the distance, the effect in the radius is 100 times as great and the number of revolutions is v'^lO times less. Then with radius 1 cm. the distance decreases by foTooo ^^ ^^^ years, and there are not so many as 80 X 10^ revolutions §, while with radius 0-001 cm. the distance decreases by ^Q QQQ in 0-1 year, and there are not so many as 80,000 revolutions §. Small particles, therefore, even of the order of 1 cm. radius, would be drawn into the sun, even from the distance of the earth, in times not large compared with geological times, and dust-particles if large enough to absorb solar radiation would be swept in in a time almost comparable with historical times. Near the sun the effects are vastly greater. The application to meteoric dust in the system is obvious. There should be a similar effect with dust and small particles circulating round the earth. If, for example, any of the Krakatoa dust was blown out so far beyond the appreciable atmosphere, and was given such motion that the particles became satellites to the earth, at no long time the dust will * [The original has 6 Iria. In what follows, the necessary alterations consequent upon tliis correction have been made. Ed.] § [The original has 80,000 and 80 respectively. The correction necessitates a modification in the views expressed in the succeeding paragraphs. Ed.] 330 - RADIATION IN THE SOLAR SYSTEM: return. A ring of dust-particles moving round a planet and receiving heat either from the sun or from the planet will tend to draw in to the planet. [Note added October 31. Since the foregoing paper was printed I have re-examined the theory of the pressure on a fully radiating surface when in motion, and have come to the conclusion that the change in pressure due to the motion is only half as great as that obtained on p. 325. In that investi- gation the pressure was assumed to be equal to the energy-density, whether the surface was at rest or in motion, whereas it appears, if the following mode of treatment is correct, that the pressure on a radiating surface moving u forward is only 1 — yy of the energy-density of the radiation emitted. Let us suppose that a surface A, a full radiator, is moving with velocity u towards a full absorber J5, which, with the surroundings, we will suppose at 0° ^. Consider for simplicity a parallel pencil issuing normal from A with velocity U towards B. Let the energy-density in the stream from AhQ E when A is at rest, and E' when it is moving. Let the pressure onAhaf^E when it is at rest, and f' when it is moving. When moving, A is emitting a stream of momentum f' per second and this momentum ultimately falls on B. Let A start radiating and moving at the same instant; let it move a distance d towards B, and then let it stop radiating and moving. It emits momentum f' per second for a time dju and therefore emits total momentum f'dju. Since B is at rest, the pressure on it, the momentum which it receives per second, is W . But since A is following up the stream sent out, B does not receive through a period as long as dju, but for a time less by djU. If we assume that the total momentum received by B is equal to the total sent out by A, we have f'dju = E' (dju - djU), or rp' =.E' (I- ujU). To find E' in terms of E we must make some assumption as to the effect of the motion on the radiation emitted. In the paper I have assumed that the emitting surface converts the same amount of its internal energy per second into radiant energy as when it is at rest, but that ^p'u of the energy of motion of the radiating mass is also converted into radiant energy. Since the radiation emitted in one second is contained in length TJ — ^/, we have E' {U - u) =^EU + p'u =EU-\-Fy (^— ) ^, whence E' = E - ^' ,, = ^ (1 + 2WC/). The same result is obtained if we assume that the amplitude of the emitted waves is the same whether the surface is moving or not, and that ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 331 the energy-density is inversely as the square of the wave-length for given amplitude. We have, therefore, if the above application of the equality of action and reaction is justified, In a similar way we can find the effect of motion of an absorber on the pressure against it due to the incident radiation. Let a stream of energy-density E be incident on a fully absorbing surface moving towards the source with velocity u. Let the surface be at 0° ^, so as to obtain the effect of the incident radiation only. When the surface is at rest, we may regard the stream as bringing up momentum E per second, or as containing momentum of density E/U brought up with velocity U to it. If the surface is moving towards the source, it takes up in one second the E momentum in length U + u, or receives yj{U + u), and the pressure on it is ,'^.(i.a = ,(i.-^). It is easy to show that when a perfect reflector is moving, the pressure 2u' upon it is altered from ^^ to ^ ( 1 + jj In the paper, the case of a full radiator in an enclosure at zero has alone 1 -IT 11 •<• . ^ u ^ u cos Y , been considered, so that the correctmg factor is 1 + yy or 1 H ^,- when the motion is at an angle x to the line of radiation. Hence the forces obtained 2u in the paper when the factor was 1 + 77 are all double those obtained with the factor now given. The process of drawing in small particles to the sun is correspondingly lengthened out. It is, perhaps, worth noting that the motion of a body round the sun produces a small aberration-effect. If the body is a sphere, the sunlight does not fall on the hemisphere directly under the sun, but on one turned round through an angle u/U. The pressure of the radiation, though still straight from the sun, does not act through the centre but through a point ^jj^ X (radius of sphere) in front of the centre. Thus, in the case of the earth, it will tend to stop the rotation. But the effect is so minute that if present conditions as to distance and radiation were maintained, it would take something of the order of 10^^ years to stop the whole of the rotation. J. H. P.] 21. NOTE ON THE TANGENTIAL STRESS DUE TO LIGHT INCIDENT OBLIQUELY ON AN ABSORBING SURFACE. [PM. Mag. 9, 1905, pp. 169-171.] [Read at Section A, British Association, Cambridge, August, 1904.] The existence of pressure on a surface due to tlie incidence of a normal beam of light, first deduced as a consequence of the electromagnetic theory by Maxwell, has been fully confirmed by the experiments of Lebedew, and quite independently by the exact work of Nichols and Hull. These experi- ments show that the pressure exists and that it is equal to the energy per c.c. or to the energy-density in the incident beam. In so far as it produces this pressure we may regard the beam as a stream of momentum, the direction of the momentum being along the line of propagation, and the amount of momentum passing per second through unit area cross-section of the beam being equal to the density of the energy in it. Let E denote this energy-density. If the beam is inclined at 9 to the normal to a surface on which it falls, the momentum-stream on to unit area of the surface is E cos 9 per second, and this is the force which the beam will exert in its own direction. If the beam is entirely absorbed, the result is a pressure E cos^ 9 along the normal and a tangential stress in the plane of incidence E sin 9 cos 9 = IE sin 29. If ^ of the incident beam is reflected, the normal pressure is {I -\- ^x) E cos^ 9 , and the tangential stress is - -^sin2^*. When there is absorption the tangential stress has a maximum value at 45° if IX is constant. When there is no absorption the tangential stress disappears. The tangential stress is much more easily detected than the normal pressure. For the action of the gas surrounding the surface is normal to it and is with difficulty disentangled from the normal light-pressure. But the gas-action is at right angles to the tangential stress, and it is merely necessary to arrange a surface free to move in its own plane to eliminate the action of the normal forces and to reveal the tangential stress. * These expressions are given in 'Radiation in the Solar System,' Phil. Trans. A, 202, p. 539. [Collected Papers, Art. 20.] NOTE ON THE TANGENTIAL STRESS DUE TO LIGHT, ETC. 333 With the assistance of my colleague Dr. Guy Barlow, to whom I am much indebted for help in the work, I have made the following experiment to show the existence of the stress. Two circular glass discs, each 2-75 sq. cm. area, were fixed at the ends of a horizontal light glass rod 5*3 cm. long, the discs being perpendicular to the rod and fixed to it at their highest points. One of the discs was lampblacked, and the other silvered. The rod was placed in a light wire cradle and suspended by a fine quartz fibre about 25 cm. long in a brass case with glazed sides. On the cradle was a mirror by which deflections could be observed with a telescope on a millimetre-scale 1-8 metres distant. The moment of inertia of the system was 2*35 gm. cm.^ and the time of vibration was 146 seconds. A deflection of 1 scale-division therefore corresponded to a tangential force on a disc of about one two-millionth of a dyne — more exactly 0-483 X 10-6 (jyne. The air was pumped from the case till the pressure was less than 1 cm. of mercury. At this pressure the irregularity of the disturbances due to the residual gas is very greatly reduced. A parallel beam of light from a Nernst lamp was then directed so as to be incident obliquely on the lampblacked disc. From the arrangement of the discs it is obvious that a uniformly distributed normal force would have no moment tending to twist the system, while a tangential force would have a moment and would twist it. In all cases the disc moved away from the source of light. The deflection was a maximum when the incidence was not very far from 45°, and fell off on each side of the maximum value. As there are various sources of error not yet removed, we have not made a complete series of measurements but have only made sure that the effect is of the order to be expected from the theory, by finding the deflection for an angle of 45°. The beam from the Nernst lamp when incident at 45° turned the rod through 16-5 scale-divisions. Assuming total absorption, the tangential force should be ^E sin 2d x area of disc == ^E x 2-75. Equating to the value of the force given by the deflection, viz., 0-483 X 10-6 X 16-5, we have E = 5-8 x 10-^ erg/cm.^. The same beam was then directed on to a small lampblacked silver disc of known heat-capacity, through a glass plate of thickness equal to that of the side of the case. The initial rise of temperature per second was measured by a ther mo junction of constantan wire soldered to the disc. The energy- density of the stream was thus found to be ^ = 6-5 x 10"^ erg/cm. 3. 334 NOTE ON THE TANGENTIAL STRESS DUE TO LIGHT, ETC. The agreement of the two values is quite as close as could be expected in so rough a determination*. When the beam was directed on to the silver disc at the other end of the torsion-rod, the deflection was much less, as was to be expected. We have also made some qualitative experiments with a blackened glass cyhnder — a ring cut from a test-tube — suspended by a quartz fibre with its axis vertical. When a beam fell on this in any direction not along a diameter, there was always a twist in the direction corresponding to the tangential stress. * [A redetermination of the various constants and a revision of the calculations gave a still closer agreement. The observed torque was 21 x 10"^ cm. dynes, while the torque calculated from the energy was 22 x 10"^ cm. dynes. This correction is given in The Pressure of Light, p. 55. (Romance of Science Series, S.P.C.K. 1910.) Ed.] Wl [Note by G. Barlow, July 1916. Particular care was taken to make the torsion-system very symmetrical th respect to the axis of suspension. The success of the experiment depended, also, on using a uniform parallel beam of light of cross-section slightly greater than the projected area of the disc. During the small oscilla- tions of the system the disc, therefore, remained uniformly illuminated. It was found that when a circular patch of light was focussed centrally on the disc the deflections were irregular. On one occasion we used a beam of sunlight reflected from a heliostat, but owing to the very variable absorption by the town atmosphere the observed pressure showed great fluctuations. The experiment with the cylinder has since been repeated, and was shown to some members of the British Association at the Birmingham meeting of 1913. The cylinder was of aluminium and was turned very accurately by a watchmaker. The ends were closed, but near the axis two small air- holes were drilled. The surface was blackened with a deposit of asphaltum. The observed torques due to the light-pressure were always of the order of magnitude expected, and variation of the gas-pressure over a considerable range did not greatly afi^ect the results. This method is suitable for lecture demonstration but it does not appear satisfactory for exact measurements.] 22. RADIATION-PKESSURE *. [Phil. Mag. 9, 1905, pp. 393-406.] [Presidential Address, delivered at the Annual General Meeting of the Physical Society, February 10, 1905.] A hundred years ago, when the corpuscular theory held almost universal sway, it would have been much more easy to account for and explain the pressure of light than it is to-day, when we are all certain that light is a form of wave-motion. Indeed, on the corpuscular theory it was so natural to expect a pressure that numerous attempts were madef in the eighteenth century to detect it. But the early experimenters had a greatly exaggerated idea of the force they looked for. Even on their own theory it would only have double the value which we now know it to possess, and their methods of experiment were utterly inadequate to show so small a quantity. But had these eighteenth-century philosophers been able to command the more refined methods of to-day, and been able to carry out the great experiments of Lebedew and of Nichols and Hull, and had they further known of the emission of corpuscles revealed to us by the cathode-stream and by radioactive bodies, there can be little doubt that Young and Fresnel would have had much greater difficulty in dethroning the corpuscular theory and setting up the wave-theory in its place. The existence of pressure due to waves, though held by Euler and used by him 160 years ago to explain the formation of comets' tails by repulsion, seems to have dropped out of sight, till Maxwell, in 1872, predicted its existence as a consequence of his Electromagnetic Theory of Light. It is remarkable that it should have been brought to the front through the investigation of such a special type, such an abstruse case, of wave-motion, and that it was not seen that it must follow as a consequence of any wave-motion, whatever the type of wave we suppose to constitute Light. I believe that the first suggestion that it is a general property of waves is due to Mr. S. Tolver Preston, who in 1876 { pointed out the analogy of the energy-carrying power * [This address is included here because it contains an account of some original work not elsewhere described. Ed.] t Some account of these methods is given by Nichols and Hull in 'The Pressure due to Radiation,' Proc. Am. Ac. vol. .38, no. 20, p. 559. See also Priestley, On Vision, p. 385. t Engineering, 1876, vol. 21, p. 83. 336 KADIATION-PRESSURE of a beam of light with the mechanical carriage by belting, and calculated the pressure on the surface of the Sun by the issuing radiation, obtaining a value equal to the energy-density in the issuing stream, without assumption as to the nature of the waves. But though the analogy is valuable, I confess that Mr. Preston's reasoning does not appear to me conclusive, and I think it still remains an analogy. There is, I suspect, some general theorem yet to be discovered, which shall relate directly the energy and the momentum issuing from a radiating source. It seems possible that in all cases of energy- transfer, momentum in the direction of transfer is also passed on, and there- fore there is a back pressure on the source. Such pressure certainly exists in material transfer, as in the corpuscular theory. It exists too, as we now know, in all wave-transfer. From the investigation below (p. 338) it appears to exist when energy is transferred along a revolving twisted shaft. In heat- conduction in gases, the kinetic theory requires a carriage of momentum from hotter to colder parts ; so that there is some ground for supposing the pressure to exist in all cases. Though we have not yet a general and direct dynamical theorem accounting for radiation-pressure. Professor Larmor* has given us a simple and most excellent indirect mode of proving the existence of the pressure, which applies to all waves in which the average energy-density for a given amplitude is inversely as the square of the wave-length. Let us suppose that a train of waves is incident normally on a perfectly reflecting surface. Then, whether the reflecting surface is at rest, or is moving to or from the source, the perfect reflection requires that the disturbance at its surface shall be annulled by the superposition of the direct and reflected trains. The two trains must there- fore have equal amplitudes. Suppose now that the reflector is moving forwards towards the source. By Doppler's principle, the waves of the reflected train are shortened, and so contain more energy than those of the incident train. This extra energy can only be accounted for by supposing that there is a pressure against the reflector, that work has to be done in pushing it forward. When the velocity of the reflector is small, the pressure is easily found to be equal to E ll + jj), where ^ is the energy-density just outside the reflector in the incident train, U is the wave- velocity, and a the velocity of the reflector. If u = 0, the pressure is E ; but it is altered 2u by the fraction -^ when the reflector is moving, and the alteration changes sign with ii. A similar train of reasoning gives us a pressure on the source, increased when the source is moving forward, decreased when it is receding. It is essential, I think, to Larmor's proof that we should be able to move the reflecting surface forward without disturbing the medium except by * Encyc. Brit. vol. 32, ' Radiation,' p. 121. RADIATION-PRESSURE 337 reflecting the waves. In the case of light-waves it is easy to imagine such a reflector. We have to think of it as being, as it were, a semipermeable membrane, freely permeable to ether, but straining back and preventing the passage of the waves. In the case of sound-waves, or of transverse waves in an elastic solid, it is not so easy to picture a possible reflector. But for sound-waves I venture to suggest a reflector which shall freeze the air just in front of it, and so remove it, the frozen surface advancing with constant velocity u. Or perhaps we may imagine an absorbing surface which shall remove the air quietly by solution or chemical combination. In the case of an elastic solid, we may perhaps think of the solid as melted by the ad- vancing reflector, the products of melting being passed through pores in the surface and coming out to solidify at the back. Though Larmor's proof is quite convincing, it is, I think, more satisfying if we can realise the way in which the pressure is produced in the different types of wave-motion. In the case of electromagnetic waves. Maxwell's original mode of treatment is the simplest, though it is not, I believe, entirely satisfactory. According to his theory, tubes of electric and of magnetic force alike, produce a tension lengthways and an equal pressure sideways, equal respectively to the electric and magnetic energy-densities in the tubes. We regard a train of waves as a system of electric and magnetic tubes transverse to the direction of propaga- tion, each kind pressmg out sideways^that is, in the direction of propagation. They press against the source from which they issue, against each other as they travel, and against any surface upon which they fall. Or we may take Professor J. J. Thomson's point of view*. ' Let us suppose that the reflecting surface is metallic; then, when the light falls on the surface, the variation of the magnetic force induces currents in the metal, and these currents pro- duce opposite effects to the incident light, so that the inductive force is screened off from the interior of the metal plate : thus the currents in the plate, and therefore the intensity of the light, rapidly diminish as we recede from the surface of the plate. The currents in the plate are accompanied by magnetic force at right angles to them ; the corresponding mechanical force is at right angles both to the current and the magnetic force, and therefore parallel to the direction of propagation of the light.' In fact, we have in the surface of the reflector a thin* current-sheet in a transverse magnetic field, and the ordinary electrodynamic force on the conductor accounts for the pressure. In sound-waves there is at a reflecting surface a node— a point of no motion, but of varying pressure. If the variation of pressure from the undisturbed value were exactly proportional to the displacement of a parallel layer near the surface, and if the displacement were exactly harmonic, then the average pressure would be equal to the normal undisturbed value. But * Maxwell's Electricity and Magnetism, 3rd edition, vol. 2, p. 441, footnote. P. c.w. 22 RADIATION-PHESSURE consider a layer of air quite close to the surface. If it moves up a distance y towards the surface, the pressure is increased. If it moves an equal distance y away from the surface, the pressure is decreased, but by a slightly smaller quantity. To illustrate this, take an extreme case, and for simplicity suppose that Boyle's law holds. If the layer advances half-way towards the reflecting surface, the pressure is doubled. If it moves an equal distance outwards from its original position, the pressure falls, but only by one-third of its original value ; and if we could suppose the layer to be moving harmonically, it is obvious that the mean of the increased and diminished pressures would be largely in excess of the normal value. Though we are not entitled to assume the existence of harmonic vibrations when we take into account the second order of small quantities, yet this illustration gives the right idea. The excess of pressure in the compression-half is greater than its defect during the extension-half, and the net result is an average excess of pressure — a quantity itself of the second order — on the reflecting surface. This excess in the compression-half of a wave-train is connected with the extra speed which exists in that half, and makes the crests of intense sound-waves gain on the troughs. Lord Rayleigh*, using Boyle's Law, has shown that the average excess on a surface reflecting sound-waves should be equal to the average density of the energy just outside ; and I think the same result can be obtained by his method if we use the adiabatic law. But the subject is full of pitfalls, and I am by no means sure that the result is to be obtained so easily as it appears to be. It is perhaps worth while to note one of these pitfalls, of which I have been a victim. It is quite easy to obtain the pressure against a reflecting surface by supposing that the motion just outside it is harmonic. But the result comes out to (y + 1) x energy-density, where y is the ratio of the specific heats. Lord Rayleigh kindly pulled me out of the pit into which I fell, pointing out that when we take into account second-order quantities the ordinary sound-equation does not hold. In fact we cannot take the disturbance as harmonic, and the simple mode of treatment is illusory. The pressure in transverse waves in an elastic solid is, I think, to be accounted for by the fact that when a square, ABCD, is sheared into the position aBCd (Fig. 1) through an angle e, the axes of the shear, aC and Bd, 6 no longer make 45° with the planes of shear AD, BC. Since ACa = -, the 6 . . pressure-line aC is inclined at 45° — - to the direction of propagation, and the tension-line at 45° + ^ to that line. The result is a small pressure perpendicular to the planes of shear, that is, in the direction of propagation ; .and this small pressure is just equal to the energy-density of the waves. * Phil. Mag. vol. 3, 1902, p. 338, 'On the Pressure of Vibrations.' RADIATION-PRESSURE 339 For let PQR (Fig. 2) be a small triangular wedge of the solid, PQ being a plane of shear perpendicular to the direction of propagation. Let this wedge have unit thickness perpendicular to the plane of the figure. Let FR be along a pressure-line and QR along a tension-line, and let pressure and tension each be P. Resolve the forces on PR and QR perpendicular to PQ. Then we have a force from right to left, P . QR cos PQR -P .PR cos QPR = P.PQ |cos2 ^45° - - cos^ ^45° + ^\i = P . PQ .e. oz/T-ccr/OAA^ Of^ f/fOPAOATWA/ Fig. 1. Thus, to prevent motion in the direction of propagation there must be a pressure on PQ equal to Pe = ne^, where n is the rigidity- modulus. But the strain-energy per unit volume is -^ , and the kinetic energy is equal to it. The total energy-density is therefore ne^, and the pressure is equal to this. The pressure of elastic-solid waves appears to be beyond experimental verification at present. But that of sound-waves has been demonstrated most successfully by Altberg*, working in Lebedew's laboratory at Moscow. A small wooden cylinder, 21 mm. diameter, was suspended at one end of a torsion-arm, with its axis horizontal and transverse to the arm. One end of the cylinder occupied a circular hole in the middle of a board, there being just sufficient clearance to allow it to move, and the plane end was flush with the outer surface of the board. When very intense sound-waves 10 cm. in length, from a source 50 cm. distant, impinged on the board, the cyHnder was pushed back, the pressure sometimes rising to as much as 0-24 dyne/cm. 2. The intensity of the sound was measured independently by the vibrations of a telephone-plate, in a manner devised by M. Wien, and through a large range it was found that the pressure on the cylinder was proportional to the intensity indicated by the telephone- manometer. * Ann. der Physik, vol. 11, 1903, p. 405. 22—2 340 RADIATION-PRESSURE Just lately Professor Wood* has devised a strikingly simple experiment to illustrate sound-pressure. The sound-waves from strong induction-sparks are focussed by a concave mirror on a set of vanes like those of a radiometer, and when the focus is on the vanes as they face the waves the mill spins round. Theory and experiment, then, justify the conclusion that when a source is pouring out waves, it is pouring out with them forward- momentum as well as energy, the momentum being manifested in the reaction, the back-pressure against the source, and in the forward pressure when the waves reach an opposing surface. The wave-train may be regarded as a stream of momentum travelling through space. This view is most clearly brought home, perhaps, by considering a parallel train of waves which issues normally from a source for one second, travels for any length of time through space, and then falls normally on an absorbing surface for one second. During this last second, momentum is given up to the absorbing surface. During the first second, the same amount was given out by the source. If it is conserved in the meanwhile, we must regard it as travelling with the train. Since the pressure is the momentum given out or received per second, and the pressure is equal to the energy-density in the train, the momentum-density is equal to the energy-density H- wave-velocity. This idea of momentum in a wave-train enables us to see at once what is the nature of the action of a beam of light on a surface where it is reflected, absorbed, or refracted, without any further appeal to the theory of the wave- motion of which we suppose the light to consist f. It is convenient to consider the energy per linear centimetre in the beam, and the total pressure-force, equal to this linear energy-density, so as to avoid any necessity for taking into account the cross-section of the beam. Thus, in total reflection, let a beam AB (Fig. 3) be reflected along BC, and let AB = BC represent the momentum in each in length V equal to the velocity of light. Produce AB to D, making BD = AB. Then DC represents the change in the momentum per second due to the reflection — the force on the beam, if such language is permissible ; and CD is the reaction, the total light-force on the surface. If there is total absorption, let AB (Fig. 4) represent the momentum of the incident beam. Eesolve AB into AE parallel and EB normal to the surface. Then, since the momentum AB disappears as light-momentum, there must be a normal force EB on the surface and a tangential force AE * Phys. Zeitschrift, 1 Jan. 1905, p. 22. •j- A discussion, on the electromagnetic theory, of the forces exerted by light is given by Goldhammer, A7in. der Phys. vol. 4, 1901, p. 483. RADIATION-PRESSUKE 341 parallel to the surface. I have lately* described an experiment which shows the existence of the tangential force AE. If there is total refraction, let AB (Fig. 5) be refracted along BC with velocity V. HE is the energy in unit length of AB, and if E' is the energy in unit length of BC, the equahty of energy in the two beams is expressed by- VE = V'E'. Fig. 3. Fig. 4. But if M is the stream of momentum passing per second along AB, and if M' is that along BC, then M=E and M' = E'. Whence and VM = F'M' M' = y, M = fxM. Let AB = M, and BC along the refracted beam = M' = fjuM = fiAB. Draw CD parallel to BA, meeting the normal BN in D. Then CD = CB sin r/sin CB = AB=M. Phil. Mag. Jan. 1905, p. 169. [Collected Papers, Art. 21.] 342 RADIATION-PRESSURE Hence, by the refraction, momentum DC has been changed to momentum BC, or momentum BD has been imparted to the light. There is therefore a reaction BB on the surface. The force DB may be regarded as a pull-out or a pressure from within, and it is along the normal*. If the refraction is from a denser to a rarer medium, CB will now represent the incident stream and BA or CD the refracted stream. BD is the stream added to CB to change it to CD, and DB is the force on the surface, again a force outwards along the normal. In any real refraction with ordinary light, there will be reflection as well as refraction. The reflection always produces a normal pressure, and the refraction a normal pull. But with unpolarised light, a calculation shows that the refraction-pull, for glass at any rate, is always greater than the reflection-push, even at grazing incidence. The following table has been calculated from Fresnel's formula for unpolarised light by Dr. Barlow : P = total pull on surface. M = momentum per second in incident beam. R = reflection-coefflcient for angle i. /x = 1-5. i R PjM •0400 •4000 20 •0402 •4240 40 •0458 •4925 50 •0572 •5310 60 •0893 •5720 65 •1205 •5771 Maximum 70 •1710 •5683 75 •2531 •5329 80 •3878 •4521 89 •9044 •0738 90 -f/^ 2^,^de 90 i"ooo6 •0000 If a ray of light passes obliquely through a parallel plate, there is a normal pull outwards at incidence and a normal pull outwards at emergence : and if the refraction were total, this would result in a couple. But since some of the light returns into the first medium, it is easy to see that the net result is a normal repulsion and a couple. An experiment which I have lately made in conjunction with Dr. Barlow will serve as an illustration of the idea of a beam of light regarded as a stream * It has been pointed out by J. J, Thomson, Electricity and Matter, p. 67, 'that even when the incidence of the light is oblique, the momentum communicated to the substance is normal to the refracting surface ' The change of momentum of a beam of light is, it may be noted, the same on the wave and on the corpuscular theory. RADIATION-PRESSURE 343 of momentum. A rectangular block of glass, 3 cm. x 1 cm. x 1 cm., was suspended by a quartz fibre so that the long axis of the block was horizontal. It hung in a case with glass windows, which was exhausted to about 15 mm. of mercury. A horizontal beam of light, from either a Nernst lamp or an arc, was directed on to one end of the block so that it entered centrally as AB in Fig. 6, and at an angle of incidence about 55°. After two internal reflections it emerged centrally as EF from the other end. Thus a stream of momentum AB was shifted parallel to itself into the line EF, or a counter-clockwise couple acted on the beam. The reaction was a clockwise couple on the block. Using mirror, telescope, and a millimetre-scale about 184 cm. distant, with the strongest light a very small deflection in the right direction could just be detected. But the quartz fibre was rather coarse, indeed needlessly strong; and as the time of vibration was only 39 seconds, the deflection was very minute. To render the effect more evident we used intermittent passage of the beam, sending it in during the half-period of vibration while B was F Fig. 6. Plan. moving from A, and shutting it off while B was moving towards A. The swings then always increased. When the beam was sent in during the approaching half and shut off during the receding half, the swings always decreased, and always rather more rapidly than they increased during the first half. For in the first case the natural damping acted against the light couple, and in the second with it. In one experiment the average increase was -55 scale-division and the average decrease -61 per period, and was fairly regular in each case. The mean was -58. The steady deflection is half this, or 0-29 division, giving a couple 11 x 10"^ cm. dyne. We made a measure- ment of the energy in the beam by means of the rate of rise of a blackened silver disc; but it was necessarily very inexact, as we had no means of securing constancy in the arc used in this experiment. This energy- measure- ment gave as the value of the couple 6 x 10"^, and the agreement is sufficient to show that the order of the result is right. An analysis of this experiment shows that the couple was really due to the pressures at the two internal reflections ; for, as we have seen, the forces at incidence at B and emergence at E are normal and produce no twist. 344 RADIATION-PRESSURE Another experiment which we have made is, I think, more interesting, in that it brings into prominence the pull outwards or push from within occurring on refraction. Two glass prisms, each with refracting angle 34°, another angle being a right angle, and with refracting edge 1-6 cm. long, were arranged as in Fig. 7 (which shows the plan) at the ends of a thin brass torsion- arm suspended at its middle point from a quartz fibre in the same case as that used in the last experiment. The two inner faces were 3 cm. apart, and their width was 1-85 cm. A mirror gave the reflection of a millimetre- scale 171-4 cm. distant. The moment of inertia of the system was 48 gm. cm.^, and the time of vibration was 317 seconds. The air-pressure was reduced as before. When a beam of light from a Nernst lamp was sent through the system, as shown in the figure, it was shifted parallel to itself through a distance about 1-64 cm. The torsion-arm moved round clockwise by an easily measurable amount. In one experiment the deflection was 3-3 scale- divisions, indicating a couple 1-84 x 10~^ cm. dyne. The same beam directed Fig. 7. Plan. on to the blackened silver disc gave the linear energy-density as 9-8 x 10"^, which should have given a couple 1-6 x 10~^. Though the agreement is perhaps accidentally close, yet, as we could use a Nernst lamp, the measure- ments were much more trustworthy than in the last experiment*. The interesting point here is that the effect could only be produced by a force outwards at B and E. Whatever forces exist at C and D would be normal to the surfaces and would give no twist. A very short experience in attempting to measure these light-forces is sufficient to make one realise their extreme minuteness — a minuteness which appears to put them beyond consideration in terrestrial affairs, though I have tried to showf that they may just come into comparison with radiometer- action on very small dust-particles. In the Solar system, however, where they have freer play and vast times to work in, their effects may mount up into importance. Yet not on the larger bodies ; for on the earth, assumed to be absorbing, the whole force of * [See The Pressure of Light, p. 61 (S.P.C.K. 1910), where these values are slightly corrected, and results are given for a similar experiment with smaller prisms. Ed.] t Nature, Dec. 29, 1904, p. 200. ^Collected Papers, Art. 65.] RADIATION-PRESSURE 345 the light of the sun is only about a 50 million-millionth of his gravitation- pull. But since the ratio of radiation-pressure to gravitation-pull increases in the same proportion as the radius diminishes if the density is constant, the pressure will balance the pull on a spherical absorbing particle of the density of the earth if its radius is a 50 billionth that of the earth — a little over a hundred-thousandth of a centimetre, say, if its diameter is a hundred- thousandth of an inch. We may illustrate the possible effects of radiation-pressure without proceeding to such fineness as this. Let us imagine a particle of the density of the earth, and a thousandth of an inch in diameter, going round the sun at the earth's distance. There are two effects due to the sun's radiation. In the first place, the radiation-push is j^^y of the gravitation-pull ; and the result is the same as if the sun's mass were only 99/100 of the value which it has for larger bodies like the earth. Hence the year for such a particle would be longer by ^^-^, or about 367 instead of 365 J days. In the second place, the radiation absorbed from the sun and given out again on all sides is crushed up in front as the particle moves forward and is opened out behind. There is thus a slightly greater pressure due to its own radiation on the advancing hemisphere than on the receding one, and this appears as a small resisting force in the direction of motion. Through this the particle tends to move in a decreasing orbit spiralling in towards the sun, and at first at the rate of about 800 miles per annum. Further, if there be any variation in the sun's rate of emitting energy, there will be a corresponding variation in the increase of the year and the decrease of the solar distance, and the particle, if we could only observe it^ would form a perfect actinometer. Though, unfortunately, we cannot observe the motion of independent small particles circling round the sun at the distance of the earth, there is good reason to suppose that some comets at least are mere clouds of dust. If we are right in this supposition, they should show some of these effects. Encke's comet at once suggests itself as of this class ; for, as everyone knows, it shortens its journey of 3 J years round the sun on every successive return, and on the average by about 2i hours each revolution. Mr. H. C. Plummer* has lately been investigating this comet's motion; and he finds that if it were composed of dust- particles, each of the earth's density and about J^j mm. or rather less than a thousandth of an inch in diameter, the resisting force due to radiation-pressure would account for its accelerating return. But the sun's effective mass would be reduced by about 1/80; and on certain sup- positions he finds that the assumed mean distance as calculated from Kepler's law, without reference to radiation, is greater than the true mean distance * Monthly Notices B.A.S. Jan. 1905, 'On the Possible Effects of Radiation on the Motion of Comets, with special reference to Encke's Comet.' 346 RADIATION-PRESSURE by something of the order of 1 in 400, and he thinks such a large error is hardly possible. So that radiation-pressure has not yet succeeded in fully explaining the eccentricities of this comet. But comets are vague creatures. As Mr. Plummer suggests, we hardly know that we are looking at the same matter in the comet at its successive returns ; and I still have some hope that the want of success is due to the uncertainty of the data. There is one more effect of this radiation-pressure which is worthy of note : its sorting action on dust-particles. If the particles in a dust-cloud circling round the sun are of different sizes or densities, the radiation-accelera- tions on them will differ. The larger particles will be less affected than the smaller, will travel faster round a given orbit, and will draw more slowly in towards the sun. Thus a comet of particles of mixed sizes will gradually be degraded from a compact cloud into a diffused trail lengthening and broadening, the finer dust on the inner and the coarser on the outer edge. Let us imagine, as an illustration of this sorting action, that a planet, while still radiating much energy on its own account, while still in fact a small sun, has somehow captured and attached to itself as satellite a cometary cloud of dust. Then, if the cloud consists of particles of different sizes, while all will tend to draw in to the primary, the larger particles will draw in more slowly. But if the larger particles are of different sizes among themselves, they will have different periods of revolution, and will gradually form a ring all round the planet on the outside. Meanwhile the finer particles will drift in, and again difference in size will correspond to difference in period and they too will spread all round, forming an inner fringe to the ring. If there are several grades of dust with gaps in the scale of size, the different grades will form different rings in course of time. Is it possible that here we have the origin of the rings of Saturn ? The Radiation Theory is only just starting on its journey. Its feet are not yet clogged by any certain data, and all directions are yet open to it. Any suggestion for its future course appears to be permissible, and it is only by trial that we shall find what ways are barred. At least we may be sure that it deals with real effects and that it must be taken into account. [Compare Rayleigh, 'On the Momentum and Pressure of Gaseous Vibrations,' Phil. Mag. vol. 10, 1905, p. 364. Ed.] 23. ON PKOF. LOWELL'S METHOD FOR EVALUATING THE SURFACE- TEMPERATURES OF THE PLANETS ; WITH AN ATTEMPT TO REPRESENT THE EFFECT OF DAY AND NIGHT ON THE TEMPERATURE OF THE EARTH*. [Phil. Mag. 14, 1907, pp. 749-760.] Prof. Lowell's paper in the July number of the Philosophical Magazine marks an important advance in the evaluation of planetary temperatures, inasmuch as he takes into account the effect of planetary atmospheres in a much more detailed way than any previous writer. But he pays hardly any attention to the 'blanketing effect,' or, as I prefer to call it, the 'green- house-effect' of the atmosphere. He assumes in fact that the fourth power of the temperature is proportional to the fraction of solar radiation reaching the surface, and he neglects both the surface-radiation reflected down again and the radiation downwards of the energy absorbed by the atmosphere. This is brought out clearly in the footnote on p. 172, where he uses a formula of Arrhenius, to which I am unable to refer, but wliich I think he must misinterpret in making it give his result. The inadequacy of his method is well shown by its application to the cloud-covered half of the earth's surface. He finds that this half only receives 0-2 of the radiation which the clear sky half receives. The surface-temperature under cloud should therefore be only V 0-2 = 0-67 of that under clear sky. If the latter is 300° A. the former is only about 200° A. Common observation contradicts this flatly, for the difference is at most but a few degrees. On another point common observation appears, at any rate at first sight, to contradict Professor Lowell. He assumes that the loss in the radiation of the visible spectrum in its passage through the atmosphere is practically all due to reflection, and he puts it down as about 0-7 of the whole in clear sky. If this were true the reflection from the sky opposite to the sun would I think be vastly greater than it is. White cardboard reflects diffusely about 0-7 of sunlight. But when a piece of white cardboard is exposed normally to the sun's rays it is several times brighter than the cloudless sky. * In Phil. Trans. A, vol. 202, p. 525 f I attempted an evaluation, in which the atmosphere was taken into account as keeping the temperature at a given point practically the same day and night. I did not then know that Christiansen {Beibldtter zu den Ann. der Physik und Chemie, vol, 10, 1886, p. 532) had nearly twenty years earlier applied the fourth-power law to calculate planetary temperatures. His work deserves recognition as the first in which this law was applied. t [Collected Papers, Art. 20.] 348 The * greenhouse- effect' of the atmosphere may perhaps be understood more easily if we first consider the case of a greenhouse with horizontal roof of extent so large compared with its height above the ground that the effect of the edges may be neglected. Let us suppose that it is exposed to a vertical sun, and that the ground under the glass is ' black ' or a full absorber. We shall neglect the conduction and convection by the air in the greenhouse. Let S be the stream of solar radiation incident per sq. cm. per sec. on the glass. Of this let rS be reflected, aS be absorbed, and tS be transmitted by the glass. Then r + a+ t = 1. Let the ground send out radiation R per sq. cm. per sec. and of this let r^R be reflected, a^R be absorbed, and t^R be transmitted by the glass. Here also rj + a^ + ^^ = L It is to be noted that since the edges are far distant R is incident on each sq. cm. of glass. The glass, then, absorbs aS + a^R, and as it is thin it may be taken as having the same temperature on each side, so that it sends down to the ground J {aS + a-^R), the other half going upwards into space. Equating receipt and expenditure of radiation by the ground, R=.tS+riR+ i {aS + a^^R), whence on putting 7\= 1 — n^ — t^ we obtain a i2 = -S. The values of t and a depend upon the glass. By way of illustration let us take t = 0-6, a = 0-3. For radiation from a surface under 100° C. Melloni found that even thin glass is quite opaque. We have then t-^ = 0, and if we neglect reflection, probably small, Oi == L Then R = ^S=l'bS. If the glass were removed we should have R^S. The temperature of the ground is therefore \/'i-D = 1-1 times as high under the glass as it is in the open. If, for instance, it is 27° C. or 300° A. in the open, it is 330° A. or 57° C. under the glass. If the glass reflects some of the radiation R then a^ is less and the ground temperature is still higher. If the ground, instead of being black, reflects a fraction p of the incident sunlight, or has total albedo p, the formula must be modified. If we take into account merely the first reflection from the ground and assume that the glass has absorption a for it, then we easily find SURFACE-TEMPERATURES OF THE PLANETS 349 t + n^sJl^ If we take p = 0-1 the numerator is 0-78 instead of 0-75, and if we assume the fourth-power law for the low-temperature radiation emitted by the surface, the temperature is about 1 per cent, higher*. But the ground will probably reflect a much smaller fraction of the whole spectrum, and the correction for total albedo becomes inconsiderable. If we replace the sun by cloud the radiation is, on the average, of much lower temperature, and t and a are much nearer to t^ and a^. The value of R/S is then much nearer to 1, and the covered ground has a temperature much ]ess raised above that of the open ground. This agrees of course with common experience. A planetary atmosphere no doubt acts in some such way as the green- house glass. Let us, for the sake of comparison with Prof. Lowell's results, assume, as he has done, that we have a steady state, with the incident radiation normal to the surface. I do not see how to estimate the distribution of the radiation from the air between the upward stream into space and the down- ward stream to the surface. Since the lower layers of air are warmer than the upper probably more than half comes down, and the truth probably lies between the assumptions that the atmospheric radiation is J {aS + a^R) as it is with the greenhouse, and that it is aS + a^R when all the radiation would be downwards. Let us suppose that - (aS + a^R) comes downward. The albedos of the surfaces of both the Earth and Mars average, according to Lowell, 0-1 for visible radiation. They must be much less for the whole spectrum. Where all the data are uncertain the effect of small albedo may be neglected, and indeed in our ignorance of the dependence of temperature on radiation in the case of a partially reflecting surface, it is safer to neglect it. If dg is the actual surface-temperature under a vertical sun, and 6 is the tempera- ture which the surface would have without atmosphere, it is easily found that t-\- a/n ^1 ^- {n— l)a-^/n Earth. If we use Lowell's figures for the Earth under a clear sky, t = 0-42, a = 0-5 X 0-65 = 0-325, t^ = 0-5, since of the invisible radiation half is transmitted, a^ = 0-5, very httle is reflected. * [It would appear that in the preceding equation for R the term ( 9 ~ P ) ^ should he(^-l]pt. This would give the numerator as 0-70, and the temperature about 2 per cent, lower. Ed.] 350 PROF. Lowell's method for evaluating the We shall suppose in succession that (a) half of the radiation is downwards or that n = 2, (b) two-thirds ^ = f , (c) all n=l. We then find {a) 1 = 0-94; (6) J=0-99; (c) f=M2. For the case of a cloud-covered earth the data are very uncertain. Lowell takes t = 0-2 of 0-42 = 0-084, assuming that the atmosphere has already reflected and absorbed 0-58 before the cloud is reached, surely an overestimate, since the cloud-surface is in the higher air. Let us guess that ^ = 0-L The absorption without cloud is according to Lowell about 0-3. With cloud much is reflected back without reaching the lower and more absorbing regions. Let us guess that a = 0-2. Of the radiation from the surface we may suppose perhaps that 0-2 passes through, that 0-7 is reflected, and that 0-1 is absorbed. Of the 0-2 passing we may suppose that 0-1 is absorbed and 0-1 goes into space. Then ^^^ = 0-1 and a^ = 0-2. With these values we get for the different values of n . («) J=l; (b) J =1-08; (c) 1=1-31. These guesses, then, make the temperature under a cloudy sky at least as great as under a clear sky. But this is certainly not true in common experience, where, however, we may have clouds accompanied by cold winds and no approach to the steady state here assumed. The results merely serve to show that with certain absorptions and transmissions clouds might actually raise the surface-temperature, and that for the present it is better to neglect them. Mars. If we apply Lowell's data for Mars we have t = 0-64, and a = 0-40 x 0-65 = 0-26, ti = 0-6, and a^ = 0-4, since R is dark radiation. With these values we get for the different values of n (a) 1 = 0-99; (b) |=]-02; (e) |=M0. Comparison of the Earth and Mars. Let us take the temperature of the Earth as 17° C. or 290° A. If it were removed to the distance of Mars its temperature would be inversely as the square-root of the distance, which is 1-524 that of the Earth, or 290/1-235 = 235° A. StlRF ACE-TEMPERATURES OF THE PLANETS 351 With the different values of n the temperature of Mars should be (a) 235 X If =247° A. or -26°C., (b) 235 X -^2. _ 242° A. or -31°C., (c) 235 X j|§ = 231° A. or - 42° C. Of course the data are very uncertain and the formula used is only an approximation. But with these data it is hard to see how the temperature of Mars can be raised to anything like the value obtained by Professor Lowell. Perhaps the data are quite wrong. It is conceivable that Mars has a quite peculiar atmosphere practically opaque to radiations from the cold surface. Those who believe that there is good evidence for the existence of intelligent beings on that planet, should find no difficulty in supposing that they have been sufficiently intelligent to cover the planet with a glass roof or its equiva- lent. Then we might easily have ^+- = 0-77 and ^^+^^ = 0-5, and then the temperature might be raised to 281° A. or 8° C. Indeed, if the glass were of such kind as to transmit solar radiation, and if it were quite opaque to dark radiation while still reflecting a considerable proportion, the temperature might easily be raised far above this. An Attempt to represent the Effect of Day and Night on the Temperature of the Earth. The 'greenhouse' formula, which has been used in the foregoing dis- cussion, would hold only if all the conditions were steady. But in reality the alternations of day and night prevent a steady state, and we can only hope that the neglect of these alternations does not greatly affect the ratios of the temperatures found for different planets or for different elevations on the same planet. I shall now attempt to represent the effect of the diurnal variation in the supply of solar heat to the Earth, or rather to an abstract Earth. For even if we could represent the actual conditions we should obtain differential equations so complicated that they would be useless for practical purposes. To simplify matters, let us suppose that we are dealing with the equatorial region of the earth at the equinox, that the air is still, that the surface is solid and black, and that the sky is clear. The temperature of the air except near the surface can change but little during 24 hours. For over each square centimetre at sea-level we have 1000 gms. of air with specific heat 0-2375, and therefore with heat capacity 237-5. Consider a band of the atmosphere 1 cm. wide round the equator. A stream of solar radiation of length equal to the diameter 2r of the earth 352 PROF. Lowell's method for evaluating the enters a band of air of length equal to half the circumference. If the solar constant is 3 the average energy entering a sq. cm. column is 2rS 2S 6 , . . = — = - cal./mm. Then in 12 hours 1375 cal. enter on the average, and if this heat were all absorbed and retained it would raise the temperature on the average about 1375/237-5 = 5°-8 C. As the absorption is only partial and as radiation takes place from the air, the rise cannot really average nearly as much as this. Again, consider the radiation during the twelve hours of night. If the air ^vere a black body and of temperature 300° A., and these are absurdly exaggerated estimates of its radiating power and of its average temperature, it would only radiate about 1-2 cal./min. per sq. cm. column from its two surfaces, or 864 calories in the twelve hours, and neglecting the radiation from the ground the temperature would only fall about 864/237-5 or 3°-6 C. Obviously, then, the air as a whole cannot undergo much variation in tempera- ture as day alternates with night. It is indeed a flywheel storing the energy of many diurnal revolutions. We may, then, in a rough estimate consider that its temperature, and therefore its radiation, remains constant during the 24 hours. If the total radiation from a sq. cm. column per second is A, there will be a stream D downwards and U upwards where Z) + U = A. We can find an expression for A by equating it to the average absorption. Considering an equatorial band 1 cm. wide, the average energy entering it per sq. cm. in the 24 hours is - . Let the average amount absorbed be - . The value of 77 77 a at sea-level varies for clear sky from perhaps 0-3 with the zenith sun to very nearly 1 with the setting sun. Let the average radiation from the surface during the 24 hours be /?, of which a^ R is absorbed by the atmosphere. Then neglecting conduction through the air, the constant-temperature assumption gives us A=^ V a^R. 77 If a fraction is radiated downwards n _ j^ dS a^R rnr n The actual surface-temperature depends not only on radiation but also on conduction both by ground and air. But we shall neglect this conduction and shall suppose that the surface has reached an equilibrium between receipt and expenditure of radiation. This is a condition to which the surface tends at or soon after noon by day, and before dawn at night. We shall suppose SURFACE-TEMPERATURES OF THE PLANETS 353 that the low-temperature radiation from the surface is either transmitted or absorbed, so that, using the previous notation, ^j + ^1 = 1 and r^ = 0. If R^ is the equilibrium surface-radiation reached, we suppose about noon, yiTT n If Rn is the equilibrium surface-radiation in the later part of the night, we have to omit tS, and _ fiTT n To proceed further, we must express R in terms of S. We can only do this by some assumption. Probably it is not very far from the truth to assume that R = i (Ra + Rn), and we shall take this value. It gives us t a n and substituting in the values of day and night radiations we get t a s a + <Li 2 + mr = t + rnr n 1- "^ n t d i-> ai2'^ Rn a + mr s rnr ^1- ~' n Though these formulae are only obtained by making large assumptions, and by neglecting important considerations, they nevertheless show the tendency of the day and night effect, and it is worth while to apply them to the Earth, taking the best data at our command. At the surface let us take t = 0-4:2 and aj = 0-5 as before. For d we have no trustworthy observations, and I doubt whether a calculation from Langley's observations is of any more value than an estimate. Since a varies from perhaps about 0-3 to 1, let us take a = 0-628 or 27r/10, a value simpHf5H[ng arithmetic. At the level of Camp Whitney 3550 metres above sea-level, with barometer about 500 mm., and therefore with about | of the atmosphere below it, we may take ^ = 0-6 and a^ = 0-4. For a we must take a value much smaller than that at sea-level. Since the most absorbing third of the atmosphere is below, I do not think it is far wrong to take d as having half the value at the p. c. w. 23 354 PROF. Lowell's method for evaluating the lower level, and I therefore put a = 0-314. But I have also examined the consequences of putting it equal to 0-419, i.e. f of its value at the lower level, and the results are given below to show how much the figures are affected by the variation in the value taken. We have no data for n. I have therefore calculated the values of R^ and Rn in terms of S for successive values of n equal to 1, |, |, f , 2 ; corresponding to D equal to A, ^A, ^A, %A, and \A respectively. In the following tables the values of RajS and RJS are given, and also the mean RjS = J (R^ + Rn) S- Then follow the ratios of the day and night temperatures, 6^ and 6^, to the temperature ^ of a black surface radiating S, and then the mean value djd. The last column gives the range 6^ — 9^ on the supposition that = 300° A. The third table is only given to show that the change in the value of a does not greatly affect the results. The value of a of Table II is much more reasonable if that of Table I is near the truth. We need, therefore, only compare the results given in the first two tables. If we take the same values of n in each table, the value of 7^ is less at the higher level than at the lower in every case except that in which n has the extreme and probably inadmissible value of 2. The value of 6 is less at the higher level in every case. But it appears most probable that 1/n or DjA is greater at the lower level than at the higher. For consider a thin layer of air at sea-level. It is radiating equally up and down, but of the half going upwards a considerable fraction will be intercepted by the superin- cumbent and strongly absorbing layers. Now consider a thin layer close to the surface at the higher level. It, too, radiates half up and half down. But of the half going upwards a less fraction will be intercepted since the superincumbent layers are now less absorbing. Thus DjA will be greater at the lower than at the higher level*. We should, therefore, compare the results for any value of DjA in Table I with the results in Table II for a some- what lower value. We may exclude the extreme cases of 9^ = 2 and n= 1, as the true value is certainly between these, and confine our examination to intermediate values. Suppose, for example, that DjA = 4/5 at the lower level, while it is 3/4 at the upper level. Then djd = 0-88 from Table I at the lower level, while O/e = 0-83 from Table II at the upper level. Or if D/A = 3/4 at the lower level, while it is 2/3 at the upper level, O/O = 0-86 below, while ejO - 0-81 above. Or in each case the mean temperature is higher at sea-level by about 5 in 87 or by about 17° in 300°. * Another consideration leading to the same conclusion is that the atmosphere acts like a plate with its lower surface much warmer than its upper. When we only have the part above an elevated region the difiference of temperature between the surfaces is much less than for the whole air, and the radiations up and down are more nearly equal. SURFACE-TEMPERATURES OF THE PLANETS 356 Table I. At sea-level, t = 0-42, a-^ = 0-5, a = 0-6^8. Range n DIA ^d/S iiJS R/S eje ^nl^ ejd about 300° A. 1 1 103 0-61 0-83 1-01 0-88 0-95 41° 5/4 4/5 0-83 0-41 0-62 0-95 0-80 0-88 51° 4/3 3/4 0-79 0-37 0-58 0-94 0-78 0-86 56° 3/2 2/3 0-72 0-30 0-51 0-92 0-74 0-83 65° 2 1/2 0-62 0-20 0-41 0-89 0-67 0-78 85° Table II. At 3550 m. above sea-level. Barometer 500 mm. t = 0-6, ai = 04, a = 0-314. Range n D/A Rd/S njs R/S Sdie Kid dfd about 300° A. 1 I 0-97 0-37 0-67 0-99 0-78 0-89 71° 5/4 4:15 0-86 0-26 0-56 0-96 0-71 0-84 89° 4/3 3/4 0-84 0-24 0-54 0-96 0-70 0-83 94° 3/2 2/3 0-80 0-20 0-50 0-95 0-67 0-81 100° 2 1/2 0-74 0-14 0-44 0-93 0-61 0-77 125° Table III. At 3550 m. above sea-level and with. ^ = 0-6, ai = 0-4, but with a = 0419 = 2/3 of 0-628. Range n D/A RdlS Rjs R/S dale eje did about 300° A. 1 1 102 0-42 0-72 101 0-81 0-91 66° 5/4 4/5 0-90 0-30 0-60 0-97 0-74 0-86 80° 4/3 3/4 0-87 0-27 0-57 0-97 0-72 0-85 88° 3/2 2/3 0-83 0-23 0-53 0-95 0-69 0-82 95° 2 1/2 0-76 0-16 0-46 0-93 0-63 0-75 120° 23—2 356 METHOD FOR EVALUATING SURFACE-TEMPERATUBES OF THE PLANETS It is to be observed that the lower mean temperature at a higher level must hold good if the higher level is so much higher that there is practically no atmosphere above. For then t = 1 and a^ = 0, so that R^^ S and i?„ = 0. Therefore djd = 1 and 6 JO = and 0/6 = 1/2. The lower mean temperature of elevated parts of the earth's surface is a well-established fact. Perhaps if it were only observed in the case of mountain peaks it might be ascribed to the cold air blowing against them. The fall of temperature in free air as we go upwards tends towards that given by convective equilibrium, though recent observations show that it is not so great as that given by the adiabatic law. Thus for a rise of 3500 metres the adiabatic law would give a fall of about 32° C. if the sea-level temperature were 300° A. ; whereas the observations of Teisserenc de Bort at Trappes show a mean annual fall of about 16° C. for this rise (Encyc. Brit. vol. 30, Meteorology, p. 695). A continual blast of air thus cooled might of course reduce the temperature on the mountain peaks, even if radiation did not tend to any such reduction. But we can hardly account in this way for the equally well-established lower temperature of elevated continental plateaus. According to Abbe {loc. cit. p. 694) 0°-5 C. must be subtracted from sea-level temperature for every 100 metres general elevation of the land-surface or about 18° for an elevation of 3500 metres, and this fall may be ascribed to radiation in some such way as that here set forth. If the atmosphere of Mars is comparable with our own atmosphere at high levels, and if the effect is of the same general character in the two cases, it appears probable that the surface-temperature of Mars is actually lower by many degrees than that which the surface of the Earth would have at the same distance from the Sun. 24. THE MOMENTUM OF A BEAM OF LIGHT. [Atti del IV Congresso internazionale dei Matematici (Rome), 3, 1909, pp. 169-174.] [The substance of this paper is contained in the Address to the French Physical Society, March 1910. Collected Papers, Art. 70. Ed.] 25. ON PKESSURE PERPENDICULAR TO THE SHEAR-PLANES IN FINITE PURE SHEARS, AND ON THE LENGTHENING OF LOADED WIRES WHEN TWISTED. [Roy. Soc, Proc. A, 82, 1909, pp. 546-559.] [Read June 24, 1909.] In the Philosophical Magazine, vol. 9, 1905, p. 397*, I gave an analysis of the stresses in a pure shear which appeared to show that if e is the angle of shear and if n is the rigidity, then a pressure ne^ exists perpendicular to the planes of shear. That analysis is, I believe, faulty in that the diagonals of the rhombus into which a square is sheared are not the lines of greatest elongation and contraction, and are not at right angles after the shear, when second-order quantities are taken into account, i.e., quantities of the order of 6^ ; I think the following analysis is more correct, and though it does not give a definite result, it leaves the existence of a longitudinal pressure an open question. The question appears to be answered in the affirmative by some experiments, described in the second part of the paper, in which loaded wires when twisted were found to lengthen by a small amount proportional to the square of the twist. I. Stresses in a Pure Shear. Let a square ABCD (Fig. 1) of side a be sheared into EFCD by motion through AE = d, the volume being constant. The angle of shear is ADE = e, and tan e = dja exactly ; neglecting e^, we may put e = dja. To find which line is stretched most by the shear, consider the Kne r drawn from B to P and making an angle 6 with DC before stretching. Let it stretch to p, making an angle 6' with DC ; we have r = a /sin 6 and p = a/sin 6' ; also p2 ^ ^2 _|_ 2rd cos 6 -{- d^; thus /)2/r2 = 1 + 2dlr . cos ^ -h d^/r^ = 14- 2dla . sin ^ cos ^ -1- d^/a^ . sin^ 6. * [Collected Papers, Art. 22, p. 338.] ON PRESSURE PERPENDICULAR, TO THE SHEAR-PLANES 359 Differentiating p^jr^ with respect to 6, it is a maximum when Mja . cos 2d + d^la^ . sin 26 = 0, or tan 29 = — 2a/d = — 2 cot e. Put 6 = 45° + 8, then tan 28 = J tan e, or 8 = Je to the second order, so that r makes an angle J e with the diagonal DB of the square through D, and on the upper side. If the same shear is now made in the opposite direction p contracts to r, and the same directions of p before, and r after shear, give the maximum contraction. It is almost obvious that p makes an angle Je with DB on the lower side, but it may be verified by putting r^ = p^- 2pd cos d' + d^, and finding the maximum value of r^/p^ after putting p = a/ sin 6' on the right. J) C A J2a. ^ Fig. 1. Hence the lines of maximum elongation and contraction are at Je with the diagonals of the square, and are at right angles before and after the strain, to the order of e^. It is noteworthy that as the shear increases the fibres which undergo maximum elongation and contraction change. To find r and p put r = a/sin 6 = a/sin (45° + \e), then r - y/2a (1 - Jc + A^') ; and changing the sign of e we get p = V2a (1 + ie + i-^e^). It is easily seen that the elongation and contraction are respectively e=(p- r)lr = ^c (1 + Jc) ; c = {r - p)/p = Jc (1 - Je). We shall now consider the stresses. We shall assume that a pressure P is put on in the direction of maximum contraction and a tension Q in the direction of maximum elongation, these being, as we have seen, at right angles ; and we shall consider the equilibrium of the wedge ABC (Fig. 2) when 360 ON PRESSURE PERPENDICULAR TO THE SHEAR-PLANES sheared, assuming that P and Q are the only forces on AC and BC, Let AB = 2a ; AC = p\ BC = r; these having the values just found. Kesolve in a direction perpendicular to the base, and let R be the pressure against the base, then R.2a = Pp cos (45° - Je) - Qr cos (45° + Je) = Pa cot (45° -i€)-Qa cot (45° + Je) = Pa (1 + Je + le2) - Qa (1 - Jc + Je^) = (P - g) a + (P + Q) . ia€ + (P - g) . lac^, where P and Q can only be taken as equal to the first order. Proceeding to the second order, we must put P ^ 7l€ + fe^ ; then Q = ne - pe^, where p is a constant to the second order. Thus P-Q = 2pe^ and P + Q = 2ne, and P = (Jn + p) e^, the third term being negligible. If we resolve parallel to the base, it is easily found that the tangential stress is T = 1 (P + Q) = nc. If the shear is produced by a tangential stress T, then it requires the system P, Q, and R to maintain equilibrium with it. It is possible that a stress exists perpendicular to the plane of the figure in Fig. 1. It can only be assumed that the changes of dimension in that direction neutralise each other to the first order when equal pushes and pulls are put on in the plane of the figure ; when the dimensions perpendicular to the figure are constrained to remain the same to the second order — and this is our supposition — it may require a tension or pressure to effect this. Let us suppose that a pressure S = qe^ is introduced, a tension if q is negative. To make P = we should require to have f = — ^w, also P would then be less than Q. If pressure perpendicular to AC is exerted alone, and then tension perpendicular to BC is exerted alone, it appears probable that for very large equal compressions and extensions P is greater than Q. If we suppose that when they are simultaneous the tendency is in the same direction, then R should have a positive value, or the longitudinal pressure perpendicular to AB should exist. Let us examine the consequences of the supposition that both R and S exist. Let a thin tube of length I and of radius a be fixed at one end, and let IN FINITE PURE SHEARS, ETC. 361 the other end be twisted through an angle 6 so that the angle of shear is € = ad/l. Let an end-pressure R = (^n + p) e^ be put on, and also a side- pressure S = qe^ so as to maintain constant dimensions. The side-pressure S may be replaced by a uniform pressure S over the whole surface, and a tension S over the ends. We have then an end-pressure R — S and a pressure S all over. Now suppose that these forces are removed. Through the removal of R — S we shall have a lengthening dl^ given by (in + p- q) £2 = Ydljh and a contraction S^ of the diameter given by Sj2a = adljl, where Y is Young's modulus and o is Poisson's ratio. Through the removal of the pressure S we shall have a lengthening dlz given by qe^ = ^Kdljl, where K is the bulk-modulus, and an expansion §2 of the diameter given by qe^ = ZKh.J2a. The end-lengthening is therefore dl = dl^ + dl^ = {{in + p-q)IY+ q/SK} k^ ; or putting 1/3Z = 3/7 - 1/n, dl = {(Jn + p)IY+ (2/F - l/n) q} k^ = sU^ = sa^d% where s is put for (Jn + p)/7 + {2/Y — l/n) 5'. The diameter decreases by S = Sj _ §2 = {{in +p-q)alY- q/SK} 2a€^ = {{in + p)cjIY- [(3 + c7)/r - l/n] q} 2a^6^ll\ It would not be easy to test this result with a thin tube. But if we suppose that a wire extends by the amount equal to the average extension of the tubes into which it may be resolved, we get dl=—J ^"""^dr = isa^e% I now proceed to describe some experiments which show that such an extension exists. II. The Lengthening of Loaded Wires when twisted. Experiments were made on several wires hung vertically from a fixed support, and loaded in order that kinks or remnants of the spiral due to the coihng to which they had been subjected might be taken out. This was considered to be effected when the stretch for a given addition of load was 362 ON PRESSURE PERPENDICULAR TO THE SHEAR-PLANES sensibly the same whether the wire was twisted or not. An account of the twisting of a steel wire before this stage was reached will be given later. Fig. 3 represents the arrangement more or less diagrammatically. The upper end of the wire to be twisted was fixed to a stout bracket B near the ceiUng. The wire was always about 231 cm. long ; its lower end was clamped in jaws in the upper end of a turned steel rod rr, 51 cm. long, which passed through a hole in the table T on which was the observing microscope M, and a parallel- plate micrometer m. One division of this micrometer was equal to 0-00974 mm. At the lower end of the rod was a horizontal iron cross-piece cc, 19 cm. long and 1-6 cm. square. From the lower end was suspended a carrier for the weights, or for the two stouter wires the weight itself, connected to the rod by a flat steel strip twisted in its middle, so that the upper and lower halves were in two vertical planes at right angles. Below the weights was a set of vanes immersed in a shallow bath of oil to damp vibrations. This bath rested on a circular turntable tt, on which were two uprights ii, ii at opposite ends of a diameter, with horizontal screws at their upper ends which could be brought to bear against the ends of the cross-piece as shown in the plan, Fig. 4. The screws ended in small steel balls and the sides of the cross-piece were polished. On rotating the turntable the screws came against the cross-piece and turned it round ; and so the wire was twisted by a couple with vertical axis. The axis of the turntable was made vertical by means of the levelling- screws I, I. To adjust this axis in the axis of the wire prolonged, the turntable could be moved over the base-plate by means of the horizontal screws s, of which only one is represented in Fig. 3. All are shown in Fig. 4. A horizontal microscope, not represented in the figure, was attached to one of the uprights and focussed on the edge of the rod rr. The adjustment by the screws s was continued until the microscope always saw the edge of the rod in the middle of the field, however the turntable might be turned. Fig. 3. Elevation of Arrange- ment for Twisting the Wires. IN FINITE PUEE SHEARS, ETC. 363 To give a definite point of view in the microscope M, in the earlier experi- ments starch-grains were put on the wire about 1 cm. from the lower end. These were illuminated, and a suitable one was selected. In the later experiments a needle, about 1 cm. long, was fixed, point upwards, on the upper end of the rod close alongside the wire, and the needle- point was viewed. This was better than the starch-grains. In the earlier work the temperature of the room was fairly steady, and the changes in length due to temperature- variations were too slow to give trouble. But in some gusty weather occurring later there were such rapid and considerable variations in the temperature of the room that it was necessary to enclose the wire in a wooden tube. After this was done tempera- ture gave no further trouble, whatever the weather. In order to observe the effect of a twist the turntable was levelled and adjusted axially when the wire and cross-piece were free. The turntable was rotated till the screws on the uprights just touched the cross-piece. Then chalk-marks were made on the turntable and on the plate below, one just over the other. The microscope was adjusted exactly to sight the upper or lower edge of a starch- grain on its horizontal cross- wire, and the micrometer was read. Then the turntable was rotated so many whole turns, and the micrometer-plate was moved till the edge of the grain was again on the cross-wire and the micrometer was read again. Except in the case of a wire stretched only by the weight of the rod and cross-piece, in some experiments described later, there was always a lengthening on twisting, of the same order whether the twist was clockwise or counter- clockwise. The lengthening was nearly proportional to the square of the twist put on. It was necessary to limit the twist to a few turns to avoid permanent set, and when such a small twist had been given and the wire was untwisted it returned sensibly to its original length. The lowering was entirely due to twisting and not to any giving of the support, for when a microscope was sighted on a point on the wire close to the upper end, no change in level could be detected, when the wire was twisted through 5. turns at its lower end. This was further verified by an experiment on a steel wire from the same piece as No. 3 below, which showed that the extension half-way down the wire was, within the limits of experimental error, half that at the lower end. A microscope and micro- meter were fixed on a table half-way up the wire, and a needle-point was fixed here as well as at the lower end. At each twist and untwist both micrometers were read. I give the observations in this experiment in full, as they will show the sort of accuracy attained. Fig. 4. Plan of the Cross- piece and Turntable. 364 ON PRESSURE PERPENDICULAR TO THE SHEAR-PIANES The lower end was twisted from a starting twist of J turn to 4 J turns. Micrometer-Readings at Lower End. J turn 4J turns Lowering 22-3 18-6 3-7 22-5 19-0 3-5 23-0 19-2 3-8 22-6 19-4 3-2 22-9 19-6 3-3 22-6 19-5 31 23-0 19-5 3-5 23-0 18-9 41 22-7 19-2 3-5 22-9 19-6 3-3 Mean lowering, 3-50 divisions. One division of micrometer = 0-00974 mm. The lowering is 0-0341 mm. Micrometer-Readings Half-way up the Wire. ^ turn 4J turns Lowering 30-4 28-3 2-1 30-5 28-4 2-1 30-2 28- 1 2-1 30-5 28-5 20 30-5 28-0 2-5 30-4 28-7 1-7 310 28-4 2-6 30-6 28-5 21 31-8* 29-9 1-9 31-6* 29-lt 2-5 Mean lowering, 2-16 divisions. One division of micrometer = 0-00751 mm. The lowering is 0-0162 mm. If the lowering at the end is accurate, that half-way up should be 0-0171 mm. The observed lowering is as nearly equal to this as could be expected. With the first wire, determinations of extension due to an addition of 520 grammes were made both in the untwisted and twisted conditions, as it was only when these became sensibly equal that the lowering on twist became equal for different loads. The extra load could be put on or taken ofi by lowering or raising a lever, not represented in Fig. 3. It is unnecessary * Another point on the needle sighted, ■f [This number has been corrected from 29-9 in accordance with the original MS. other small shps in the results which follow have been similarly corrected. Ed.] A few IN FINITE PURE SHEARS, ETC. 365 to describe the details of this arrangement. The experiments with the other wires were made with such loads that it was not considered necessary to observe the stretch due to addition of load. Results. la. Steel piano- wire, diameter 0-720 mm. (mean of 10 measurements at different points), length to observing point in this and all cases 230 cm. Permanent set, after putting on eight turns twist and then untwisting, only a very few degrees. Total load, 7081 grammes. The twist is termed clockwise when the turntable as viewed from above is moved clockwise. Clockwise twist, 0-4 turns ; lowering 0-0181 mm., mean of 10 observations. „ 0-8 „ „ 0-0732 The ratio of these is 4-04 : 1. The extension due to an addition of 520 grammes was : No twist on the wire 0-143 mm., mean of 10 observations. 4 turns „ 0-141 „ „ „ 8 ,, ,, u-14o „ ,, ,, lb. Same wire. Total load, 9081 grammes. Clockwise twist, 0-4 turns; lowering 0-0180 mm., mean of 20 observations. „ 0-8 „ „ 0-0749 The ratio of these is 4-15 : 1. The extension due to an addition of 520 grammes was : No twist on the wire 0-142 mm., mean of 10 observations. 8 turns „ 0-144 „ „ „ Taking the mean lowering for the two loads of 7081 and 9081 granmies for eight turns twist, viz., 0-074 mm., and taking it as proportional to the square of the twist, the lowering for one turn is 0-00116 mm., and s = 2ldllaW^ = 1-043. The moduli of elasticity of this wire were found to be n = 0-769 X 1012, y _ 2-013 x 10^2, whence n/Y = 0-382. The value of n, found for loads of 1081 grammes and 9081 grammes respectively, was identical. 2. The same wire was raised to a red heat, by an electric current, with the load of 9081 grammes on it. It lengthened about 3 cm., and this length was cut off. The surface oxidised, and when the oxide was rubbed off the diameter was 0-696 mm. (mean of 10 measurements). 366 ON PRESSURE PERPENDICULAR TO THE SHEAR-PLANES The permanent set after twisting and untwisting was greater, and so only- three turns were given. Total load, 9081 grammes. Clockwise twist, 0-3 turns ; lowering 0-0129 mm., mean of five observations. The extension due to an addition of 520 grammes was : No twist on the wire 0-155 mm., mean of 10 observations. 3 turns „ 0-154 „ „ „ The lowering for one turn according to the square-law is : 0-00143 mm., whence 5 = 1-376 mm. The five values of the lowering were : 1-5, 1-2, 1-2, 1-2, 1-5 divisions, mean 1-32 divisions. With such small lowering no accuracy could be expected, and it would be difficult to verify the square-law. The moduli of elasticity for the softened wire were : n = 0-809 X 1012 and Y = 2-06 x IO12, whence n/ 7 = 0-393. 3. Steel piano- wire, diameter 0-970 mm. (mean of 10 measurements). A needle-point fixed at the side of the wire was viewed in the microscope. After twisting and untwisting, a slight permanent set threw the point out of focus if the start was from no twist. A quarter- turn was therefore put on initially, and the twisting was from this, and the untwisting was back to it. Total load, 19,504 grammes. Clockwise twist, J-2J turns; lowering 0-0088 mm., mean of 10 observations. 55 55 4~^4 '5 " 0-0343 ,, ,, ,, Counter-clockwise twist, ^-2^ turns; lowering 0-0090 mm., mean of 10 obs. 1-41 0-0844 Mean lowering, J-2J turns, 0-0089 mm. 1-41 turns, 0-0344 „ By the square-law the lowerings for 4|, 2J, and J should be as 289 : 81 : 1, and the difference should be as 288 : 80 = 18 : 5. The observed differences are as 19-4 : 5. The lowering for one turn deduced from the difference between J and 4J is 0-00191 mm., whence s = 0-946. Comparing the lowerings for one turn of this wire with the hard wire No. 1, if the lowering is proportional to the square of the diameter, we ought to have for No. 1 a lowering of 0-00191 x (72/97)2 = 0-00105 mm. The observed lowering was 0-00116 mm., which is as near the calculated value as could be expected. EST FINITE PURE SHEARS, ETC. 367 4. The same wire was then raised to a red heat by an electric current with the load on. After being rubbed down its diameter was 0-947 mm. (mean of 10 measurements). Same load as in experiment 3. Clockwise twist, J-3J turns, lowering 0-0207 mm., mean of 10 observations. The deduced lowering for one turn is 0-00197 mm. The value of s is 1-025 mm. Comparing the lowerings for one turn of this wire with the softened wire No. 2, the square-law for the diameter should give for No. 2 a lowering 0-00197 X (696/947)2 = 0-00106 mm. The observed lowering was 0-00143 mm., a considerable divergence. 6. Copper wire, diameter 0-655 mm. (mean of 10 measurements). Load, 7081 grammes. Clockwise twist, J-2J turns; lowering 0-0066 mm., mean of 10 observations. Counter-clockwise twist, ^-2^ turns ; lowering 0-0083 mm., mean of 10 obs. It was not safe to give a greater twist owing to the largeness of the permanent set. With 2^ turns the set was still small. The larger value of the lowering for the counter-clockwise twist is almost certainly real, and not merely error of observation. Some other observations showed an even greater excess, though they were very irregular owing to temperature- variations, and are not worth recording. The extension due to an addition of 520 grammes was : No twist on the wire 0-268 mm., mean of 20 observations. 3 turns „ 0-269 ,, „ ,, Taking the mean for clockwise and counter-clockwise twist, the lowering for one turn is 0-00149 mm., and s = 1-62 mm. 6. Brass wire, diameter 0-928 mm. (mean of 10 measurements). Load 19,504 grammes. Clockwise twist, i-2J turns; lowering 0-0169 mm., mean of 10 observations. 1-41 0-0540 Counter-clockwise twist, j-2J turns; lowering 0-0135 mm., mean of 10 obs. „ i-41 „ „ 0-0479 The difference between clockwise and counter-clockwise twisting is too large for errors of observation. For the square-law the lowerings for J-4J and for J-2 J turns should be in the ratio 18 : 5. They are in the ratios 16 : 5 for clockwise, and 17-7 : 5 for counter- clockwise twisting. The lowering for one turn clockwise, as deduced from i-4J turns, is 0-00300 mm., and for one turn counter-clockwise is 0-00265 mm. The mean value of s = 1-537 mm. 368 ON PRESSURE PERPENDICULAR TO THE SHEAR-PLANES Experiments with Smaller Loads. When the piano- wire diameter 0-72 mm. was loaded only with the rod and cross-bar weighing 1081 grammes, there was a rise on twisting. Clockwise twist, 0-4 turns, rise 0-041 mm. 0-8 „ 0-139 „ Counter-clockwise twist, 0-4 „ 0-023 „ „ 0-8 „ 0-108 „ The extension for an addition of 520 grammes was : No twist on wire 0-137 mm.. mean of 6 observations. 4 turns clockwise 0-170 jj 55 6 55 8 „ „ 0-237 ?j 55 3 55 4 turns counter-clockwise 0-156 5J 55 6 55 8 „ .. ,. 0-238 55 55 3 55 If by means of the observed extensions we calculate the positions of the point viewed, when the load of 1081 grammes is taken off, we find that the total rise for clockwise twist would be: for four turns 0-110 mm., and for eight turns 0-347 mm. The rise appears to be due to coiling up of the wire on twisting, through some remnant of the spiral condition in which it existed before suspension. This is confirmed by the very large increase in extension, due to addition of load as the twist on the wire is increased. It may be a coincidence that the rise on twisting and the increase of stretch are both nearly proportional to the square of the number of turns. Experiments were then made with greater loads to find how the lowering and extension changed. Only clockwise twist was observed. Load 3081 grammes, the rise changed to lowering. Twist, 0-4 turns; lowering 0-0131 mm., mean of 20 observations. 55 0-8 „ „ 0-0498 The extension due to an addition of 520 grammes was : No twist on the wire 0-144 mm., mean of 10 observations. 4 turns „ 0-143 „ „ „ ^ 55 55 U-l-iy ,, ,, ,, Showing a still slight excess of extension in the most twisted condition. Load 5081 grammes. Twist, 0-4 turns; lowering 0-0164 mm., mean of 20 observations. „ 0-8 „ „ 0-0660 m FINITE PURE SHEARS, ETC. 369 The extension due to an addition of 520 grammes was : No twist on the wire 0-141 mm., mean of 10 observations. 4 turns „ 0-142 „ „ 15 8 „ „ 0-144 „ „ 15 The results for loads of 7081 and 9081 grammes are already recorded under 1 a and 1 h. There is obviously a tendency for the lowering to increase with load until the extensions under different twists become more nearly equal with equal added load. When the same wire was softened and loaded with 3081 grammes the lowering for three turns was 0-0093 mm. (mean of 10 observations). The extension due to an addition of 520 grammes was : No twist on the wire 0-149 mm., mean of 10 observations. 3 turns „ 0-147 With load 9081 grammes the same wire gave the results recorded under 2, which show a greater lowering for an equal twist but the same extension with added load. " The copper wire diameter 0-655 mm. (No. 5 above) with load 4081 grammes gave: Clockwise twist, 0-3 turns, 0-00965 mm., mean of 10 observations. Counter-clockwise twist, 0-3 turns, 0-0156 mm., mean of 10 observations. Taking the mean of these, the lowering for one turn is 0-0014 mm. The extension due to an addition of 520 grammes was : No twist on the wire 0-268 mm., mean of 5 observations; 3 turns „ 0-270 extensions agreeing very nearly with those recorded above for a load of 7081 grammes on the same wire. Remarks on the Results of Measurements. The lowering was never so much as 0-1 mm. and was usually much less. The accuracy attained could hardly be expected to be great. The measure- ments, however, appear to show that when a wire is sufficiently loaded to be straightened, it is lengthened by twisting by an amount proportional to the square of the twist and, with a given number of turns, inversely as the length. It might be thought possible that the effect observed was due to rise of temperature, either through adiabatic strain or through dissipation of strain- energy as heat. But the observations give no support to this explanation. When the wire was extended by twisting, it remained extended, and when untwisted it returned. Temperature-effects would be a maximum the instant after twisting, and would then gradually subside. It may be noted that p. c.w. 24 370 ON PRESSURE PERPENDICULAR TO THE SHEAR-PLANES the adiabatic change of temperature is proportional to a^B^jl, but it is a cooling, and its amount is such as to shorten the wire, in the case of steel, by something of the order of 1/100 of the observed extension. If we suppose that some definite fraction of the strain-energy put in is dissipated, again the change, now a warming, is proportional to a^O^jl. The whole strain-energy, in the case of steel, would only raise the temperature by an amount accounting for something of the order of 1/10 the observed extension, and, in fact, only an exceedingly minute fraction of the strain-energy is dissipated. A comparison of the wires (1) and (3) appears to show that the lengthening for a given twist is proportional to the square of the radius. If we put the lengthening s for steel is in the neighbourhood of 1. For copper and brass s is in the neighbourhood of 1-5. The lowering for the copper and brass wires tested for twists in opposite directions is not the same. With a hard steel wire with small load the end of the wire rises on twisting, probably through coiling. The value of s = (\n + p)/Y — (21 Y — Ijn) q appears to be measurable, but its value gives us no clue to the values of f and q. If we could assume q = 0, then for steel we should have f about 2n, but I see no justification for the assumption. If we could measure the decrease in diameter, we should obtain the value of (\n + f — q) (jjY — {(?> + g)IY — Ijn) q, and knowing n, Y and a we should be able to find p and q. But a thin wire is quite unsuitable for this measurement. The decrease is probably of the order of 2a/? x lengthening. With the wires I have used this is of the order 1/1000 x lengthening, and an accuracy of measurement of 10"^ mm. would be required at least. With a shaft of considerable diameter it might be possible to measure the quantity, though the experimental difficulties are obviously very great*. The Effect of the Lengthening of a Wire on its Torsional Vibration. If a wire is loaded with mass M having moment of inertia /, when M is set vibrating torsionally it falls and rises as it swings, its distance below the highest point being given by X = isaW/l. The kinetic energy is T = ^16^ + iMx\ The last term is easily found to be negligible. The potential energy is F = Inira^e'^ll - Mgx = Inira'^d^ll - iMgsaW% * [For the experimental carrying out of these measurements see Collected Papers, Art. 30. Ed. ] IN FINITE PURE SHEARS, ETC. 371 The equation of motion is • 10 + (JwTraV^ - Mgsa^/l) d = 0. Whence T^ - ^^^^^ ~n7Ta^{l-2Mgsln7Ta^)' and T is greater than it would be if s were 0, by the factor 1 + Mgslmra^. If Y is Young's modulus and if e is the elongation of the wire due to the load Mg, Mg/ira^ = Ye, so that the factor may be conveniently written as 1 + seY/n. If the vibrations are used to determine the modulus of rigidity n, then the value of n will be greater than that deduced by neglect of s, and by the factor 1 + 2seY/n, To give an idea of the effect on the determination of the modulus of rigidity, let us suppose that a quite straight steel wire, diameter 0-7 mm., has a load of 2000 grammes. For steel Y/n is about 2-6. For the given diameter e is about 2 x 10"*. We have found that s is about 1. The correcting factor is then about 1-001, or the true rigidity exceeds the value calculated in the ordinary way by about 1 in 1000. If the wire is not sufficiently loaded to be straight the value of s is less. If very lightly loaded the sign of s may be changed and the true rigidity may be less than the value as ordinarily calculated. The correction is hardly needful in practice, as the modulus of rigidity is probably not measurable to three figures. Distortional Waves. In purely distortional waves in a medium of great extent it is evident that the pressure S perpendicular to the axes of shear, if it exists, will not produce any motion. To keep the waves purely distortional, i.e. with motion perpendicular to the direction of propagation only, a force must be applied from outside dRjdx per cubic centimetre in the direction of propagation. If this force is not applied then longitudinal motion must result, obviously of the second order, unless Jn + p = 0. This is probably the condition for an incompressible medium. If \n + p is not zero it appears possible that dispersion may exist. If the longitudinal motion is neglected the pressure in the direction of propagation is (|n + f) e^ and all that we can say, at present, is that it is probably of the order of ne^. 24—2 2t). THE WAVE-MOTION OF A EEVOLVING SHAFT, AND A SUG- GESTION AS TO THE ANGULAR MOMENTUM IN A BEAM OF CmCULAELY POLARISED LIGHT. [Roy. Soc. Proc. A, 82, 1909, pp. 560-567.] [Read June 24, 1909.] When a shaft of circular section is revolving uniformly, and is transmitting power uniformly, a row of particles originally in a line parallel to the axis will lie in a spiral of constant pitch, and the position of the shaft at any instant may be described by the position of this spiral. Let us suppose that the power is transmitted from left to right, and that as viewed from the left the revolution is clockwise. Then the spiral is a left-handed screw. Let it be on the surface, and there make an angle e with the axis. Let the radius of the shaft be a, and let one turn of the spiral have length A along the axis. We may term A the wave-length of the spiral. We have tan e = IrrajX. If the orientation of the section at the origin at time t is given by ^ = 27rNt, where N is the number of revolutions per second, the orientation of the section at x is given by e = 'lirNt - - tan e = ^ (iVA^ - x), (1) a A which means movement of orientation from left to right with velocity iVA. The equation of motion for twist-waves on a shaft of circular section is ^^^U^^ (2) where JJ^^ = modulus of rigidity/density = njp. Though (1) satisfies (2), it can hardly be termed a solution, for d^6/dt^, and dW/dx^ in (2) are both zero. But we may adapt a solution of (2) to fit (1) if we assume certain conditions in (1). The periodic value e = esm^{Uj-x) satisfies (2), and is a wave-motion with velocity Un and wave-length I. Make WAVE-MOTION OF A REVOLVING SHAFT 373 t SO great that for any time or for any distance under observation Unt/l and x/l are so small that the angle may be put for the sine. Then e=9^{Uj-x) (3) This is uniform rotation. It means that we only deal with the part of the wave near a node, and that we make the wave-length I so great that for a long distance the 'displacement-curve' obtained by plotting 6 against t coincides with the tangent at the node. We must distinguish, of course, between the wave-length I of the periodic motion and the wave-length A of the spiral. We can only make (1) coincide with (3) by putting 0/?=l/A and NX= C/„. Then it follows that for a given value of N, the impressed speed of uniform rotation, there is only one value of A or one value of € for which the motion may be regarded as part of a natural wave-system, transmitted by the elastic forces of the material with velocity = \/(nlp). There is therefore only one 'natural' rate of transmission of energy. The value of e is given by tan e = 27Ta/X = 27TaNINX = 27raiV/t/'„ = 27TaN^{pln). Suppose, for instance, that a steel shaft with radius a = 2 cm., density /) = 7-8, and rigidity n = 10^^ ig making iV = 10 revs, per sec. We may put tan € = €, since it is very small. The shaft is twisted through 2?? in length A or through 27r/A per centimetre, and the torque across a section is smce A- U^ _ 1 In The energy transmitted per second is 27rNG ^ 27T^a^NWM- Putting 1 H.p. = 746 x 10^ ergs per second, this gives about 38 h.p. But a shaft revolving with given speed A^ can transmit any power, subject to the limitation that the strain is not too great for the material. When the power is not that 'naturally' transmitted, we must regard the waves as 'forced.' The velocity of transmission is no longer j[7„, and forces will have to be applied from outside in addition to the internal elastic forces to give the new velocity. Let H be the couple applied per unit length from outside. Then the equation of motion becomes ^ _ p 2 ^ ^ 2^ dt^ ' "" dx^ TraV 374 WAVE-MOTION OF A REVOLVING SHAFT, AND A SUGGESTION AS TO where ^Tra* is the moment of inertia of the cross-section. Assuming that the condition travels on with velocity TJ unchanged in form, or H has only to be applied where dW/dx^ has value, that is where the twist is changing. The following adaptation of Rankine's tube-method of obtaining wave- velocities* gives these results in a more direct manner. Suppose that the shaft is indefinitely extended both ways. Any twist- disturbance may be propagated unchanged in form with any velocity we choose to assign, if we apply from outside the distribution of torque which, added to the torque due to strain, will make the change in twist required by the given wave-motion travelling at the assigned speed. Let the velocity of propagation be U from left to right, and let the dis- placement at any section be 6, positive if clockwise when seen from the left. The twist per unit length is dd_ I d_d^ I dx" Udt~' U' The torque across a section from left to right in clockwise direction is 1 ^ dd mra'^ X Let the shaft be moved from right to left with velocity U ; then the dis- turbance is fixed in space, and if we imagine two fixed planes drawn perpendicular to the axis, one, ^, at a point where the disturbance is 6 and the other, B, outside the wave-system, where there is no disturbance, the con- dition between A and B remains constant, except that the matter undergoing that condition is changing. Hence the total angular momentum between A and B is constant. But no angular momentum enters at B, since the shaft is there untwisted and has merely linear motion. At A, then, there must be on the whole no transfer of angular momentum from right to left. Now, angular momentum is transferred in three ways : 1. By the carriage by rotating matter. The angular momentum per unit length is ^p-na'^O, and since length V per second passes out at A, it carries out IpTra^dU. 2. By the torque exerted by matter on the right of A on matter on the left of A. This takes out - nTra^d/^U. 3. By the stream of angular momentum by which we may represent the forces applied from outside to make the velocity U instead of Un . * Phil. Trans. 1870, p 277. ANGULAR MOMENTUM IN A BEAM OF CIRCULARLY POLARISED LIGHT 375 If H is the couple applied per unit length, we may regard it as due to the flow of angular momentum L along the shaft from left to right, such that H = — dL/dx. There is then angular momentum L flowing out per second from right to left. Since the total flow due to (1), (2), and (3) is zero, ifma*dU - n7ra^e/2U - L = 0, and X = '?(.^-^) = '^(^-"^„^)=-^|(^-.A li H = 0, either U^ = C7„2 when the velocity has its 'natural value,' or d^d/dx^ = 0, and the shaft is revolving with uniform twist in the part con- sidered. Now put on to the system a velocity U from left to right. The motion of the shaft parallel to its axis is reduced to zero, and the disturbance and the system H will travel on from left to right with velocity U. A 'forced' velocity does not imply transfer of physical conditions by the material with that velocity. We can only regard the conditions as reproduced at successive points by the aid of external forces. We may illustrate this point by con- sidering the incidence of a wave against a surface. If the angle of incidence is i and the velocity of the wave is F, the line of contact moves over the surface with velocity v = 7/sin i, which may have any value from F to infinity. The velocity v is not that of transmission by the material of the surface, but merely the velocity of a condition impressed on the surface from outside. Probably in all cases of transmission with forced velocity, and certainly in the case here considered, the velocity depends upon the wave-length, and there is dispersion. With a shaft revolving N times per second U = NX, and it is interesting to note that the group- velocity, U — XdU/dX, is zero. It is not at once evident what the group-velocity signifies in the case of uniform rotation. In ordinary cases it is the velocity of travel of the 'beat' pattern, formed by two trains of slightly different frequencies. The complete 'beat' pattern is contained between two successive points of agreement of phase of the two trains. In our case of superposition of two strain-spirals with constant speed of rotation, points of agreement of phase are pomts oi intersection of the two spirals. At such points the phases are the same, or one has gained on the other by 27T. Evidently as the shaft revolves these points remain in the same cross- section, and the group- velocity is zero. With deep-water waves the group- velocity is half the wave- velocity, and the energy-flow is half that required for the onward march of the waves*. * 0. Reynolds, Nature, August 23, 1877 ; Lord Rayleigh, Theory of Sound, vol. 1, p. 477. 376 WAVE-MOTION OF A REVOLVING SHAFT, AND A SUGGESTION AS TO The energy-flow thus suffices for the onward march of the group, and the case suggests a simple relation between energy-flow and group- velocity. But the simpHcity is special to unforced trains of waves. Obviously, it does not hold when there are auxiliary working forces adding or subtracting energy along the waves. For the revolving shaft the simple relation would give us no energy-flow, whereas the strain existing in the shaft implies trans- mission of energy at a rate given as follows. The twist per unit length is dd/dx, and therefore the torque across a section is — ^rnra^ dd/dx, or ^nTra^d/U, since dO/dx = — 6/U. The rate of working or of energy-flow across the section is ^mra^d^lU. The relation of this to the strain and kinetic energy in the shaft is easily found. The strain- energy per unit length being J (couple x twist per unit length) is ^rnra^ (dO/dx)'^, which is ^mra'^b^lU^. The kinetic energy per unit length is Ipna^O^, or, putting p = n/Un^, is l^iTra^d^/UJ. In the case of natural velocity, for which no working forces along the shaft are needed, when U = Un = s/in/p), the kinetic energy is equal to the strain- energy at every point and the energy transmitted across a section per second is that contained in length [/"„ . But if the velocity is forced this is no longer true*, and it is easily shown 2U that the energy transferred is that in length fj^Tfj—2> which is less than U a U > Un, and is greater than U ii U < 17^. It appears possible that always the energy is transmitted along the shaft at the speed Un • If the forced velocity U > 11^, we may, perhaps, regard the system in a special sense as a natural system with a uniform rotation super- posed on it. Let us suppose that the whole of the strain- energy in length U^ is trans- ferred per second while only the fraction ijl of the kinetic energy is transferred, the fraction 1 — /x being stationary. The energy transferred : strain-energy in U„ : kinetic energy in 11^ = IIU:VJW^:UJ2U„\ Put U = fU.n , and our supposition gives pu: 2im,jiu„ " /" /* p-^ V p) ■ If the forced velocity U < 11^, we may regard the system as a natural one, with a uniform stationary strain superposed on it. * In the Sellmeier model illustrating the dispersion of light, the particles may be regarded as outside the material transmitting the waves and as applying forces to the material which make the velocity forced. The simple relation between energy-flow and group- velocity probably does not hold for this model. ANGULAR MOMENTUM IN A BEAM OF CIRCULARLY POLARISED LIGHT 377 We now suppose that the whole of the kinetic energy is transferred, but only a fraction v of the strain-energy, and we obtain It is perhaps worthy of note that a uniform longitudinal flow of fluid may be conceived as a case of wave-motion in a manner similar to that of the uniform rotation of a shaft. A Suggestion as to the Angular Momentum in a Beam of Circularly Polarised Light. A uniformly revolving shaft serves as a mechanical model of a beam of circularly polarised light. The expression for the orientation 6 of any section of the shaft distant x from the origin, d = 27t\-'^ (Ut — x), serves also as an expression for the orientation of the disturbance, whatever its nature, constituting circularly polarised light. For simpUcity, take a shaft consisting of a thin cylindrical tube. Let the radius be a, the cross-section of the material s, the rigidity n, and the density p. Let the tube make N revolutions per second, and let it have such twist on it that the velocity of transmission of the spiral indicating the twist is the natural velocity 11^ = \^{n/p). Repeating for this special case what we have found above, the strain- energy per unit length is ^neh, or, since e = adO/dx = — aO/Un, the strain- energy is Inahe^jTIr? = Ipahd^. But the kinetic energy per unit length is also ^pahd^, so that the total energy in length Z7„ is pahO^Un. The rate of working across a section is nesaO = na^sO^/Ur, = pa^sd^U^, or the energy transferred across a section is the energy contained in length ?/„ . If we put E for the energy in unit volume and G for the torque per unit area, we have GsO=EsUn, whence G = EUJO = ENXjIirN = EXj^tt. The analogy between circularly polarised light and the mechanical model suggests that a similar relation between torque and energy may hold in a beam of such light incident normally on an absorbing surface. If so, a beam of wave-length A containing energy E per unit volume will give up angular momentum ^A/27r per second per unit area. But in the case of light- waves E = P, where P is the pressure exerted. We may therefore put the angular momentum delivered to unit area per second as PA/27r. 378 WAVE-MOTION OF A REVOLVING SHAFT, AND A SUGGESTION AS TO In the Philosophical Magazine, 1905, vol. 9, p. 397*, I attempted to show that the analogy between distortional waves and light-waves is still closer, in that distortional waves also exert a pressure equal to the energy per unit volume. But as I have shown in a paper on ' Pressure Perpendicular to the Shear-Planes in Finite Pure Shears, etc.f,' the attempt was faulty, and a more correct treatment of the subject only shows that there is probably a pressure. We cannot say more as to its magnitude than that if it exists it is of the order of the energy per unit volume. When a beam is travelling through a material medium we may, perhaps, account for the angular momentum in it by the following considerations. On the electromagnetic theory the disturbance at any given point in a circularly polarised beam is a constant electric strain or displacement / uniformly revolving with angular velocity 6. In time dt it changes its direction by dd. This may be effected by the addition of a tangential strain /c?^; or the rotation is produced by the addition of tangential strain fO per second, or by a current fO along the circle described by the end of /. We may imagine that this is due to electrons drawn out from their position of equilibrium so as to give/, and then whirled round in a circle so as to give a circular con- vection-current/^. Such a circular current of electrons should possess angular momentum. Let us digress for a moment to consider an ordinary conduction-circuit as illustrating the possession of angular momentum on this theory. Let the circuit have radius a and cross-section s, and let there be N negative electrons per unit volume, each with charge e and mass m, and let these be moving round the circuit with velocity v. If i is the total current, i = Nsve. The angular momentum will be Ns27ra . mva = ^iraHmje = 2Aim/e = 2Mm/e. where A is the area of the circuit and M is the magnetic moment. This is of the order of 2M/10'. It is easily seen that this result will hold for any circuit, whatever its form, if A is the projection of the circuit on a plane perpendicular to the axis round which the moment is taken, and if M = Ai. If we suppose that a current of negative electrons flows round the circuit in this way and that the reaction while their momentum is being established is on the material of the conductor, then at make of current there should be an impulse on the conductor of moment 2M/W. If the circuit could be suspended so that it lay in a horizontal plane and was able to turn about a vertical axis in a space free from any magnetic field, we might be able to detect such impulse if it exists. But it is practically impossible to get a space free from magnetic intensity. If the field is H, the couple on the circuit due to it is proportional to HM. It would require exceedingly careful construction and adjustment * [Collected Papers, Art. 22, p. 338.] j [Collected Papers, Art. 25.] ANGULAR MOMENTUM IN A BEAM OF CIRCULARLY POLARISED LIGHT 379 •Pi ■Pi [of the circuit to ensure that about the vertical axis the component of the jouple due to the j&eld was so small that its effect should not mask the effect Fof the impulsive couple. The electrostatic forces, too, might have to be [considered as serious disturbers. Returning to a beam of circularly polarised light, supposed to contain electrons revolving in circular orbits in fixed periodic times, the relations between energy and angular momentum are exactly the same as those in a revolving shaft or tube, and the angular momentum transmitted per second per square centimetre is EXJ^tt = PXj2tt, where P is the pressure of the light per square centimetre on an absorbing surface. The value of this in any practical case is very small. In light-pressure experiments, P is detected by the couple on a small disc, of area A say, at an arm h and suspended by a fibre. What we observe is the moment APh. If the same disc is suspended by a vertical fibre attached at its centre and the same beam circularly polarised in both cases is incident normally upon it. according to the value suggested the torque is APXI27T. The ratio of the two is A/27r6. Now b is usually of the order of 1 cm. Put A = 6 x 10~^ cm., or, say, 27r/10-^, and the ratio becomes 10~^. . It is by no means easy to measure the torque APb accurately, and it appears almost hopeless to detect one of a hundred- thousandth of the amount. The effect of the smaller torque might be multiplied to some extent, as shown in accompanying diagram. Let a series of quarter wave plates, Pi, ^2' Vs^ ..., be suspended by a fibre above a Nicol prism N, through which a beam of light is transmitted upwards, and intermediate between these let a series of quarter wave plates, g'l, g'2^ ?3> •••> be fixed, each with a central hole for the free passage of the fibre. The beam emerges from N plane polarised. If N is placed so that the beam after passing through 2?i is circularly polarised, it has gained angular momentum, and there- fore tends to twist p^ round. The next plate q^^ is to be arranged so that the beam emerges from it plane polarised and in the original plane. It then passes through ^2 > which is similar to pi , and again it is circularly polarised and so exercises another torque. The process is repeated with q^ and f^, and so on till the beam is exhausted. By revolving iV through a right angle round the beam, the effect is reversed. But, even with such multiplications, my present experience of light-forces does not give me much hope that the effect could be detected, if it has the value suggested by the mechanical model. 27. PKELIMINARY NOTE ON THE PRESSURE OF RADIATION AGAINST THE SOURCE: THE RECOIL FROM LIGHT. By J. H. PoYNTiNG and Guy Barlow, D.Sc. [British Association Report, 1909.] [This is merely a preliminary account of the following paper Art. 28.] • 28. THE PEESSUKE OF LIGHT AGAINST THE SOURCE: THE RECOIL FROM LIGHT. BAKERIAN LECTURE. By J. H. PoYNTiNG and Guy Barlow, D.Sc. [Roy. Soc. Proc. A, 83, 1910, pp. 534-546.] [Read March 17, 1910.] All experiments on the pressure of radiation have hitherto been made on the force exerted by light or radiation on a receiving surface. The experiment now to be described shows the pressure of radiation against the source from which it starts, and from analogy with a gun we may term this the recoil from hght. It does not appear practicable to show this effect by using a source in which heat is developed intrinsically. But if radiation falls on an absorbing body it heats the body and the heat so developed issues again as radiation, and it is possible to detect the efEect of this issuing radiation. Theory. We may see the nature of the action to be looked for by considering an ideal case in which we allow a beam of light with energy P per cubic centimetre to faU normally in a perfect vacuum in turn on each of four discs, the front and back surfaces of these discs being respectively as in Fig. 1, BB BS SS SB ' BACK Fig. 1. where B represents a fully absorbing or * black' surface, and S a fully reflecting or non-radiating surface. When the radiation falls on an absorbing face, as in the case of either of the discs (1) and (2), the temperature of the disc rises till a steady state is reached in which emission equals absorption. We may suppose that the discs are so thin that the two faces are sensibly at the same temperature. If we did not take into account the pressure due to issuing radiation, or if we 382 THE PRESSURE OF LIGHT AGAINST THE SOURCE: only considered the initial effects before heating took place, the pressures on the first two discs would be P in each case, due to the incident beam alone, and on the last two would be 2P, due to the sum of the incident and reflected beams. We should have, therefore, pressures respectively (1) (2) (3) (4) P P 2P 2P But when a steady state is reached, the discs (1) and (2) must be giving out as much radiant energy as they receive. The first disc gives out equal amounts on the two sides, producing equal and opposite pressures. All the radiation from the second disc is given out at the front side and is equal in energy to that of the incident beam. Assuming this emitted radiation is distributed according to the cosine-law, the pressure resulting from it is easily shown to be |P, so that the total pressure on this disc is fP. Since there is no absorption by discs (3) and (4), we still have the pressures 2P ; hence we have now (1) (2) (3) (4) P fP 2P 2P In a real case these results are modified in two ways : (i) By the possession of some small reflecting power by surface B, and of some small absorbing and radiating power by surface S. (ii) By an inequality of temperature between front and back surfaces conditioned by the energy which is carried through from front to back to be radiated thence. The vacuum is not perfect, and there is radiometer-action due to the residual gas, which, owing to the inequality of temperature, is not the same on the two sides. This is probably the only way in which gas- action is sensible, for the effects due to ordinary convection and conduction in the residual gas are negligible. The temperature-diflerence, though sufficient to produce a differential radiometer-action, is so small that in estimating the radiation from the two sides of a disc we may take them as being at the same temperature. In the experiment to be described the diffusion is so sHght that we dis- regard it. Considering, then, only the reflection and absorption, let r be the coefficient of reflection of the surface B for the incident radiation, p that of S, a the coefficient of emission of B for the emitted radiation, a that of S. It is then easy to show that the total radiation-pressures on the four discs are respectively (1) (2) (3) (4) The emitted radiation is not of the same quality as the incident radiation ; and strictly we are not justified in assuming that the emissive powers for the THE RECOIL FROM LIGHT 383 two surfaces for the one quality of radiation are in the same ratio as the absorbing powers for the other. But we shall for simpUcity suppose that the ratios are the same, a supposition which enables us to proceed, and which is probably not very far from the truth. We have, therefore, a=l— r, a=l— /). On this assumption Table I has been constructed, giving the four pressures for different values of r and p. The pressure-ratios in the last two columns are of interest in connection with the experimental results, and will be referred to later. Table I. Pressures on Discs, calculated for Different Values of the Reflection- Coefficients . Reflecting power of B Reflecting power of 8 Pressures taking P = 1 Pressure-ratios r P BB BS 88 8B B8 BB BS \{88 + 8B) 1-00 1-00 1-67 200 2-00 1-67 1-67 0-95 1-00 1-60 1-95 1-92 1-60 1-66 0-90 1-00 1-54 1-90 1-85 1-54 1-65 0-05 1-00 105 1-68 200 200 1-60 1-68 005 0-95 105 1-62 1-95 1-92 1-54 1-67 005 0-90 1-05 1-56 1-90 1-85 1-49 1-66 010 100 110 1-70 2-00 200 1-55 1-70 010 0-95 MO 1-64 1-95 1-92 1-49 1-70 010 0-90 MO 1-58 1-90 1-85 1-44 1-69 The modification of the values in the above table by radiometer- action will be greater for the BB disc than for the others. In that disc energy proportional to JP has to be carried through the disc, and therefore the temperature-difference is the greatest. It is only possible to guess at the relative magnitudes of the radiometer-actions for the different discs. We have therefore sought to make the vacuum so high that the action nearly disappeared. The Experiment. In the final form of the experiment each disc consisted of a pair of circular cover-glasses, 1*2 cm. in diameter and about 0-1 mm. thick, between which was squeezed a layer of asphaltum also about 0*1 mm. thick, the temperature being first raised sufficiently to render the asphaltum molten. It is difficult to make the discs of uniform thickness and free from bubbles of gas ; but a great number were made and the four best were selected for use. Such a compound disc appears to be perfectly opaque, and its surface is the blackest black and the least diffusing that we have yet been able to obtain. 384 THE PRESSURE OF LIGHT AGAINST THE SOURCE The reflecting surface was made by depositing silver on the outside of the compound disc by means of the discharge from a silver cathode in an exhausted receiver. A similar deposit on clear glass just allowed an arc light to be seen through it. Four holes the size of the discs were cut in a stout plate of mica ABCD, the centres of the holes being at the corners of a 2 cm. square (Fig. 2). The discs were then fixed in these holes by a minute amount of celluloid varnish. The suspension-rod E and the mirror-holder F were attached to H :i <}. the mica plate, at the middle points of its top and bottom edges respectively, by very small copper clips without any cement. A platinised mirror was cut in half, and the two portions M^, M^ were mounted back to back in a suitable clip of copper foil at the extremity of the rod F, the plane of the mirrors being perpendicular to the mica plate. This system was suspended by a quartz fibre G, 9 cm. long, in the centre of a glass flask of 16 cm. diameter. The upper end of the quartz fibre was fixed to a brass collar H held by friction in the neck of the flask. Both ends of the fibre were silvered and coppered, so that they could be soldered to E and to the support. After THE RECOIL FROM LIGHT 385 suspension the mouth of the flask was sealed off, a lateral tube in the neck being still available for connection with the exhausting apparatus. To carry out the exhaustion of the flask to a very high degree, the general arrangement shown in Fig. 3 was adopted, and the successive stages in the process were as follows : (1) Prehminary exhaustion by aid of a Gaede pump. The experimental flask A was kept hot by gas-burners below it, and the charcoal bulbs Cj and Cg were strongly heated electrically by enclosing them in asbestos tubes con- taining coils of platinum wire. From time to time the whole apparatus was washed out with dry oxygen generated from the manganese dioxide in the bulb E. A small discharge tube G indicated the state of the vacuum, and examination of the spectrum gave useful information as to the gas given off from the bulbs C^ and C2. At the end of three days these bulbs appeared to Fig. 3. have ceased to give off any gas, and the Gaede pump would bring the vacuum down to a hard X-ray stage in about 10 minutes after admitting oxygen. The temperatures of C^ and C2 were then somewhat reduced, and the apparatus was sealed off at Si. B and F in the figure represent phosphorus pentoxide drying tubes, and H is a manometer. (2) The charcoal bulb Cj was put in liquid air, and the temperature of the second bulb Cg and of the flask A was then slowly reduced to the room temperature. After 30 hours the apparatus was finally sealed off at 82- During both these stages the U-tube D was kept always immersed in liquid air. This arrangement formed a perfect trap for mercury vapour, which otherwise diffused from the pump into the flask and attacked the silver mirrors. A roll of silver foil was placed in one limb of the U-tube, with the object of making the trap more effective, but this precaution was probably unnecessary. p. c. w. 386 THE PRESSURE OF LIGHT AGAINST THE SOURCE: (3) In the final stage of the exhaustion, the charcoal bulb Cg was surrounded by liquid air, which was boiled off continuously at the reduced pressure of about 2 cm, of mercury for several hours before and during the whole of the measurements. Experience showed that the highest vacuum was obtained probably two hours after the application of fresh liquid air, and that it was necessary to renew the liquid air after every four or five hours. The source of light S (Fig. 4) was an Ediswan 50- volt ' Focus lamp,' which was fed from accumulators. By means of an adjustable resistance in series with the lamp, the voltage was maintained exactly at 60 volts, the lamp then taking a current of 5-37 amperes. The light was so steady for hours at a B o s Li. La. ; ,' G : ::©----- -:::::::: .■:::;:;i:: --'€ o so I 1 :. IOC A Cms- Fig. 4. time that adjustment of the resistance was seldom necessary. A photo- graphic lens Li, of 15 cm. focal length, provided with an iris diaphragm, was arranged to throw an image of the lamp filament on an achromatic lens Zg of 19 cm. focal length. This seeond lens then formed a uniformly illuminated image of the iris diaphragm on the disc to be worked with. By adjustment of the diaphragm this image was made rather smaller than the disc, so that when the beam was centred on the disc an unilluminated margin about | mm. wide was left all round it. The lamp and lenses were fixed to a board which could be moved parallel to itself vertically and horizontally between guides, so that the beam could be easily directed on to the four discs in succession. The centring of the image on the disc was made by eye. The flask was mounted on an iron turntable with the quartz fibre accurately in the axis of rotation. By rotation of the flask through 180° it was possible to experiment on the reverse sides of the four discs. We 9 THE RECOIL FROM LIGHT 387 shall refer to the observations raade in the two positions as 'direct' and 'reverse.' Thus the 'direct' BS disc becomes the 'reverse' SB disc. The flask was shielded from electrification and from extraneous radiation by enclosure in a cylindrical case of tinned iron, blackened inside and provided with windows to admit the beam of light and to allow the deflections to be observed. For reading the deflections the image of an electric lamp B (Fig. 4) on a millimetre scale C, at a distance of 113 cm., was used. The definition of the image was sufficiently good to allow deflections to be read accurately to 0-2 mm., although the optical irregularities of the glass flask rendered a telescope useless. In finding the centre of swing from the deflections, a curious periodic motion was observed in its position, the centre moving to and fro with simple harmonic motion. The complete period of the torsional vibrations of the suspended system was 74-6 seconds with no appreciable damping; and the period of the motion of the centre of swing was found to be about seven minutes, that is nearly 11 torsional half-periods. The periodic motion was ultimately traced to pendulum-motions of the system set up by external disturbances, for the amplitude of the displaced centre of swing was found to increase with such pendulum-motion. The pendulum-period of the system was slightly longer for motion in the plane of the mica plate than for motion in a perpendicular plane. Hence a pendulum-motion once set up changed periodically from motion in a straight line to motion in an ellipse, and the complete cycle of these changes was actually gone through in seven minutes. This meant a periodic change in the angular momentum of the system about the axis of suspension, and to neutralise this the mica plate tended to turn with equal and opposite angular momentum about the vertical axis, and so to give a twist to the fibre. Accordingly, we should expect the maximum twist to take place when the vibrations are lineat, and this was observed to be the case. To eliminate this effect, 12 consecutive turning points were always taken (i.e. observations over a period of seven minutes), and the mean centre of swing calculated from these. When the beam was allowed to fall on a disc, some initial effects were observed, of which we shall give an account below, and shall then suggest a tentative explanation. The Results. The observed deflections for the four discs are given in Table II. These results are divided into two series, A and B. In Series A the beam of Hght was kept on the disc under experiment until the deflection seemed nearly constant, the time of exposure being generally about 20 or 30 minutes before the deflection was read. Attention was chiefly given to the BB and 25—2 388 THE PRESSURE OF LIGHT AGAINST THE SOURCE *. BS discs ; the deflections for the SS and SB discs are not so reUable, as it became afterwards evident that these discs require even longer time to recover from the initial disturbances (see below). Table II. i BB BS SS SB Series A Light on for about 20 or 30 mins. ; zero observed be- tween each exposure to light D 14-91 15-56 15-03 20-45 23-29 20-77 32-32 27-22 Mean D 15-17 21-50 32-32 27-22 R 17-15 17-01 22-89 23-94 27-36 26-19 Mean R 17-08 23-42 27-36 26-19 Mean D and i? 16-13 22-46 29-84 26-71 1 Series B Light on for 1 hour; zero observed only at beginning and end of observations on several discs D (13-78)* 15-20 14-71 2107 20-44 30-71 28-51 29-15 27-95 Mean I) 14-96 20-76 29-61 28-55 R 1711 23-37 27-72 27-42 Mean D and R 16-04 22-07 28-67 27-99 Final Values BB and BS taken as mean for Series A and B ! SS and SB from Series B alone 16-1 22-3 28-7 28-0 Deflections in scale-divisions for the four discs. Observations on the 'direct' and 'reverse' sides are denoted by D and R respectively. * Rejected in taking the mean as, owing to a breakdown of the pump for exhausting the liquid air, the vacuum had doubtless deteriorated. In the Series B, made a month later, the beam was kept on each disc for one hour before reading the deflection, with the object of obtaining a still closer approach to a steady state. One hour was also allowed before taking the zero after cutting off the light. The much greater time now required made it impracticable to observe the zero more often than twice while making observations on all four discs. Experience showed that, provided the vacuum was well maintained, the zero seldom changed more than one or two tenth- THE RECOIL FROM LIGHT 389 divisions during five hours. There appeared, therefore, no objection to this course. Examination of the table shows that in most cases rather different values are given by the * direct' and 'reverse' observations. This is particularly- evident in the case of the BB disc, where the values, roughly 15 and 17 divisions respectively, differ by about 13 per cent. The explanation of this appears to be that the radiometer-action was still sensible and acted differently in the ' direct ' and ' reverse ' positions of the disc. The difference was probably due to inequality in the thickness of the two cover-glasses enclosing the asphaltum layer. Some observations were made without using liquid air, i.e. with the charcoal bulb at room-temperature. The deflections were then always very unsteady, and the zero showed erratic behaviour. A few deflections obtained under these conditions are given in Table III. They merely serve to indicate the general effects of the gas-action. On the BB disc the want of symmetry referred to above is now greatly exaggerated, and points to the gas-action being a suction on the 'direct' side and a pressure on the 'reverse.' It is also to be noted that the action is a suction on the BS disc. Any tendency towards this action in the high-vacuum experiments would, therefore, tend to mask the recoil-pressure sought for. Table III. BB BS SS SB Direct 5 37 - 19 - 77 no obs. 45 93 58 Reverse Deflections for the four discs with charcoal bulb at room-temperature. (The minus sign indicates suction.) On account of the uncertain values of the various corrections required, and on account of the existence of outstanding disturbances, it was felt that mere multiplication of observations would not lead to more exact results. As the final values, in scale-divisions, for the pressures on the four discs given by the experiment we therefore take BB BS ss SB 16-1 22-3 28-7 28-0 The values for BB and BS are the means given by both Series A and B, the values for SS and SB are from Series B alone, since for these discs the steady state was not attained properly in the observations of Series A. Hence we have BS BB = 1-39, BS J (SS + SB) = 1-58. 390 THE PRESSURE OF LIGHT AGAINST THE SOURCE: We select for comparison with these ratios those calculated for r = 0-05, p = 0-95 in the last columns of Table I, i.e. the values m~^'^^' UssTsB)- ''"''• The latter ratio agrees better with the experiment than the former. This is what we might expect, since the latter ratio does not involve the pressure on the BB disc, and that is the pressure which is most affected by the radiometer- action. As to the actual reflection-coefficients for the surfaces of the discs, we can do but httle more than make a guess. For the black surface we may take the glass surface as reflecting 4 per cent, of the incident Ught, and allowing another 1 per cent, for the asphaltum — probably a reasonable estimate — we have, finally, r =- 0-05. In the case of the silver surface the reflection was tested by means of a thermopile, and it was concluded that at least 96 per cent, of the beam used was reflected. The Energy of the Beam. A determination of the energy of the beam used was made and afforded a means of calculating the absolute values of the pressures to be expected. As in the experiments of Nichols and Hull, and in other experiments on light-pressure which we have made, the energy was measured by allowing the beam to fall on a blackened disc of pure silver (2 cm. diameter and 0-28 cm. thick), and by observing the initial rate of rise of temperature by means of a constantan-silver thermo-electric junction soldered to the disc. A Rubens panzer-galvanometer was used and was adjusted to have a period of about two seconds and was then made dead-beat. The transit of each centimetre- division of the scale across the field of view of the observing telescope was recorded on the drum of an electrically driven chronograph. Just before and just after each series of transits, a set of 5-second intervals were also recorded in order to give the peripheral speed of the drum. The number of micro- volts per scale-division was then determined, and the thermo-electric power of the couple being known from a separate experiment, it was possible, by means of a graphical representation of the chronograph-record, to calculate the rate of rise of temperature, and hence the energy of the beam was calculated to be 33 x 10~^ erg per centimetre length. This would be the force in dynes on a fully absorbing surface. The moment of inertia of the suspended system was 0-770 gramme-cm. 2, and its period was 74-6 seconds. The arm was 1-00 cm. Hence the beam should give a deflection of 13-6 divisions of the scale used when falling on a disc fully absorbing on both sides. Assuming that the BB disc reflects 5 per cent., the deflection should be 14-3 divisions. This is in close agreement with half the value obtained with SS and SB, and the excess over 14-3 of the observed value 16-1 obtained THE RECOIL FROM LIGHT 391 with BB is probably to be ascribed to residual radiometer-action. The smallness of the excess shows that the radiometer-action was reduced to a very small amount. Initial Effects. We have already referred to the necessity for exposing the discs in general to the beam for a long time before taking the observations of the deflections. On the BB disc, however, there was no marked initial effect, and after a few periods the deflection was nearly constant. When the beam was first allowed to fall on one of the other discs an initial disturbing effect was evident, and in some cases even half an hour seemed insufficient to produce a steady state. On the BS disc there were sometimes indications of a pressure for the first few seconds, but a strong suction always set in, and this suction, after reaching a maximum, rapidly subsided, giving place finally to a pressure increasing to the Umiting value corresponding to the steady state. The SS disc showed want of symmetry. On the direct side there was suction followed by pressure, as in the last case, but on the reverse side there was at first excessive pressure, reaching a maximum and then slowly falling to the value given in the steady state. The SB disc showed an initial excess pressure very similar to that on the reverse side of the SS disc. For both the SS and SB discs the duration of these initial effects was so drawn out that an appreciably steady state was not attained in much less than an hour. In Fig. 5 the effects on all four discs are represented by curves in which the time is taken as abscissa and the deflection as ordinate. The residual effects which follow the cutting off of the Hght are also indicated. As an explanation we suggest that these effects are due to the heat of the beam driving off occluded gas from the silver films, the expulsion of the gas causing a back pressure on the film. In the case of the SS disc it was known that the silver film on one side was decidedly thicker than that on the other, so that there is no difficulty in accounting for the want of symmetry observed in that case. This explanation is supported by two observations : first, that with a stronger beam the steady state is sooner reached ; secondly, that in the case of the BS disc it was noticed that if after the estabhshment of the steady state one cut off the light for a minute or two, and then put it on again, an initial suction took place but was much less than originally. Assuming that the effects observed were really due to the expulsion of occluded gas as suggested above, it is possible to form an estimate of the total mass of gas given off by calculating from the curves (Fig. 5) the total impulse given to the disc. Thus, if we assume the gas to be oxygen and suppose, 392 THE PEESSURE OF LIGHT AGAINST THE SOURCE: further, that the molecules leave the film normally with the ordinary molecular velocity, we find that for the BS disc the total mass given off was about 1*7 X 10-' gramme. Taking the volume of the experimental flask as 2 litres. BB fi a SB mr/7S Fig. 5. Initial Effects. The ordinate represents the deflection, plotted against the time as abscissa. The direction of the deflection for pressure is shown by the arrow. For the SS disc curves D and R correspond to the ' direct ' and ' reverse ' sides respectively. The light is put on at a and cut off at ^. this quantity of gas would give a partial pressure of about ^^ dyne/cm. 2. So that unless the gas is rapidly absorbed by the charcoal it would appear that the vacuum might be sensibly affected. Unfortunately we had no means of forming even a rough idea of the actual pressure of the residual gas in the apparatus. THE RECOIL PROM LIGHT 393 The Temperature of the Discs. In the steady state all the energy of the beam absorbed by a disc must be ! radiated from the faces. By assuming the fourth-power law of radiation, we may therefore estimate roughly the rise in temperature of the disc. The results are 55° C. and 90° C. for the rise in temperature of the BB and BS discs respectively. For the BB disc it was also estimated that the temperature-difference between the two faces was probably less than J^° C. The temperature- differences for the other discs should be still less. It should be noticed that the glass is black for the issuing radiation, hence the asphaltum layer alone counts in the case of the BB disc. Early Experiments with Platinum Discs. In some early experiments we used two discs, BB and BS, oi platinum foil J mm. thick, the black surfaces being formed by depositing platinum black. Results were obtained somewhat like what we expected as to the ratios, but the deflections depended very greatly on the state of the vacuum, and under the best conditions were about 50 to 100 per cent, greater than the values calculated from the energy of the beam. This disagreement was doubtless chiefly due to radiometer-action. The black surface, being flocculent, is obviously badly conducting, the temperature-slope is therefore increased, and we have in consequence a big differential radiometer-action on the faces of the discs. Moreover, the polished platinum is a poor reflector, so that such discs quite fail to approach the ideal conditions. These considerations led to the use of the asphaltum discs. 29 ON SMALL LONGITUDINAL MATERIAL WAVES ACCOMPANYING LIGHT-WAVES. [Roy. Soc. Proc. A, 85, 1911, pp. 4'74-476.] All experiments on the pressure of light agree in showing that there is a flow of momentum along the beam. This flow is manifested as a force on matter wherever there is a change of medium. When the light is absorbed, the momentum is absorbed by matter. When the beam is shifted parallel to itself there is a torque on the matter effecting the shift. The momentum would therefore appear to be carried by the matter and not merely by the ether. Though there is an obvious difficulty in accepting this view when the density of the matter is so small as it is in interplanetary space, it appears to be worth while to follow out the consequences of the supposition that the force equivalent to the rate of flow of momentum across a plane perpendicular to a beam of light acts upon the matter bounded by the plane. This rate of flow per square centimetre is equal to the energy-density or energy per cubic centimetre in the beam. Of course, in experiments, only the average of the rate of flow during many seconds and the average energy per cubic centimetre in a length of beam of millions of miles is actually measured. But on the electromagnetic theory of light, which suggested the experiments and which gives the right value for the pressure, this pressure is equal to the energy-density at every point of a single wave. Let us suppose that we have a train of plane polarised electromagnetic waves of sine-form, the magnetic intensity being given by H = H^ sin ^ (x — vt), A where H^ is the amplitude of H. The magnetic energy per cubic centimetre at any point is hH^/Stt, and as the electric energy is equal at each point to the magnetic energy, the total energy is jjlH^I^^tt. The energy per unit volume is ~ — dx = ijuH-^j^tt. SMALL LONGITUDINAL MATERIAL WAVES ACCOMPANYING LIGHT- WAVES 395 The pressure p across a transverse surface is AfIT p = fiH^I4^ = ^ sin2 ^(x- vt) &7T 1 — COS -y (iC — Vt) The force on an element of length dx is ^ dx = dx '-^ J- sm jr [x — vt) dx fJiH^^ .477. .. = — ^-^ sm -Y (x — vt) dx. If ^ is the linear longitudinal displacement of the element there will be a force due to elastic change of volume d^$. where q is the elastic constant for compression or extension. If p is the density of the material, the equation of motion is d^ M^l^,.47^ ^' dx^ d^i Pd^ 2A smy (a;-i;0. 477 A sin -Y (x — vt — e). Assume Then, substituting, ( p^ . -^ v^ - qA -^ j sm -j-(x-vt-e)= ^^^ sm y (^ - ^'0- Putting X =^ and ^ = 0, we see that e = 0. Putting q = pu^, where u is the velocity of free elastic waves of the q type, and assuming that the longitudinal waves are forced waves, keeping exact time with the waves of light, we have Xp^H ,^ 32tt^P (v^ - u^) ' As u/v is negligible for all ordinary matter, . XuHi' .477. ,, . uHi^ 477, . The potential energy in these waves is neghgible in comparison with the kinetic. We have then Energy per unit volume = J p^hlx •' 25677^^^ ' 396 SMALL LONGITUDINAL MATERIAL WAVES ACCOMPANYING LIGHT- WAVES As the electromagnetic energy per unit volume is ^xH-^j^, Energy in longitudinal waves _ ^E^ _ 1 ij^H^ jpv^ Electromagnetic energy S27Tpv^ 8 Stt / 2 ' which is one-eighth of the electromagnetic energy divided by the energy which the matter would have if it were moving with the velocity of light in that matter. This shows how infinitesimal is the fraction of the energy of the beam which is located in these waves of compression of the material. The fraction is proportional to the intensity of the beam. As an example, take a beam of the intensity of full sunlight just outside the earth's atmosphere, in which the energy-flow is about 1-4 x 10^ ergs/sec. The energy-density ixH^^j^rr is therefore 1-4 x 10^ ^ v. Put v = 2> x lO^^jn, where n is the refractive index. The fraction is 1 1'4 y 10^ ^3 At the surface of the sun it would be about 40,000 times as much, say, 5 X 10-22^7^. It is interesting to note that if a beam of light is incident on any reflecting or absorbing surface and if the pressure of light is periodic with the waves it must give rise to ordinary elastic waves in the material of frequency double that of the light-waves. 30. ON THE CHANGES IN THE DIMENSIONS OF A STEEL WIKE WHEN TWISTED, AND ON THE PRESSUKE OF DISTORTIONAL WAVES IN STEEL. [Roy. Soc. Proc. A, 86, 1912, pp. 534-561.] [Read March 21, 1912.] In the Proceedings of the Royal Society* there is an account of some experiments which I made to show that wires when twisted lengthen by an amount proportional to the square of the angle of twist, a result expected from an analysis of the strains in a finite pure shear. In those experiments it was necessary to put considerable loads on the wires. I have now succeeded in measuring the change in the diameter of a wire when twisted, as well as the longitudinal extension, and have found that the change, a contraction, is also proportional to the square of the angle of twist. It has been now found that the change is sensibly the same for large loads and for the smallest load which could be used, when the wire was sufficiently straightened before being twisted, so that apparently the only function of the load is to straighten the wire. To measure change in the diameter the wire was fastened at the bottom of a long narrow tube, the 'wire- tube,' filled with water. It passed out from the top of the wire- tube through a water-tight leather washer. A capillary glass tube rose vertically from an orifice in the side of the tube, into which it was cemented, and the change of the water level in the capillary when the wire was twisted indicated the change in the volume of the wire within the wire-tube. Description of the Apparatus. The apparatus used for the measurement of the effects is shown in Fig. 1, where, for convenience of representation, various parts are put into the plane of the figure, though actually they were in different planes. An iron bracket B projected from the wall of the laboratory, and a tripod rested on it, on three levelHng-screws. The tripod carried a conical bearing * Series A, 1909, vol. 82, p. 546. [Collected Papers^ Art. 25.] 398 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED, for the twisting head-piece T. When the axis of this was exactly vertical, the tri- pod was fixed to the bracket by clamping- screws s, s. The twisting-head was pro- vided with a circular plate P, with marks at 90° intervals, which could be set against a fixed index i. In practice only whole turns were given, so that only one mark was used, except in one experiment de- scribed later. At the lower end of T, there was a chuck into which the upper end of the wire was inserted, and a tightening-screw made a firm grip. The wire in all cases was very nearly 160-5 cm. long. At its lower end it was gripped by a similar chuck attached to a steel cross-piece C, about 29 cm. long, seen endwise in the figure. Polished steel plates were screwed on to the vertical sides of this cross-piece near its ends. Four horizontal screws, working in brackets projecting from the wall, and with small steel balls at their ends, were screwed up so as just not to touch the steel plates when there was no twist on the wire. But when a twist was put on, the cross-piece moved up against two of the screws, and was thus fixed in position. Below the cross-piece there was a rod to which was attached another rod carrying a platform p, and on this weights could be placed. Each weight was in two semicircular halves. Below the platform was a lead weight S, which I call the sinker, with a volume of 1020 c.c. This hung in a can, and near the can was a water cistern, not shown, connected to it by a rubber tube. When the cistern was pulled up water flowed into the can so as just to cover the sinker and lessen the load by 1020 grm. When the cistern was Fig. 1. AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL 399 let down the water flowed back into it and the load increased to its full value. This was used to determine Young's modulus and Poisson's ratio. A table t was fixed to the wall independently of the bracket B, to carry the observing microscope, and the observer sat on a platform built up about 1-5 metres from the floor. The brass wire- tube had an internal diameter about 2 mm. The top of the tube was fixed in a horizontal brass plate, in which was a hole about 0-25 mm. wider than the wire. On this plate rested a well-vaseHned leather washer about 1-5 mm. thick, drilled so that it was fairly tight round the wire. On the washer was another brass plate, with a hole in it about 0-35 mm. wider than the wire. Four screws passed freely through holes in this upper plate, and were screwed into the lower plate. Springs^ between the heads of the screws and the upper surface of the upper plate gave sufficient pressure on the washer. It was found necessary to have the holes in the plates somewhat larger than the wire, in order to adjust the wire and the wire- tube both vertical. An arm, not shown in Fig. 1, projecting from the lower part of the apparatus, with a sliding weight on it in the plane of the upper side-tubes, sufficed to make this vertical adjustment. At first I tried india-rubber washers. They were quite good when first put in, but they deteriorated rather rapidly, and, when they began to perish, they let a small quantity of water out of the tube when the wire moved. The leather washer only required renewal once, when it began to let water escape, and then, on examination, it appeared to be due to action on the wire, which was perceptibly rough on the surface where it emerged from the tube. A short length was cut off the lower end of the wire, and an equal length was let down through the upper chuck, so that once more a smooth part of the wire passed through the washer. There was no further difficulty, and no evidence again of any escape of water. At the upper end of the wire-tube there were two side-tubes. A glass capillary tube was cemented into one, and bent as shown on the right in the figure, the vertical branch being about 10 cm. long. When the tube was filled with water, the level in the capillary was adjusted at the level of the micro- scope about 7 cm. above the level of the washer, as this was about the rise of water in the capillary due to surface-tension, and there would therefore be no hydrostatic pressure on the water at the level of the washer. The tendency to leak would thereby be lessened, but the precaution was probably needless. Into the tube on the left a plunger passed through a leather stuffing-box. The plunger had a diameter of 0-2060 cm., and it was driven to or fro by a micrometer-screw of J mm. pitch. On the head of this screw was a 10 cm. plate, with 500 divisions on its circumference. This plunger was used ordinarily to adjust the level of the water in the capillary. But it was also used to calibrate the capillary. For this caHbration, the usual observing microscope was replaced by a microscope-cathetometer, and the change of 400 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED, water-level in the capillary was measured for one turn in or one turn out of the screw. Turning always in or always out, it was hardly possible to hit exactly on a whole turn, but a correction could be made, of course, for the fraction of a division in the micrometer head in excess or defect of a whole turn. The mean of 10 measurements when the micrometer was driven inwards 0-5 mm. gave a rise of 11-135 mm., with a range of 0-155 mm. between greatest and least. The mean of 10 measurements when the micrometer was drawn out 0-5 mm. gave a fall of 11-190 mm., with a range of 0-045 mm. The value was taken as 1 1 - 1 6 mm. This gives the cross-section of the capillary as 0-001493 sq. cm., and its diameter as 0-0436 cm. At the lower end the wire-tube was soldered on to a screw-cap which could be screwed over the chuck gripping the lower end of the wire. Below the chuck was a side-tube used to fill the wire-tube with water. For this purpose the side-tube was connected with a flask in which water Avas boiled. The steam passed up through the crevices in the chuck and out at the plunger-tube, from which the plunger was removed. A funnel con- taining water was connected on to the plunger-tube, and when the water in this was boiling freely, through the passage of the steam, the flask was allowed to cool and water was sucked back into the wire-tube. When it was full the flask was detached and a cap was screwed on to the lower tube. The plunger was replaced and the capillary, which had been closed meanwhile, was opened. By driving in the plunger the water was raised up to the level of the washer and to any desired point in the capillary. When the apparatus was not being used the open end of the capillary was under water in a beaker, the plunger being driven in so that the capillary was entirely filled with water. The apparatus thus remained full of water whatever change of temperature might occur. When required for work the beaker was withdrawn and the plunger was screwed out till the meniscus was in the field of view of the microscope. The wire-tube was surrounded by an outer tube about 2-5 cm. diameter, filled with water. This merely served as a means of reducing the effect of outside or inside temperature-changes. A wooden casing covered with tin foil surrounded the whole from the floor up to the table to lengthen out still further any effects due to temperature-change. To observe the changes of level due to twisting, a microscope with a 1-inch objective and provided with a parallel-plate micrometer was used. The micrometer-scale was calibrated by means of a millimetre divided to tenths on a standard invar bar. Twelve determinations of 0-4 mm, gave 107*2 micrometer-divisions equal to 1 mm., the determinations falling within about 1 per cent, range. Then 1 micrometer-division = 0-00933 mm. Since the cross-section of the capillary is 0-001493 sq. cm., one division of the AND ON THE PRESSURE OF DISTORTION AL WAVES IN STEEL 401 micrometer signifies a change of volume of the water in the wire-tube of 1-393 X 10-6 c.c. When it was desired to read the height of the water in the tube the micrometer- plate was moved till the cross- wire in the microscope just touched the image of the lowest point of the capillary meniscus. The field was well illuminated by a small lamp behind the capillary, but the image was not always very distinct, and settings of the micrometer could not be trusted to, I think, two or three tenths of a division in some cases, though usually they were more exact. Close to the tube and between it and the micrometer-plate a small vertical plate of glass was fixed to the tube at 45° to the line of sight, and this reflected the point of a needle which was also fixed to the tube, so that its image was in the same plane as, and close to, the image of the meniscus. This enabled the observer to note the position of either the meniscus or the needle-point without moving the microscope. When the wire lengthened the wire-tube was let down by an equal amount, and the needle-point fell. Let us call this fall NP. At the same time the wire contracted laterally, and the meniscus fell in the tube, and the fall relative to the tube gave the change in volume. The fall observed was that relative to the tube plus that of the tube or NP. Hence, if the fall observed in the microscope is T the fall relative to the tube is T - NP. The Wires and their Preparation. Two piano-steel wires were used in the experiments here described. No. I with a mean diameter 0-0986 cm., the diameters in two planes at right angles being measured with a micrometer every decimetre of its length. The measurements ranged from 0-0980 to 0-0989 at different points. No. II had a mean diameter of 0-1210 cm., measured in the same way, with a range at different points from 0-1207 to 0-1212. It was found necessary to straighten these wires, for, unstraightened, they showed the effect with light loads noticed in the previous paper *, an apparent shortening on twisting, due, I think, to coiling. To straighten them they were loaded and an electric current was passed through them. No. I was loaded with 50 kgrm., and received a current of 10 amperes. No. II was loaded with 60 kgrm., and received a current of 16 amperes. In each case the wire drew out slightly and then stopped, acquiring a blue temper without rising to a red heat. Of course the wires became circularly magnetised, but the magnetisation can hardly have contributed to the results here to be described, as these results are of the same character and order as results obtained with heavily * [Collected Papers, Art. 26.] p. c. w. 26 402 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED, loaded unannealed wires in a number of preliminary experiments made before the experiments took their final shape. A few experiments were made on a hard-drawn copper wire, mean diameter 0-1219 cm. (with a range from 0-1216 to 0-1224 cm.). The Method of Measuring the Lowering on Twisting. In making any determination of NP or T the following plan was adopted. Suppose, for instance, that the value of T was to be found for four turns of the wire clockwise as seen from above. The position of the meniscus was read for no twist at a given minute, then my assistant put on four turns clockwise — denoted by C^ — then he gave a signal just before, and again exactly at the next half minute, and I set the cross- wire on the meniscus at the half minute. The micrometer was read, and the twist was taken off. At the next half minute the micrometer was set as before. Again C^ was put on, and so on, usually for 32 observations. The first two or three readings were not taken into account, as initially there was usually some irregularity, due probably to settling down in the bearing. The readings were combined in threes in the usual way to give T = \ {a -\- c) — h to eliminate as far as possible any march of the zero reading. With the meniscus there was almost always a march, due chiefly to temperature- change, for, of course, the arrangement was a very sensitive thermometer. The mean result of the 32 observations was equivalent to 15 or 16 inde- pendent determinations. To determine NP the same course was followed, except that the time was not noted. For, though there was often a march in the zero, it was very much smaller, and the observations were made at suflQ-ciently nearly equal intervals of time without noting exact times. This small march was doubtless partly due to temperature-change, but also partly due, I believe, to further settling down of the cone of the twisting-head into its bearing through slow squeezing out of the oil. In reconstruction I should try the effect of replacing the conical bearing by a ball-bearing. Before the reading was made it was found to be absolutely necessary to move the head-piece some ten or fifteen times to and fro through a small angle — perhaps diminishing from 20° — on each side of the final position. If this was not done the wire did not sink down or rise up to its final position, probably owing to some small friction in the leather washer. After the alternating motion of the head-piece had been given ten or fifteen times no further alternation made any difference in the reading. It was only after finding the necessity of this that I obtained consistent readings. The maximum and minimum values of NP in a set of 30 determinations usually differed by about 0-4 division, only once rising to 0-8 division. The maximum and minimum values of T in a set of 30 differed more, as might be expected; the difference averaging 1-7 divisions, and once rising to AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL 403 nearly 5 divisions. This, however, was in the case of the least load, when the weight was probably insufficient to keep the wire-tube quite vertical. Subsidiary Experiments, The three following subsidiary experiments were made in order to justify the methods of measuring the changes of dimensions on twisting : 1. To show that the lengthening on twisting is not due to a change in Young's modulus, Y. This is satisfactorily proved by the experiments on twisting described below. For, suppose that we have a stress P applied to the end of the wire by a load stretching length I by dl when the wire is not twisted, we have dl = Pl/Y. Now let the wire thus loaded be twisted through, say, four turns and let the lowering through twisting be 8. If this is due to a change in Young's modulus to F, dl + S = Pl/Y'. Whence S = PI (F-i - F-i), and S should be proportional to P, whereas it is found to be very nearly the same for loads varying from 5 to 50 kgrm. (approximately). It appeared worth while, however, to test the question directly, by finding the extension of wire I for very different loads when 1-02 kgrm. was added, first with the wire untwisted, then with the wire twisted through four turns clockwise. The following results, in micrometer- divisions, were obtained, each the mean of a number of measurements : Table I. Load No twist Lowering for 1-02 kgrm. C4 twist Lowering for 1-02 kgrm. 18-5 38-5 48-5 10-55 10-68 10-50 1 10-52 10-80 10-39* Means 10-58 10-57 Thus a twist of four turns produced no measurable change in Young's modulus. 2. To show that the rise and fall of the liquid meniscus were due to, and measured, the change in volume of the wire in the wire-tube. The most satisfactory way of showing this appeared to consist in using the apparatus to measure Poisson's ratio a. The load was altered by 1-02 kgrm. by alternately immersing the sinker in water and letting the water run out. * [This corrected value is that given in the original manuscript. Ed.] 26—2 404 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED, The rise and fall of the needle-point gave the end-extension, and Young's modulus Y could then be calculated ; while the rise and fall of the meniscus gave the volume-change, and the side-contraction and Poisson's ratio a could be calculated. The rigidity-modulus n could be obtained from n = ^ Y/(l + a). When the load was increased there was some yield of the supporting bracket. To determine its amount, a needle-point was fixed on the bracket close to the upper chuck and sighted by the microscope. A series of loads up to 50 kgrm. showed that the lowering per kilogramme was 0-06 division. An addition of 1-02 kgrm., therefore, lowered the bracket by 0-061 division, and this had to be subtracted from the NP reading when used to find Y. In the lowering of the meniscus T — NP, it obviously did not come into con- sideration. The observed change of volume given by T — NP had to be corrected by a factor about 160-5/156, since only 156 cm. of wire were within the wire-tube and 4-5 cm. were outside. The actual length outside varied from 4-2 to 4-6, and the factor was varied accordingly. No doubt better values of Y and a might have been obtained with a larger change of load, but, to test the apparatus, it was important to observe lower- ings of the same order as those observed in the twisting. In the following Table II the values of T — NP and NP, due to an addition of 1-02 kgrm., are given in micrometer-divisions corrected as above described. Each value is the mean of 30. The range between maximum and minimum in a set averaged 1-7 divisions for T and 0-57 division for NP : Table II. Elastic Moduli and Poisson's Ratio. Load in kgrm. NP T-NP a 10-12 y 10-12 n Steel Wire I, Diameter 0-0986 cm. 18-5 28-5 38-5 48-5 Mean values 10-65 10-60 10-74 10-44 29-86 29-57 29-37 29-05 0-272 0-271 0-265 0-270 2-11 2-12 2-09 2-16 0-830 0-835 0-828 0-849 — 0-270 2-12 0-835 Same Wire with C^ Twist on it 48-5 10-27 1 28-04 1 0-265 j 2-18 Steel Wire II, Diameter 0-1210 cm. 0-861 48-5 7-05 1 31-22 1 0-287 | 2-12 [ Hard-drawn Copper Wire, Diameter 0-1219 cm 0-825 18-5 11-20 61-52 0-331 1-31 0-493 AND ON THE PEESSURE OF DISTORTIONAL WAVES IN STEEL 405 The values found for a steel wire after annealing, given in the paper already referred to*, were Y = 2-06 x lO^^, n = 0-809 x 10^2 (by vibration), whence or = 0-273. The values for the steel wires are sufficiently near to each other and to the values previously found to show that the tube-readings gave, at any rate, very nearly the true changes in volume. 3. To show that the changes were very nearly isothermal. The change in temperature of a solid sheared adiabatically through e is de=- Xnd€y2JC^p, where An is the decrease in rigidity per degree rise, 6 is the absolute tem- perature, Cp is the specific heat, and p is the density. Let us suppose that a steel wire 156 cm. long and 0-05 cm. radius — nearly wire I — is twisted through one turn. It is sufficient to investigate the effect for one turn, for both the adiabatic temperature-change and the twisting- effects are proportional to the square of the shear, and therefore in a ratio independent of the shear. Then e = 27rr/156, where r is the distance of an element from the axis. The mean change in temperature of such a wire is [^ 27Trdedr _ Tr'^a^Xnd where a = 0-05. For steel we may put n = IO12, A = 2 x lO"*, p = 7-8, C^ = 0-112. Taking 6 as 300° A. we find the heat in calories developed by the twist to be about — 18 X 10-* calories. Or on untwisting + 18 x 10"* calories. If this heat were confined to the steel it would alter its temperature by about 1/600° C. and its linear dimensions by about 1-8 x 10"^ in 1. The twisting through one turn, as will be seen below, alters the radius by about 3-19 X 10-*^ in 1 and the length by about 1-72 x 10"^. The effects of an adiabatic change of temperature would, therefore, be appreciable compared with the effects of twisting, especially on the radius. But the wire shares its heat, positive or negative, with the water in the wire-tube, and here it may produce a serious effect, if it gets no farther than the wire-tube, owing to the considerable coefficient of expansion of the water. Let us suppose that the heat or cold is shared with the water in the wire-tube so rapidly that both are at one temperature. The water, having approximately three times the volume of the steel, has about 7/9 the heat-capacity of steel plus water. So that the water would receive about 14 x 10"* calories. If a mass of water at a tempera- ture at which its cubical expansion is a receives H calories its change of volume is Ha, whatever the total volume. In our case the temperature was usually about 12° C, at which a is about 10"*. Then the volume-change would be about 14 X 10~^ c.c. and since one micrometer-division is about 14 x 10~' c.c. * Loc. cit. p. 554. [Collected Paper s^ p. 366.] 406 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED, along the capillary, one turn, through thermal effect alone, would produce a fall on twisting and a rise on untwisting of about 0-1 division. The actual change observed on twisting through one turn was about 0-56 division. If then the heat or cold only slowly spread from the wire, or again if it were rapidly shared with the water in the wire-tube but only slowly spread thence, the measurements would be seriously affected. But it is obvious that there must, in reality, be a rapid adjustment of temperature between the wire-tube and the outer water-jacket, and it was important to find out how rapidly the adjustment progressed. Fortunately the wire was insulated from the tube where it passed through the washer, so that it was easy to pass an electric current through it by connecting the terminals of a battery, one to the bracket, the other to the wire-tube. Heating-currents of the order of 1 to 2 amperes were thus passed along the wire. The current was put on for 2 seconds, the meniscus rushing up meanwhile fairly uniformly, and the point to which it rose was read on the micrometer. Then, 15 seconds after the cut off, the position of the meniscus was read again and the mean of a number of determinations showed that after 15 seconds only 0-032 of the original rise remained. The original rise varied from 7 to 18 divisions with different currents. If the twisting were made instantaneously and the reading of the fall in the tube were made 15 seconds later, about 0-032 x 0-1 -^0-56 = 0-006 of the fall would be due to the cooling on twisting. But this is a very con- siderable over-estimate. The twisting was usually begun 25 seconds before reading and ended more than 15 seconds before. The effect of temperature change may, I think, be estimated at less than 1/300 of the whole. It was impossible to assign even an approximate value to it and as it proved to be so small it was neglected. The effect would have been reduced altogether beyond consideration if the readings had been taken at intervals of one minute, but this would have introduced errors, probably much worse, through irregularities in the march of temperature. In the experiments on Poisson's ratio the adiabatic change of temperature on adding a load which stretches length I by dl is dd = - aYddl/JC^pl, where a 7 is the change in Young's modulus per degree. The value of a for steel is about 1/4000. This gives the heat for a stretch of 10 divisions as about — 9 X 10-^ calories. With uniform temperature of steel and water in the wire- tube the water would have about 7 x 10~^ calories, and its effect would be about 0-5 division. After 15 seconds it would be about 0-016 division. As the change of level observed was about 30 divisions the effect is negligible. The direct effect in lengthening or shortening the wire is easily shown to be very much smaller. AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL 407 Measurement of the Changes of Dimensions on Twisting, The method of making the measurements has been described abeady. In the case of wire I, NP, or the lengthening of the wire, was observed for various loads for four turns and for two turns clockwise twist, denoted by C4 and Cg, and for four turns and for two turns counter-clockwise twist, denoted by CC^ and CC 2i each value being the mean of 30 determinations — once or twice of 40 — made as described above. As permanent set came in with five turns, four turns was the maximum twist employed. The mean values of T were also determined for the same four twists and T — NP was corrected for the length of wire outside the wire-tube. For one load on wire I the lowerings for Og and COg were also observed. In all cases the lowering w could be represented very nearly by the parabola L (n -\- cY = w -\- h, where n is the number of turns put on and L, c, and b are constants, not, of course, the same for NP and T — NP. The constant c represents the fraction of a turn, always on the counter-clockwise side of the point of no twist, about which the lowering is symmetrical. Putting n = — c, b = — w is a small shortening, or for a counter-clockwise twist c the wire has a minimum length. The existence of c and b is due to want of homogeneity in the wire. They may be explained by supposing that the wire in the apparently neutral condition consists of a core and a sheath twisted against each other, as will be shown in the theory given later. Owing to want of exact centering the image * wobbled' somewhat in the field during twisting, and only returned to the same vertical line after a whole number of turns, so that it was futile to attempt to measure b. But there was fairly conclusive evidence that it had a real existence. According to the theory given, L is the all-important quantity. The internal strain only shifts the vertex of the parabola without altering its size. To find the constants of a parabola which should fairly represent the results, it was assumed that the curve went through the point w = 0,n = 0, so that Lc^ = b. The equation then becomes L (n^ -\- 2nc) = w. Let w be the lowering for C„, and w' that for CCn, then L (n2 - 2nc) = w', whence L== {w + w')/2n^ and c = {w - w')linL = ^n (w - w')l(w + w'). The errors are given by hL = h{w-\- w')l2n% and Sc =^ S (w - w')linL, assuming that L is without error in c. 408 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED, If then we find the value of L, say L^, from the lowerings at C^ and OC4, and the value of L, say L^ , from those at G^ and CG^ , the value of the former should have four times the weight of the latter and we may take the best value of X as 1^ (4X4 + L^. The value of c determined from C4 and CC4 should have twice the weight of that determined from C^ and CC2, and we may take the best value as J (2C4 + C2). In the following tables the results are set out. In Table III the lengthening of the wire I is given in micrometer-divisions, and below each lengthening the difference, calculated — observed, is put in italics, the calculated values being those given by the parabolas of which the constants are given in Table IV. Similar tables are given for wire II, and for the hard-drawn copper wire, but for a single load only, sufficient to secure good centering. After the experience with various loads with wire I, it appeared unnecessary to vary the load in the other cases. With wire II it was not thought advisable to go beyond three turns, and with the copper wire beyond one turn owing to permanent set, which began to be very considerable beyond those limits. The mode of calculating the best parabola was modified accordingly. Table III. Lowering NP for Steel Wire I, Dimneter 0-0986 cm. Load c. C2 CC.^ \ CC, i kgrm. 48-5 5-095 1-452 0-768 4-233 1 + 0-06 - 0-03 1 + 0-13 - 0-13 38-5 5-353 1-422 0-818 4-152 - 0-05 + 0-06 + 0-06 - 0-05 28-5 5-265 1-500 0-858 4-259 - 0-01 + 0-03 + 0-04 - O'lO 18-5 5-382 1-680 0-905 4091 + O'U - 0-13 - 0-04 + 0-06 4-7 5-407 1-742 0-797 3-865 1 1 + 0-13 - 0-15 - 0-03 + 0-04 For load 28-5, C3 was 3-048, and CC3 was 2-187, and these were taken into account in calculating the parabola, the difference in each case being + 0-05. Table IV. Constants of Parabolas for Table III. Load 48-5 0-289 c 0-226 b 0-015 38-5 0-294 0-256 0-019 28-5 0-295 0-237 0-017 18-5 0-302 0-281 0-024 4-7 0-295 0-345 0-035 AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL 409 There was probably some permanent set given in the last two owing to accidental over-twisting of the wire. The mean value of L is 0-295, and within errors of observation it is inde- pendent of the load. It is hardly likely that this is strictly true. Table V. Lowering of the Meniscus T — NP for Wire /, corrected for length Outside the Tube. Load c. c^ CO, CC^ 48-5 11-69 3-94 0-07 6-65 ■\-0-33 - 0-17 + 0-63 - 0-78 38-5 11-63 3-94 0-84 6-36 + 0-22 - 0-28 + 0-06 - 0-22 28-5 11-82 4-00 1-13 6-38 - 0-25 - 0-2H - 0-23 + 0-04 18-5 10-99 2-84 1-00 6-17 + 0-36 + 0-37 - 0-02 - 0-02 4-7 1017 2-54 0-99 8-24 - 0-14 + 0-30 + 0-57 - 0-72 For load 28-5, Cg was 7-33 and CCg was 2-86. These were used in calcu- lating the parabola, and the differences were respectively — 0-01 and + 0-23. Table VI. Constants of Parabolas for Table V. Load L c b 48-5 0-559 0-69 0-26 38-5 0-569 0-61 0-21 28-5 0-578 0-61 0-21 18-5 0=525 0-54 0-15 4-7 0-548 0-29 0-05 As the errors of observation with the last two loads were about double those for the earlier loads, they are only given half the weight in finding L. The value of L is taken as 0-561. Table VII. Lowering NP for Steel Wire II, Diameter 0-1210 cm. Two Independent Sets. Load Cs c. ca. cc. I 48-5 4-922 2-368 1-740 4000 + 0-05 - 0-06 - 0-04 + 0-05 II 48-5 4-983 2-243 1-708 4049 - 0-06 + 0-05 - 0-01 0-00 1 410 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED, Table VIII. Constants of Parabolas for Table YII. L c 6 I II 0-501 0-499 0154 0147 0012 0011 The mean value of L is 0-500. Table IX. Lowering of the Meniscus T — NP for Wire II, corrected for length Outside the Tube. Load c^ G, CO., cc^ ■ I 48-5 15-16 7-73 3-90 9-22 + 0-20 - 0-25 - 0-31 + 0-33 II 48-5 15-48 7-61 3-71 9-55 + 0-04 - 0-06 - 0-07 + O'lO Table X. Constants of Parabolas for Table IX. L c b I II 1 1-385 1-398 0-35 0-35 0-17 017 The mean value of L is 1-392. Copper Wire. Diameter, 0-1219 cm. I was only able to use C^ and CC^ owing to permanent set. The values of NP were 1-043 and 0-415, and the parabola going through these points and the origin is 0-73 {n + 0-22)^ = iv + 0-03. The values of T — NP, corrected for 4-2 cm. outside the tube, were 8-007 and 1-433, and the parabola is 4-72 {n + 0-35)^ ^w+ 0-58. The work given in the former paper appears to justify the assumption of the parabolic law for copper. The End- Elongation, Side-Contraction, and Volume-Increase. Steel Wire I. Diameter, 0-0986 cm. If w is the end-lowering for one turn from the position of minimum length assumed to be L divisions, w = L X length of one micrometer-division = 0-295 X 933 x 10-^ = 2-75 x lO"* cm. AND ON THE PRESSURE OF DISTORTIONAIj WAVES IN STEEL 411 The length is I = 160-5. Then w/l = 1-71 X 10-8. If u is the decrease in the radius a for one turn, 27raul = L X volume of one micrometer-division of capillary, u = (0-561 X 1-393 X 10-«) -r (2tt x 0-0493 x 160-5) = 1-57 X 10-8 cm. The radius is 0-0493 cm. Then u/a = 3-19 X 10-7. The ratio side-contraction/end-elongation, namely, u/a ^ w/l = 0-187. If dv is the volume-increase in total volume v, dv/v = (irahv — '2malu)/iTaH = w/l - 2u/a = 1-07 X 10-6. All the quantities w/l, u/a, dv/v are proportional to the square of the twist from the point of minimum length. The ratio u/a -^ w/l is the same for all twists. Steel Wire II. Diameter, 0-1210 cm. Using the values given in the tables, we have for one turn w = 4-66 X 10-4 cm., u/a = 5-24 x 10-', w/l = 2-90 X 10-6, u/a -^ w/l = 0-181, u = 3-17 X 10-8 cm., dv/v = 1-85 x lO-^. Copper Wire. Diameter, 0-1219 cm. The corresponding quantities given by the single set of observations are not of such weight as those for wires I and II, but I add them here : w = 6-81 X 10-4 cm. u/a = 1-75 x 10-^, w/l = 4-25 X 10-6, .j^ja ^ w/l = 0-41, u = 10-7 X 10-8 cm., dv/v = 0-75 x 10-6. On comparing the results for wires I and II we see that side-contraction ~ end-elongation is very nearly the same for both. The theory given below makes both w/l and u/a proportional to the square of the radius for wires of the same material undergoing the same twist. But as far as these two wires are concerned they are very nearly proportional to (radius)^'^. I do not think the discrepancy is to be ascribed to experimental error. Perhaps the theory is inadequate, but I think that it is more probable that slight differences in the material, not greatly affecting the ordinary elastic moduli, may produce 412 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED, very considerable changes in what we may term the secondary moduli, which, in the theory below, are denoted by f and q. I should like to have taken observations on several more steel wires with a wider range of diameters, but I am not able to continue the work at present. Experimental Verification of a Reciprocal Relation. In the Philosophical Magazine for November, 1911, vol. 22, p. 740, Dr. R. A. Houston has expressed the reciprocal relation between the stretching and twisting of a wire (confined within limits of reversibility) in the form dd\ fdw\ (S) =(|) . (1) \aj^ / Q const. \aix/ p const. where F is the end pull and w the increase in length, G the torque and 6 the twist on the wire (I use letters for length and torque differing from Dr. Houston's). As the apparatus only needed small modification it appeared to be worth while to see how nearly this relation was verified, and wire II was used for the purpose. Incidentally, the value of the rigidity was obtained by the method of statical torque. When the observations needed are worked out it is found that they are identical, as of course was to be expected, with those needed to verify the relation m ^C^l) , (2) \du J ,^, const. \^^^/0 const. which is the more direct expression of the Conservation of Energy in these phenomena. Taking equation (1) we require to know on the left the extra twist dd which must be put upon the wire to keep G the same when a load dF is added. For this purpose the wire was initially loaded with 18-5 kgrm., and the head was turned through a right angle. The bar at the bottom was also turned through a right angle from its usual position. On the cross-bar a mirror was fixed reflecting into a telescope a millimetre-scale 156-5 cm. away. The ends of the cross-bar were rounded into arcs of a circle with centre in the axis of the wire and radius 14-70 cm. Horizontal threads passed ofi these arcs to two very light horizontal spiral springs which stretched very uniformly in proportion to the pull up to 40 or 50 grm. These springs were attached to the bases of two travelling microscopes, of which the horizontal scales merely were used to measure any change in stretch. Initially, the wire was without twist, and the position of the cross-bar on the scale was read. It would have been at least very difficult to determine directly the total stretch of the springs required to keep the cross-bar in position when the head was twisted, so the following plan was adopted : AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL 413 A half -turn CC was put on by the head-piece, and the springs were stretched so as to bring the cross-bar to its original position. Then a further turn and a half CC was put on, and the additional stretch of each spring needed to keep the cross-bar in position was read. This additional stretch multiplied by 4/3 gave the total stretch of the springs, and thus the pull exerted at each end of the cross-arm to maintain two turns twist on the wire. The full stretches thus computed were 12-340 cm. on the left and 12-384 cm. on the right, corresponding to pulls, according to previous calibration, of 49*24 grm. and 46-87 grm., mean 48-06 grm. The torque was therefore G = 48-06 X 981 x 29-4 = 1-38 x 10^ dyne-cm. From this the rigidity is n = 0-838 X 1012. The tube-method gave 0-825 x 10^^^ ^^d the nearness of the two values appears to show that the springs could be trusted fairly well. A load dF = 30 kgrm. was then added, and the torque for two turns was thereby diminished. The springs therefore contracted, and it was observed that they pulled the cross-bar round through 15-85 mm. on the scale — the mean of five different observations ranging from 15-45 to 16-55, or through an angle 0-00507 radian. Denoting this angle by SO, and the radius of the cross-bar arm by k, and the decrease of torque by SG, f=^>:^ (3) where s is the whole mean stretch of the springs for two turns. But we require the twist dO, which must be put on the wire from its initial two turns, and in the opposite direction to Sd, to restore the torque to G. This is given by G + dd G-hG d-W here = i^r. whence, on substituting for SG/G from (3), we get dd^[^--l)W, and jp=[~-l)j^ W Taking the right hand of equation (1), we require to know the lowering dw for a change dG in the torque under constant load. We get the lowering from the equation L{n-cY = w-\-h, giving dw = 2 L {n — c) dn = L [n - c) —. Also dir = —K-. dw L(n-c)e , . Ti^^^ m^^-^rG-' *^^ 414 ON CHANGES IN DIMENSIONS OV A STEEL WIRE WHEN TWISTED, Equating (4) and (5), we ought to find ttG fkd \ SO Substituting the known values on the right, viz., G = 1-38 x 10^, n = 2, c = 0-15, h = 14-7, e = 4^,s= 12-37, 86 = 0-00507, dF = 30 x 981000, we get L = 4-48 X 10-*. The observed value of L is given as w on p. 411, viz., L = 4-66 X 10-4, showing as close an agreement as could be expected, considering the errors of observation. Taking the second reciprocal relation (2), to find [-jn] we must twist \aC7/^ const. through dd and observe dw, and then calculate what load dF must be removed to restore the original length. We have dw -= L [n — c) — , and dF = 7ra^Ydw/l. dF L(n-c)a^Y To find If-) , we put on a load W and observe W. If hG is the \dwJ0const. diminution in torque at this point and dG the diminution in torque with the original twist 6, G-hG _d-W G-dG" 6 ' Substituting for SG/G from (3), this gives the change in torque for addition W when 6 is constant. The value of dw for this load is dw=lWl7ra^Y, dG\ fhO ^\iTa^YG and ^ ^^ ^ I const Vc^Weconst. V5 / IWd dF for ^ \ s J Equating (7) and (8) and putting dF for W, we get ttG fkd ^\ SI . OdF the same equation as before. If we could use a wire without any internal strain when untwisted, c would be zero, and we could calculate L, the lowering for one turn, from observations on the torque and load alone. AND ON THE PRESSURE OF DISTORTION AL WAVES IN STEEL 415 Fig. 2. A Theory of the Changes of Dimension on Twisting: The Stresses in a Finite Pure Shear. In the paper already referred to* I showed that in a finite pure shear € such as is represented in Fig. 2, in which a cube of section ABCD is sheared into a figure of section ABKL through an angle CBK = €, the thicknesses perpendicular to AB and to the plane of the figure remaining constant, the lines of maximum elongation and contraction are, to the order of c^, at right angles before the shear, making an angle e/4 with the diagonals of the square, as AE and BG. After the shear they are again at right angles to the order of €^, and make an angle c/4 with the diagonals on the other side as AF and BH. Since we have elongation in one direction AF, and contraction in a direction BH at right angles, the shear may be maintained by a pressure P along BH and a tension Q along AF as far as forces in the plane of the figure are concerned. If we go to the first order of € only, If we go to the second order we must put P = ne + pe^, where ^ is a constant to that order. If we reverse e, P becomes equal to — Q, so that we have — Q = — ne -\- fe^ or Q = ne — pe^ We can only assume that there is no pressure or tension perpendicular to the plane of the figure, if we neglect e^. Going to the second order, we have to allow the possibility of a pressure of that order, which we may put as S = qe% where if q is negative the force is a tension. Considering the equilibrium of the wedge ABC, Fig. 3, with AC in the direction of greatest elongation and BC in that of greatest contraction, I showed that the tangential stress along AB is, to the second order, T = n€, and that a pressure is required perpendicular to AB given by R=(in + p) e\ * Loc. cit. p. 546. ICollected Papers, p. 368.] 416 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED, The analysis stopped here and was incomplete, as no account was taken of the stresses on the plane CD, Fig. 3, perpendicular to AB. It requires to be supplemented as follows : Considering the equilibrium of the wedge CDB, let us suppose that on CD there is a tangential stress T' along CD, and a pressure R' perpendicular to it. C P R = (^n+p)e2 T=ii£ Ii--(-trL+p)&2 *-T=Tl& R=(-Jn+p)£^ T=ne -t E=(-|n+p)&^ Fig. 4. Resolving all the forces on CDB in a direction parallel to DB, R' . CB sin (45 + Je) - T . C5 cos (45 -^le) + Q.CB sin (45 + Je) = 0, whence R' = T cot (45 + Je) - Q ; or, since T = ne and cot (45 + Je) = (1 — Je), neglecting e^, as it is multi- plied by e, R' = n€(l- ic) -n€ + pe^ = {- ^n + f) eK AND ON THE PRESSURE OF DISTORTION AL WAVES IN STEEL 417 Kesolving in a direction parallel -to CD, T . CB sin (45 + ^e) - R . CB cos (45 -\- ie) - Q . CB cos (45 + Jc) = 0, whence T = (R + Q) cot (45 + Jc) = {^n + p) c^ + w€ - j>€^} (1 - Jc) = n€ to the second order. On a unit cube of the material in the sheared condition then, we have, a& in Fig. 4, Tangential stresses along AB and CD each • ne. Tangential stresses along ^Z) and 50 each n€. Pressures perpendicular to ^5 and CD each (\n -\- f) €^. Pressures perpendicular to AD and BC each (— \n + p) e^. And pressures perpendicular to the plane of the figure each ^e^ or, in more convenient form... (q — f) e^ -{- peK The Strains in a Finite Shear-Stress consisting of Tangential Stress T, T' only. If an element is subjected to the system of stresses just investigated, when we put on to it a system of tensions equal and opposite to the second order pressures we have just found, we leave only the tangential stresses T = T' = ne. The strains due to these tensions must be superposed on the shear €, and we shall then have the strains due to the tangential stresses only. We have then to examine the strains due to tensions iin + p) e2 on AB and CD (Fig. 4). {- in + p) e2 on AD and BC. {q ~ p) €^+ pe^ perpendicular to the plajie of the figure. Through the tension pe^ on every face we get an extension in all directions pe^/SK, where K is the bulk-modulus. The tensions ^ne^ on AB and CD and the pressures Jwe^ on AD and BC constitute a shear-stress giving an elongation parallel to BC of ^ne^/n = Je^, and a contraction parallel to AB also Je^. The tensions {q — p) e^ perpendicular to the plane of the figure give an elongation perpendicular to that plane y (q — p) e^, and contractions at right angles, viz., along AB and AD, y {q — p) e% where Y is Young's modulus and a is Poisson's ratio. Collecting the results, we have secondary strains accompanying the shear € as follows : 1 pe^ o An elongation parallel to BC = 7 ^^ + o^ — y (9' — /^) ^^ • i>e2 1 ,, perpendicular to the plane = f^- + y (5' — V) ^^• p. c. w. 27 418 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED, As in the experiments, described abovp, Y and a were determined directly, it will be convenient to replace K from the equation ^ = — y — , and the secondary strains become /I 1-(T C7\, f 1 1-G (^ \ 2 /2(7 I \ . Equations Representing the Changes in the Dimensions of a Wire Subject to a Torque. I am indebted to Sir Joseph Larmor for his kindness in indicating how the following equations should be formed and solved. Let us assume that we put on to a wire of length I and radius a a pure shear-stress proportional to the distance r from the axis, and twisting the length I through angle 6. Then in addition to the shear € = rd/l, this stress would produce in an element un- constrained by neighbouring material what we may term 'free strains' with the values just found, which we may write as ar^ radial, ^r^ transverse to the radius, and yr^ longitudinal ; where / 2g 1 \ -=[-Y^^Y^) ^=(-4+ F^^-T^j /I 1 - cr o \ r=(+4 + ^^P-y^^j (1) If u is the actual radial displacement, and if w is the actual longitudinal displacement, the strains in addition to the shear e are, in cylindrical co-ordinates, dujdr, u/r, and dwjdz. The differences between these actual strains and the 'free strains,' viz., du 2 ^ w o 2 ^^ 2 /o\ ^^~dr~ ' f = r~^ ' ^^cfo~^ ' *^ imply 'secondary stresses' in the wire due to adjustment of strain in neigh- bouring elements. Let these be denoted hj R, 0, W. To find R, 0, and W, we treat e, /, g as if they were strains in an inde- pendent system. Putting A = e -\-f + g, the equations are R=XA + 2fjLe, = AA + 2/x/, W = XA + 2fig, (3) where X = K — %n = ,- — -, — ^^^ ^^-^ and jx = n (1 + 0-) (1 - 2(7) ^ 2(1+ C7) AND ON THE PRESSURE OF DISTORTION AIj WAVES IN STEEL 419 The forces R, 0, and W must form a system in equilibrium, there being no external forces to balance. Considering the equilibrium of the element ABCD, Fig. 5, d{RrSd) = eSddr, whence r^ + R=e. ...(4) We obtain another equation by assuming that the wire is so gripped at each end that sections perpendicular to the axis \ ^dr remain perpendicular to the axis after twisting. Indeed, we ^ 1^ have already assumed this in omitting equations for shear- stress in (3). Hence w is independent of r and dw/dz is con- \^d/^ stant over a section for a given wire with a given twist. Let us put dw/dz = h. ' Fig. 6. Further, the load is constant, so that r Wrdr - (5) •0 Substituting in (4) from (3) we obtain ^,du_u_ 2A (g + i8 + y) + 6/xa - 2ju^ ^ ^ dr^^ dr r~ X + 2fji ^ ^^^ By putting u/r = v, we easily find the solution u = Ar^-\-Br-i- Gr-\ (7) where ^ ^ 2A (a 4- ^ + y) 4- 6/.a - 2^^ 8 (A + 2/x) and B and C are arbitrary constants to be determined by the boundary- conditions. If the wire is unstrained in all parts before twisting, the solution applies with the same constants for all parts. In order that u = when r = 0, we must have = 0, so that u = Ar^ + Br (8) When T = a,R = 0, Substituting from (8) in the value of R in (3), and putting J? = when r = a, we get 2 (A + /x) 5 + AA = {A (a + )8 + y) + 2/xa - (4A + 6/x) A} a\ ...(9) From equation (5) we obtain another relation between B and h, when we substitute for u from (8) in W from (3) and integrate from r = to r = a, viz., A5 + (iA + /^) /^ = ttA (a + i8 + y) + J/xy - A^} a\ (10) and from (9) and (10) we can find B and h, 27—2 420 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED, Since ^ is a linear function of a, ^, and y, and each of these is proportional to 6^, h and B are proportional to 6^. Substituting for B in (8), u is also proportional to 6^. The theory, then, gives the paraboHc law for the twisting of a wire initially unstrained both for lengthening and for side-contraction. It also gives the lengthening and side-contraction w/l and u/a for different wires of the same material as proportional to a^. So far the theory does not, of course, give any account of the fact that the wires examined are always unsymmetrical, that the effects always date from a point c, on the counter-clockwise side in the wires examined, c being different for w and u. This want of symmetry imphes initial internal strain, probably, in reahty, very comphcated. Let us examine a simple case in which there is a core, radius a, twisted initially against a sheath, outer radius b, and let the opposing twists be respectively 6^ and 6^. When we put a twist 6 from outside on to the core as a whole the core is twisted through 6 + d^., and the sheath through 6 — Og- For the core and sheath respectively we have Uc = Ar^ + Br, u, = A'r^ + B'r + C'r-\ where ^ is a linear function of a, ^, y, and therefore proportional to {6 + 9^)^ and A' is the same function of a, ^' , y, say, and therefore proportional to To find the constants we have Uc = Ug when r = a, R. = „ r = L and 1 W^rdr^l Wgrdr=--0. These give us four equations to find B, B', C , h of the form (it appears needless to give the detailed work) B' = p2 (0 + e,Y +Q,{e- e,Y, h = p,(e + e,Y+ q, (b -e,f; and, substituting for A' , B', C in u^, and putting r = 6, we get Both h and Wj, are of the form Dd^ + Ee + F, where D does not contain 6^ or 6^. As the parabolas depend only on D, E and F merely giving the position of the vertex, Oc and Og only affect that position. AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL 421 To find that position we may put dhjdd = for the one, dujdJd = for the other, and since h and u are different functions of dc and dg the vertices will be at different points for the two quantities h and n. Taking this simple case as a guide we shall assume that internal strain only affects the position and not the size of the parabola representing the change of linear dimensions on twisting. Hence if we could obtain a wire without internal strain we should have Ln^ = w, where L has the value found in the experiments on the actual, initially strained wire, and we may regard the values uja and wjl for one turn as the values for a wire initially without internal strain. The Values of p and q in the Secondary Stresses. We are now able to find the values of p and q. For the known values of Y and a enable us to find a, ^ and y in equations (1) in terms of p and q for a known twist, which we shall take as one turn, or as 2?? in length I = 160-5 cm. We also know A and /x, since A = (7 7/(1 + 0-) (1 - 2(7) and ^i = n = 7/2 (1 + a). Substituting for A, fx, a, p, and y we can determine A in terms of p and q. Then from equations (9) and (10) we can find B and h in terms of p and q. Equating Aa^ + B to the observed value of u/a (which is negative), and h to the observed value of w/l, we have two linear equations in p and q. The arithmetic is straightforward, though very lengthy, and may be omitted. I have used a slide rule in the calculations. Using the values of the elastic constants Y and a from Table II, and the values of u/a and w/l, on p. 411, I find for wire I p = 1-67 X 1012, q= _ 0-70 x lO^^, so that the force perpendicular to the plane of the figure in Fig. 4 is a tension and not a pressure. The Pressure in the Direction of Propagation in Distortional Waves and the Longitudinal Waves Produced by the Pressure. If we had a train of waves purely distortional, that is, a train in which the strain could be represented by a pure shear e, there would be a pressure in the direction of propagation (Jw + p) e^. But as e varies from point to point in the train, the pressure due to the shear -strain varies, and there must be longitudinal disturbance, longitudinal waves, accompanying the distortional waves. The longitudinal strain implies that the material yields under the 422 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED, pressure, and the pressure will, in general, have a different value from that in a pure shear. Let us represent the distortional train by € = 7; sm ^ (a:; — vt), where v^ = n/p and tj is the ampHtude of the shear. If f is the longitudinal displacement at the point where the shear is €, d^/dx is the elongation of the element about the point. Now if we shear a cube, and remove the pressure (Jw + p) e^, the cube elongates in that direction, and if the dimensions in the two directions at right angles are maintained the same, the removal of the pressure produces elongation y-^ {^n + f) e^, where v = X + 2fjL = K + ^n. This we may term the 'free elongation' in the direction of propagation on the supposition that there is no change of length at right angles to it. The pressure due to the shear falls from its full value {^n + p) e^ to while the elongation increases from to its full value v~^ (Jn + f) e^. When the elongation is d^jdx the pressure remaining is = (\n + f) rf sin^ X ^^ ~ ^^^ "" ^^^1^^- The equation of motion for the longitudinal waves is ^ d^ ^~dx^~ '^^ + ?^) "?' X ^^^ T ^^ ~ "^'^^ "^ vd^i/dx^ an equation similar in form to that for the longitudinal waves which I have attempted to show must accompany light- waves *. 4:77 If we put f = ^ sin -y {x — vt — a), and substitute in the above equation, we find on putting a; = 0, i = 0, that a = 0, and _ (jn + v) rj^X Stt {pv^ - v) ' or if v' is the velocity of free longitudinal waves, since pv'^ = v and v'^ > v^y If we substitute for d^/dx in P, we get P = J (^n + p)7]^ h — cos -Y {^ — vtn — vA ^ cos -y (x — vt), * Roy. Soc. Proc, A, vol. 85, p. 474. [Collected Papers, Art. 29.] AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL 423 We may regard tkis as made up by a steady pressure J (Jw + p) r\^ and a purely periodic pressure, of which the average is zero. If B is the energy per cubic centimetre at any point in the distortional waves, it is half kinetic energy, half strain-energy. The latter is Jw€^, so that the total is wc^ or E = ntf sin^ -y (ic — vii) = \n'rf y- — cos -^ (a; — v^) k Then the average value is E = In-qK If we denote the average pressure by P, p^jro + y^ n If we use the values of n and p found for wire I, we find P = 2-50^. If we put the energy per cubic centimetre in the longitudinal waves as Average energy in longitudinal waves 1 v'^ + v^ {^n + f) 2 2 . -n' Average energy in distortional wav^s 8/o [v"^ — v^)'' so that the ratio is proportional to tf and therefore in any actual waves it is very small. The pressures at right angles to the line of propagation will not produce any disturbance in a wave-front where t] is constant. Round the edges of the wave-front, however, where t] is diminishing as we go outwards, they may have effects, and it appears likely that they may give rise to disturbances propagated sideways. I have much pleasure in recording my hearty thanks to Mr. 6. 0. Harrison, mechanic in the laboratory workshop, for his great help in planning the apparatus used in the experiments described in this paper, for his skill in constructing it, and for his assistance in making the observations. 31. THE CHANGES IN THE LENGTH AND VOLUME OF AN INDIA-RUBBEK COED WHEN TWISTED. [The India-Rvhber Jomnal, October 4, 1913, p. 6.] In some investigations on the way in which pressure might be produced by transverse waves in a solid, analogous to the minute pressure produced by light-waves, the author was led to expect that a wire with a constant load on it would lengthen, when twisted, by an amount proportional to the square of the twist, and he gave at the Winnipeg meeting of the British Association an account of experiments which fully verified the expectation. According to the theor}- used, there should also be accompanying the lengthening a diminution in the radijus, and a description has been published in the Proceedings of the Royal Society* of experiments which show that the diminution exists and follows the same law. The changes are very minute, of the order of a millionth in the length and in the diameter when a steel wire 160 cm. long and 1 mm. diameter is twisted through one turn. The volume is also slightly increased. If instead of allowing the length to increase it had been kept constant by reducing the load, there would with steel have been a slight outstanding increase in the volume. The author thought it might be interesting to look for similar effects in india-rubber. To investigate the lengthening, he used a rubber cord 118 cm. long and 1-2 cm. diameter, of which the upper end was attached to a vertical axis which could be rotated in a bearing. The lower end was attached to a horizontal cross-piece between four stops, which allowed the cross-piece, and therefore the end of the cord, to rise or fall, but prevented rotation. To the cross-piece there was attached the ordinary wheel-barometer device for magnifying up and down motion. There was a lengthening on twisting, somewhat irregular, not proportional to the square of the twist, but increasing rather less rapidly. Two turns of twist gave an average lengthening of 0-088 cm., or about 750 in a million, vastly greater than the lengthening of the steel wire with a similar twist. But, as with steel wire, the lengthening is rather more than proportional to the square of the diameter. A rubber * [Collected Papers, Art. 30.] CHANGES IN AN INDIA-RFBBER CORD WHEN TWISTED 425 cord 1 mm. in diameter, and with the length and twist of the steel, would probably have increased in length by an amount of the same order as that observed with steel. To find whether there was a change in diameter, a cord of the same length and diameter as that used for the lengthening was enclosed in a vertical glass tube with brass ends, the lower end of the cord being attached to the lower brass end, and the upper end to a vertical axis coming into the tube through as close fitting a bearing in the upper brass end as could be made. This axis could be rotated, and so any twist could be put on the cord while it remained of constant length. The tube was filled with water, and as it was provided at one side with a capillary tube which issued through a hole near the top and rose above the upper end, any change in the volume of the rubber on twisting would have been indicated by a rise or fall of the water surface in the capillary, and this was viewed by a measuring microscope. When two turns were put on the rubber, small changes of volume were observed, now one way, now the other, probably due to errors of experiment. But the changes were very small, and the mean change so minute that it appears safe to say that the real change in volume was not so much as one in two millions. It was therefore, if it existed at all, of an order not greater than for the steel wire above described. If the cord had been only 1 mm. in diameter like the steel, and had been of the same length, and had been subjected to the same twist, the change in volume would have been vastly less than in the case of steel. [This appears to be the only published notice of an account of this work which was given at the meeting of the British Association at Birmingham in 1913. Ed.] APPENDIX BY SIE J. LAKMOR ON THE MOMENTUM OF RADIATION. [The following extract from a lecture 'On the Dynamics of Radiation' by Sir Joseph Larmor, read before the Fifth International Congress of Mathematicians at Cambridge in x4.ugust 1912, is inserted here, after consultation with the Author (whose permission was requested), in further elucidation and illustration, chiefly from the side of the electric theory, of Poynting's experiments resting on the momentum of radiation. Ed.] General theory of pressure exerted hy waves. If a perfectly reflecting structure has the property of being able to advance through an elastic medium, the seat of free undulations, without producing disturbance of structure in that medium, then it follows from the principle of energy alone that these waves must exert forces against such a reflector, constituting a pressure equal in intensity at each point to the energy of the waves per unit volume. Of. p. 432, infra. The only hypothesis, required in order to justify this general result, is that the velocity of the undulations in the medium must be independent of their wave-length; viz., the medium is to be non-dispersive, as is the free aether of space. This proposition, being derived solely from consideration of conservation of the energy, must hold good whatever be the character of the mechanism of propagation that is concerned in the waves. But the elucidation of the nature of the pressure of the waves, of its mode of operation, is of course concerned with the constitution of the medium. The way to enlarge ideas on such matters is by study of special cases : and the simplest cases will be the most instructive. Let us consider then transverse undulations travelling on a cord of linear density pQ , which is stretched to tension Tq . Waves of all lengths will travel with the same velocity, namely c = (Tq/pq)^, so that the condition of absence of dispersion is satisfied. A solitary wave of limited length, in its transmission along the cord, deflects each straight portion of it in succession into a curved arc. This process implies increase in length, and therefore increased tension, at first locally. But we adhere for the present to the simplest case, where the cord is inextensible or rather the elastic modulus of extension is indefinitely great. The very beginnings of a local disturbance of tension will then be equalised along the cord with speed practically infinite ; and we may therefore take it that at each instant the tension stands adjusted to be the same (Tq) all along it. The pressure or pull of the undulations at any point is concerned onlv with the component of this tension in the direction of the cord ; this is where 77 is the transverse displacement of the part of the cord at distance x ON THE MOMENTUM OF RADIATION 427 measured along it; thus, up to the second order of approximation, the pull of the cord is T -*^o(S The tension of the cord therefore gives rise statically to an undulation pressure The first of these three equivalent expressions can be interpreted as the potential energy per unit length arising from the gathering up of the extra length in the curved arc of the cord, against the operation of the tension Tq ; the last of them represents the kinetic energy per unit length of the undulations. Thus there is a pressure in the wave, arising from this statical cause, which is at each point equal to half its total energy per unit length. There is the other half of the total pressure still to be accounted for. That part has a very different origin. As the tension is instantaneously adjusted to the same value all along, because the cord is taken to be inextensible, there must be extra mass gathered up into the curved segment which travels along it as the undulation. The mass in this arc is or to the second order is approximately In the element Sx there is extra mass of amount which is carried along with the velocity G of the undulatory propagation. This implies momentum associated with the undulation, and of amount at each point equal to \pqgI^\ per unit length. Another portion of the un- dulation pressure is here revealed, equal to the rate at which the momentum is transmitted past a given point of the cord; this part is represented by IPqG"^ i-^A or \pq (-^ j , and so is equal to the component previously determined* In our case of undulations travelling on a stretched cord, the pressure exerted by the waves arises therefore as to one half from transmitted intrinsic stress and as to the other half from transmitted momentum. The kinetic energy of the cord can be considered either to be energy belonging to the transverse vibration, viz., \\p {-^\ ds, or to be the energy 4?8 APPENDIX BY SIR J. LARMOR of the convected excess of mass moving with the velocity of propagation G*, viz., I 2/° (t^ ) <^^^^ ; for these quantities are equal by virtue of the condition of steady propagation -jl = <^ ^ • On the other hand the momentum that propagates the waves is transverse, of amount p ~ per unit length ; it is the rate of change of this momentum that appears in the equation of propagation dtV'dt) dx\ dx)' But the longitudinal momentum with which we have been here specially concerned is J/jf-—] c per unit length, which is |-p.p-7^. Its ratio to the transverse momentum is very small, being \j^', it is a second-order phenomenon and is not essential to the propagation of the waves. It is in fact a special feature, and there are types of wave-motion in which it does not occur. The criterion for its presence is that the medium must be such that the reflector on which the pressure is exerted can advance through it, sweeping the radiation along in front of it, but not disturbing the structure; possibly intrinsic strain, typified by the tension of the cord, may be an essential feature in the structure of such a medium. If we derive the dynamical equation of propagation along the cord from the Principle of Action 8 | (T - 110 dt = 0, where [i.(5/..a„d>F=|ir„(g) 2 dx. the existence of the pressure of the undulations escapes our analysis. A corre- sponding remark applies to the deduction of the equations of the electro- dynamic field from the Principle of Action f. In that mode of analysis the forces constituting the pressure of radiation are not in evidence throughout the medium ; they are revealed only at the place where the field of the waves affects the electrons belonging to the reflector. Problems connected with the Faraday-Maxwell stress lie deeper; they involve the structure of the medium to a degree which the propagation of disturbance by radiation does not by itself give us means to determine. We therefore proceed to look into that problem more closely. We now postulate Maxwell's statical stress system; also Maxwell's magnetic stress system, which is, presumably, to be taken as of the nature of a kinetic reaction. But when we assert the existence of these stresses, there remain over uncompensated terms in the mechanical forcive on the electrons which * [This specification is fictitious; indeed a factor J has been dropped in its expression just following. There is however actual energy of longitudinal motion; as it belongs to the whole mass of the cord, which moves together, it is very small in amount, its ratio to the energy of transverse vibration being \ {d-rjldc^^. J. L.] t Cf. Larmor, Trans. Camb. Phil. Soc. vol. 18 (1900), p. 318; or Aether and Matter, Chapter vi. ON THE MOMENTUM OF RADIATION 429 may be interpreted as due to a distribution of momentum in the medium*. The pressure of a train of radiation is, on this hypothetical synthesis of stress and momentum, due entirely (p. 431) to the advancing momentum that is absorbed by the surface pressed, for here also the momentum travels with the waves. This is in contrast with the case of the cord analysed above, in which only half of the pressure is due to that momentum. The pressure of radiation against a material body, of amount given by the law specified by Maxwell for free space, is demonstrably included in the Maxwellian scheme of electrodynamics, when that scheme is expanded so as to recognise the electrons with their fields of force as the link of communica- tion between aether and matter. But the illustration of the stretched cord may be held to indicate that it is not yet secure to travel further along with Maxwell, and accept as realities the Faraday-Maxwell stress in the electric field, and the momentum which necessarily accompanies it; it shows that other dynamical possibilities of explanation are not yet excluded. And, viewing the subject from the other side, we recognise how important have been the experimental verifications of the law of pressure of radiation which we owe to Lebedew, too early lost to science, to Nichols and Hull, and to Poynting and Barlows The law of radiation-pressure in free space is not a necessary one for all types of wave-motion ; on the other hand if it had not been verified in fact, the theory of electrons could not have stood without modification. The pressure of radiation, according to Maxwell's law, enters fundamentallv in the Bartoli-Boltzmann deduction of the fourth-power law of connection between total radiation in an enclosure and temperature. Thus in this domain also, when we pass beyond the generalities of thermodynamics, we may expect to find that the kAvs of distribution of natural radiant energy depend on structure which is deeper seated than anything expressed in the Maxwellian equations of propagation. The other definitely secure relation in this field, the displacement-theorem of Wien, involves nothing additional as regards structure, except the principle that operations of compression of a field of natural radiation in free space are reversible. The most pressing present problem of mathematical physics is to ascertain whether we can evade this further investigation into aethereal structure, for purposes of determination of average distribution of radiant energy, by help of the Boltzmann-Planck expansion of thermodynamic principles, which proceeds by comparison of the probabilities of the various distributions of energy that are formally conceivable among the parts of the material system which is its receptacle. Momentum intrinsically associated with Radiation. We will now follow up, after Poynting f, the hypothesis thus implied in modern statements of the Maxwellian formula for electric stress, namely that the pressure of radiation arises wholly from momentum carried along by the waves. Consider an isolated beam of definite length emitted obliquely from a definite area of surface A and absorbed completely by another area B. The * For the extension to the most general case of material media cf. Phil. Trans, vol. 190 (1897).. p. 253. t Cf. Phil. Trans, vol. 202, A (1903). [Collected Papers, Art. 20.] 430 APPENDIX BY SIR J. LARMOR automatic arrangements that are necessary to ensure this operation are easily specified, and need not detain us. In fact by drawing aside an impervious screen from A we can fill a chamber AA' with radiation ; and then closing A and opening ^', it can emerge and travel along to B, where it can be absorbed without other disturbance, by aid of a pair of screens B and B' in like manner. Let the emitting surface ^ be travelling in any direction while the absorber B is at rest. What is emitted by A is wholly gained by B, for the surrounding aether is quiescent both before and after the operation. Also, the system is not subject to external influences; therefore its total momentum must be conserved, what is lost by A being transferred ultimately to B, but by the special hypothesis now under consideration, existing mean- time as momentum in the beam of radiation as it travels across. If v be the component of the velocity of A in the direction of the beam, the duration of emission of the beam from A is (1 — v/o)-^ times the duration of its absorption by the fixed absorber B. Hence the intensity of pressure of a beam of issuing radiation on the moving radiator must be affected by a factor (1 — vJG) multiplying its density of energy; for pressure multiplied by time is the momentum which is transferred unchanged by the beam to the absorber for which v is null. We can verify readily that the pressure of a beam against a moving absorber involves the same factor (1 — vjc). If the aA^ / B' emitter were advancing with the velocity of light this factor would make the pressure vanish, because the emitter would keep permanently in touch with the beam : if the absorber were receding with the velocity of light there would be no pressure on it, because it would just keep ahead of the beam. There seems to be no manner other than these two, by altered intrinsic stress or by convected momentum, in which a beam of limited length can exert pressure while it remains in contact with the obstacle and no longer. In the illustration of the stretched cord the intrinsic stress is transmitted and adjusted by tensional waves which travel with velocity assumed to be prac- tically infinite. If we look closer into the mode of this adjustment of tension, it proves to be by the transmission of longitudinal momentum ; though in order that the pressure may keep in step, the momentum must travel with a much greater velocity, proper to tensional waves. In fact longitudinal stress cannot be altered except by fulfilling itself through the transfer of momentum, and it is merely a question of what speeds of transference come into operation. In the general problem of aethereal propagation, the analogy of the cord suggests that we must be careful to avoid undue restriction of ideas, so as, for example, not to exclude the operation, in a way similar to this adjustriient of tension by longitudinal propagation, of the immense but unknown speed of propagation of gravitation. We shall find presently that the phenomena of absorption lead to another complication. ON THE MOMENTUM OF RADIATION 431 So long, however, as we hold to the theory of Maxwellian electric stress with associated momentum, there can be no doubt as to the validity of Poynting's modification of the pressure formula for a moving reflector, from which he has derived such interesting consequences in cosmical astronomy. To confirm this, we have only to contemplate a beam of radiation of finite length I advancing upon an obstacle A in which it is absorbed. The rear of it moves on with velocity c; hence if the body A is in motion with velocity whose component along the beam is v, the beam will be absorbed or passed on, at any rate removed, in a time l/io — v). But by electron theory the beam possesses a distribution of at any rate quasi-moTonentum. identical with the distribution of its energy, and this has disappeared or has passed on in this time. There must therefore be a thrust on the obstructing body, directed along the beam and equal to e (1 — n/o), where € is the energy of the beam per unit length which is also the distribution of the quasi-moraentum. along the free beam. The back pressure on a radiating body travelling through free space, which is exerted by a given stream of radiation, is by this formula smaller on its front than on its rear; so that if its radiation were unaffected by its motion, the body would be subject to acceleration at the expense of its internal thermal energy. This of course could not be the actual case. The modifying feature is that the intensity of radiation, which corresponds to a given temperature, is greater in front than in rear. The temperature determines the amplitude and velocity of the ionic motions in the radiator, which are the same whether it be at rest or in uniform motion: thus it determines the amplitude of the oscillation in the waves of aethereal radiation that are excited by them and travel out from them. Of this oscillation the intensity of the magnetic field represents the velocity. If the radiator is advancing with velocity v in a direction inclined at an angle d to an emitted ray, the wave-length in free aether is shortened in the ratio 1 cos 9 ; thus the period of the radiation is shortened in the same ratio ; thus the velocity of vibration, which represents the magnetic field, is altered in the inverse ratio, and the energy per unit volume in the square of that ratio, viz., that energy is now € (l — - cos ^) ; and the back pressure it exerts involves a further factor 1 cos 6 owing to the convection ; so that that pressure is e(l — -cos^) , where € is the energy per unit volume of the natural radiation emitted from the body when at rest. The pressural reaction on the source is in fact E'/o, where E' is the actual energy emitted in the ray per unit time. 432 ''-'■ APPENDIX BY SIR J. LARMOR Limitation of the analogy of a stretched cord. In the case of the inextensible stretched cord, the extra length due to the curved arc in the undulation is proportional to the energy of the motion. The loss of energy by absorption would imply slackening of the tension ; and the propositions as to pressure of the waves, including Poynting's modification for a moving source, would not hold good unless there were some device at the fixed ends of the cord for restoring the tension. The hypothesis of convected momentum would imply something of the same kind in electron structure. It is therefore worth while to verify directly that the modified formula for pressure against a moving total reflector holds good in the case of the cord, when there is no >v absorption so that the reflection is total. This analysis will also contain the proof of the generalisa- tion of the formula for radiant pressure that was enunciated on p. 426 sw^m*. Let the wave-train advancing to the reflector and the reflected wave-train be represented respectively by 7^1 = Ai cos Ml {x -f ct), 7]2 = A^ cos mo {x — ct). At the reflector, where x = vt, we must have jvidt^ ji^.,dt; this involves two conditions, — = — - and m^ (c + u) = m.. (c — v). Now the energies per unit length in these two simple wave-trains are ipAj^ and ipA^^; thus the gain of energy per unit time due to the reflection is 8E=(c- v) Jp^2' - {c + ^) \p^i = iM.^{,c-.)(^^-J 1 J 2 O ^ + ^ (c + v) C — V * See Larmor, Brit. Assoc. Report, 1900. [The statement that follows here is too brief, unless reference is made back to the original, especially as a minus sign has fallen out on the right of the third formula below. The reflector consists of a disc with a small hole in it through which the cord passes ; this disc can move along the cord sweeping the waves in front of it while the cord and its tension remain continuous through the hole — the condition of reflection being thus T7j + 7/2=0 when x = vt. In like manner a material perfect reflector sweeps the radiation in front of it, but its molecular constitution is to be such that it allows the aether and its structure to penetrate across it unchanged. For a fuller statement, see Encyclopaedia Britannica, ed. 9 or 10, article 'Radiation.' J. L.] ON THE MOMENTUM OF RADIATION 433 This change of energy must arise as the work of a pressure P exerted by the moving reflector, namely it is Pv ; hence c + V P = lpA,K2 C — V The total energy per unit length, incident and reflected, existing in front of the reflector is E^ + E^ = yA,^ + yA^^ C^ 4- 7j2 Hence finally P - {E^ + E (c - vf 2^ n2 ' ,.2^ becoming equal to the total density of energy E-^ -\- E2, in accordance with Maxwell's law, when v is small. If we assume Poynting's modified formula for the pressure of a wave-train against a travelling obstacle, the value ought to be and the truth of this is readily verified. It may be remarked that, if the relation connecting strain with stress contained quadratic terms, pressural forces such as we are examining would arise in a simple wave-train*. But such a medium would be dispersive, so that a simple train of waves would not travel without change, in contrast to what we know of transmission by the aether of space. The question is then suggested how far a cognate momentum can be regarded as arising from change of aethereal inertia produced by travelling electric strain. It will be represented by inertia attached