Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924012319145 QC 3.M46 V.1 Cornell University Library Scientific papers. 3 1924 012 319 145 .„n,,„2 THE SCIENTIFIC PAPEKS OF JAMES CLEBK MAXWELL. aotiDon: 0. J. CLAY & SONS, CAMBEIDGE UNIVEESITY PEESS WAEEHOUSB, AVE MAEIA LANE. HambriSge: DEIGHTON, BELL AND CO. Efiyjtfl: F. A. BEOCKHAUS. l^m^ ^yi^;fA£&>d2U'^n,'a' t>:. 'AomJ e/'ui^^niHi/c . THE SCIENTIFIC PAPEES OF JAMES CLEEK MAXWELL^ M.A., LL.D. EDIN., D.C.L., F.R.SS. LONDON and EDINBURGH, HONORARY FELLOW OF TRINITY COLLEGE, CAVENDISH PROFESSOR OF EXPERIMENTAL PHYSICS IN THE UNIVERSITY OF CAMBRIDGE. EDITED BY W. D. NIVEN, M.A., F.KS., DIRECTOE OF STUDIES AT THE BOYAL NAVAL COLLEGE, GREENWICH ; FORMERLY FELLOW OF TRINITY COLLEGE. VOL. I. CAMBKIDGE : AT THE UNIVERSITY PRESS. 1890 l^AU Eights reserved^ CAMBBIDGE : PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. TO HIS GRACE THE DUKE OF DEVONSHIEE K.G. CHANCELLOR OF THE UNIVERSITY OF CAMBRIDGE FOUNDER OF THE CAVENDISH LABORATORY THIS MEMORIAL EDITION OF THE SCIENTIFIC PAPERS OF THE FIRST CAVENDISH PROFESSOR OF EXPERIMENTAL PHYSICS IS BY HIS GRACE'S PERMISSION RESPECTFULLY AND GRATEFULLY DEDICATED ^HORTLY after the death of Professor James Clerk Maxwell a Committee was f^ formed, consisting of graduate members of the University of Cambridge and of other friends and admirers, for the purpose of securing a fitting memorial of him. The Committee had in view two objects : to obtain a likeness of Professor Clerk Maxwell, which should be placed in some public building of the Uni- versity ; and to collect and publish his scattered scientific writings, copies of which, so far as the funds at the disposal of the Committee would allow, should be presented to learned Societies and Libraries at home and abroad. It was decided that the likeness should take the form of a marble bust. This was executed by Sir J. E. Boehm, R.A., and is now placed in the apparatus room of the Cavendish Laboratory. In carrying out the second part of their programme the Committee obtained the cordial assistance of the Syndics of the University Press, who willingly consented to pubhsh the present work. At the request of the Syndics, Mr W. D. Niven, M.A., Fellow and Assistant Tutor of Trinity College and now Director of Studies at the Royal Naval College, Greenwich, undertook the duties of Editor. The Committee and the Syndics desire to take this opportunity of acknowledging their obligation to Messrs Adam and Charles Black, Publishers of the ninth Edition of the EncyclopcBdia Britannica, to Messrs Taylor and Francis, Publishers of the London, Edinburgh, and Dublin Philosophical Maga- zine and Journal of Science, to Messrs Macmillan and Co., Publishers of Nature and of the Cambridge and Dublin Mathematical Journal, to Messrs Metcalfe and Co., Publishers of the Quarterly Journal of Pure and Applied Mathematics, and to the Lords of the Committee of Council on Education, Proprietors of the Handbooks of the South Kensington Museum, for their courteous consent to allow the articles which Clerk Maxwell had contributed to these publications to be included in the present work; to Mr Norman Lockyer for the assistance which he rendered in the selection of the articles re-printed from Nature; and their further obligation to Messrs Macmillan and Co. for permission to use in this work the steel engravings of Faraday, Clerk Maxwell, and Helmholtz from the Nature Series of Portraits. VIU Numerous and important Papers, contributed by Clerk Maxwell to the Transactioiis or Proceedings of the Eoyal Societies of London and of Edinburgh, of the Cambridge Philosophical Society, of the Koyal Scottish Society of Arts, and of the London Mathematical Society; Lectures delivered by Clerk Maxwell at the Eoyal Institution of Great Britain pubhshed in its Proceedings; as well as Communications and Addresses to the British Association published in its Reports, are also included in the present work with the sanction of the above mentioned learned bodies. The Essay which gained the Adams Prize for the year 1856 in the University of Cambridge, the introductory Lecture on the Study of Experimental Physics delivered in the Cavendish Laboratory, and the Rede Lecture delivered before the University in 1878, complete this collection of Clerk Maxwell's scientific writings. The diagTams in this work have been re-produced by a photographic process from the original diagrams in Clerk Maxwell's Papers by the Cambridge Scientific Instrument Company. It only remains to add that the footnotes inserted by the Editor are enclosed between square brackets. Cambridge, August, 1890. PEEFACE. /~1LERK MAXWELL'S biography has been written by Professors Lewis Campbell and ^-^ Wm. Garnett with so much skill and appreciation of their subject that nothing further remains to be told. It would therefore be presumption on the part of the editor of his papers to attempt any lengthened narrative of a biographical character. At the same time a memorial edition of an author's collected writings would hardly be complete without some account however slight of his life and works. Accordingly the principal events of Clerk Maxwell's career will be recounted in the following brief sketch, and the reader who wishes to obtain further and more detailed information or to study his character in its social relations may consult the interesting work to which reference has been made. James Clerk Maxwell was descended from the Clerks of Penicuick in Midlothian, a well-known Scottish family whose history can be traced back to the 16th century. The first baronet served in the parliament of Scotland. His eldest son, a man of learning, was a Baron of the Exchequer in Scotland. In later times John Clerk of Eldin a member of the family claimed the credit of having invented a new method of breaking the enemy's line in naval warfare, an invention said to have been adopted by Lord Rodney in the battle which he gained over the French in 1782. Another John Clerk, son of the naval tactitian, was a lawyer of much acumen and became a Lord of the Court of Session. He was distinguished among his Edinburgh contemporaries by his ready and sarcastic wit. The father of the subject of this memoir was John, brother to Sir George Clerk of Penicuick. He adopted the surname of Maxwell on succeeding to an estate in Kirkcud- brightshire which came into the Clerk family through marriage with a Miss Maxwell. It cannot be said that he was possessed of the energy and activity of mind which lead to distinction. He was in truth a somewhat easy-going but shrewd and intelligent man, whose most notable characteristics were his perfect sincerity and extreme benevolence. He took an enlightened interest in mechanical and scientific pursuits and was of an essentially practical turn of mind. On leaving the University he had devoted himself to law and was called to the Scottish Bar. It does not appear however that he met with any great success in that profession. At all events, a quiet life in the country VOL. I. i> X PREFACE. presented so many attractions to his wife as well as to himself that he was easily induced to relinquish his prospects at the bar. He had been mai'ried to Frances, daughter of Robert Cay of N. Charlton, Northumberland, a lady of strong good sense and resolute character. The country house which was their home after they left Edinburgh was designed by John Clerk Maxwell himself and was built on his estate. The house, which was named Glenlair, was surrounded by fine scenery, of which the water of Urr with its rocky and wooded banks formed the principal charm. James was born at Edinburgh on the 13th of June, 1831, but it was at Glenlair that the greater part of his childhood was passed. In that pleasant spot under healthful influences of all kinds the child developed into a hardy and courageous boy. Not precociously clever at books he was yet not without some signs of future intellectual strength, being remarkable for a spirit of inquiry into the causes and connections of the phenomena around him. It was remembered afterwards when he had become distinguished, that the questions he put as a child shewed an amount of thoughtfulness which for his years was very unusual. At the age of ten, James, who had lost his mother, was placed under the charge of relatives in Edinburgh that he might attend the Edinburgh Academy. A charming account of his school days is given in the narrative of Professor Campbell who was Maxwell's schoolfellow and in after life an intimate friend and constant correspondent. The child is father to the man, and those who were privileged to know the man Maxwell will easily recognise Mr Campbell's picture of the boy on his first appearance at school, — the home- made garments more serviceable than fashionable, the rustic speech and curiously quaint but often humorous manner of conveying his meaning, his bewilderment on first undergoing the routine of schoolwork, and his Spartan conduct under various trials at the hands of his schoolfellows. They will further feel how accurate is the sketch of the boy become accustomed to his surroundings and rapidly assuming the place at school to which his mental powers entitled him, while his superfluous energy finds vent privately in carrying out mechanical contrivances and geometrical constructions, in reading and even trying his hand at composing ballads, and in sending to his father letters richly embellished with gi'otesquely elaborate borders and drawings. An event of his school-days, worth recording, was his invention of a mechanical method of drawing certain classes of Ovals. An account of this method was printed in the Proceedings of the Royal Society of Edinburgh and forms the first of his writings collected in the present work. The subject was introduced to the notice of the Society by the celebrated Professor James Forbes, who from the first took the greatest possible interest in Maxwell's progress. Professor Tait, another schoolfellow, mentions that at the time when the paper on the Ovals was written. Maxwell had received no instruction in Mathematics beyond a little Euclid and Algebra. PREFACE. XI In 1847 Maxwell entered the University of Edinburgh where he remained for three sessions. He attended the lectures of Kelland in Mathematics, Forbes in Natural Philosophy, Gregory in Chemistry, Sir W. Hamilton in Mental Philosophy, Wilson (Christopher North) in Moral Philosophy. The lectures of Sir W. Hamilton made a strong impression upon him, m stimulating the love of speculation to which his mind was prone, but, as might have been expected, it was the Professor of Natural Philosophy who obtained the chief share of his devotion. The enthusiasm which so distinguished a man as Forbes naturally inspired in young and ardent disciples, evoked a feeling of personal attachment, and the Professor, on his part, took special interest in his pupil and gave to him the altogether unusual privilege of working with his fine apparatus. What was the nature of this experimental work we may conjecture from a perusal of his paper on Elastic Solids, written at that time, in which he describes some experiments made with the view of verifying the deductions of his theory in its application to Optics. Maxwell would seem to have been led to the study of this subject by the following cir- cumstance. He was taken by his uncle John Cay to see William Nicol, the inventor of the polarising prism which bears his name, and was shewn by Nicol the colours of unan- nealed glass in the polariscope. This incited Maxwell to study the laws of polarised light and to construct a rough polariscope in which the polariser and analyser were simple glass reflectors. By means of this instrument he was able to obtain the colour bands of unannealed glass. These he copied on paper in water colours and sent to Nicol. It is gratifying to find that this spirited attempt at experimenting on the part of a mere boy was duly appreciated by Nicol, who at once encouraged and delighted him by a present of a couple of his prisms. The paper alluded to, viz. that entitled " On the Equilibrium of Elastic Solids," was read to the Royal Society of Edinburgh in 1850. It forms the third paper which Maxwell addressed to that Society. The first in 1846 on Ovals has been already mentioned. The second, under the title "The Theory of Rolling Curves," was presented by Kelland in 1849. It is obvious that a youth of nineteen years who had been capable of these efforts must have been gifted with rare originality and with great power of sustained exertion. But his singular self-concentration led him into habits of solitude and seclusion, the tendency of which was to confirm his peculiarities of speech and of manner. He was shy and reserved with strangers, and his utterances were often obscure both in substance and in his manner of expressing himself, so many remote and unexpected allusions perpetually obtruding themselves. Though really most sociable and even fond of society he was essentially reticent and reserved. Mr Campbell thinks it is to be regretted that Maxwell did not begin his Cambridge career earlier for the sake of the social intercourse which he would have found it difiScult to avoid there. It is a question, however, whether in losing the opportunity of using Professor Forbes' apparatus he would not thereby have lost what was perhaps the most valuable part of his early scientific training. 62 XU PREFACE. It was originally intended that Maxwell should follow his father's profession of advocate, but this intention was abandoned as soon as it became obvious that his tastes lay m a direction so decidedly scientific. It was at length determined to send him to Cambridge and accordingly in October, 1850, he commenced residence in Peterhouse, where however he resided during the Michaelmas Term only. On December 14 of the same year he migrated to Trinity College. It may readily be supposed that his preparatory training for the Cambridge course was far removed from the ordinary type. There had indeed for some time been practically no restraint upon his plan of study and his mind had been allowed to follow its natural bent towards science, though not to an extent so absorbing as to withdraw him from other pursuits. Though he was not a sportsman, — indeed sport so called was always repugnant to him — he was yet exceedingly fond of a country life. He was a good horseman and a good swimmer. Whence however he derived his chief enjoyment may be gathered from the account which Mr Campbell gives of the zest with which he quoted on one occasion the lines of Burns which describe the poet finding inspiration while wandering along the banks of a stream in the free indulgence of his fancies. Maxwell was not only a lover of poetry but himself a poet, as the fine pieces gathered together by Mr Campbell abundantly testify. He saw however that his true calling was Science and never regarded these poetical efforts as other than mere pastime. Devotion to science, already stimulated by successful endeavour, a tendency to ponder over philosophical problems and an attachment to English literature, particularly to English |)oetry, — these tastes, implanted in a mind of singular strength and purity, may be said to have been the endowments with which young Maxwell began his Cambridge career. Besides this, his scientific reading, as we may gather from his papers to the Royal Society of Edinburgh referred to above, was already extensive and varied. He brought with him, says Professor Tait, a mass of knowledge which was really immense for so young a man but in a state of disorder appalling to his methodical private tutor. Maxwell's undergraduate career was not marked by any specially notable feature. His private speculations had in some measure to be laid aside in favour of more systematic study. Yet his mind was steadily ripening for the work of his later years. Among those with whom he was brought into daily contact by his position, as a Scholar of Trinity College, were some of the brightest and most cultivated young men in the University. In the genial fellowship of the Scholars' table Maxwell's kindly humour found ready play, while in the more select coterie of the Apostle Club, formed for mutual cultivation, he found a field for the exercise of his love of speculation in essays on subjects beyond the lines of the ordinary University course. The composition of these essays doubtless laid the foundation of that literary finish which is one of the characteristics of Maxwell's scientific writings. His biographers have preserved several extracts on a variety of subjects chiefly of a specu- lative character. They are remarkable mainly for the weight of thought contained in them but occasionally also for smart epigrams and for a vein of dry and sarcastic humour. PREFACE. Xlll These glimpses into Maxwell's character may prepare us to believe that, with all his shyness, he was not without confidence in his own powers, as also appears from the account which was given by the late Master of Trinity College, Dr Thompson, who was Tutor when Maxwell personally applied to him for permission to migrate to that College. He appeared to be a shy and diffident youth, but presently surprised Dr Thompson by producing a bundle of papers, doubtless copies of those we have already mentioned, remarking " Perhaps these may shew you that I am not unfit to enter at your College." He became a pupil of the celebrated William Hopkins of Peterhouse, under whom his course of study became more systematic. One striking characteristic was remarked by his contemporaries. Whenever the subject admitted of it he had recourse to diagi-ams, though his fellow students might solve the question more easily by a train of analysis. Many illustrations of this manner of proceeding might be taken from his -writings, but in truth it was only one phase of his mental attitude towards scientific questions, which led him to proceed from one distinct idea to another instead of trusting to symbols and equations. Maxwell's published contributions to Mathematical Science during his undergraduate career were few and of no great importance. He found time however to cany his investigations into regions outside the prescribed Cambridge course. At the lectures of Professor Stokes* he was regular in his attendance. Indeed it appears from the paper on Elastic Solids, mentioned above, that he was acquainted with some of the writings of Stokes before he entered Cambridge. Before 1850, Stokes had published some of his most important contri- butions to Hydromechanics and Optics ; and Sir W. Thomson, who was nine years' Maxwell's senior in University standing, had, among other remarkable investigations, called special attention to the mathematical analogy between Heat-conduction and Statical Electricity. There is no doubt that these authors as well as Faraday, of whose experimental researches he had made a careful study, exercised a powerful directive influence on his mind. In January, 1854, Maxwell's undergraduate career closed. He was second wrangler, but shared with Dr Routh, who was senior wrangler, the honours of the First Smith's Prize. In due course he was elected Fellow of Trinity and placed on the staff of College Lecturers. No sooner was he released from the restraints imposed by the Trinity Fellowship Examination than he plunged headlong into original work. There were several questions he was anxious to deal with, and first of all he completed an investigation on the Trans- formation of Surfaces by Bending, a purely geometrical problem. This memoir he presented to the Cambridge Philosophical Society in the following March. At this period he also set about an enquiry into the quantitative measurement of mixtures of colours and the causes of colour-blindness. During his undergraduateship he had, as we have seen, found time for the study of Electricity. This had already borne fruit and now resulted in the first of his important memoirs on that subject, — the memoir on Faraday's Lines of Force. * Now Sir George Gabriel Stokes, Bart., M.P. for the University. XIV PREFACE. The number and importance of his papers, published in 1855—6, bear witness to his assiduity during this period. With these labours, and in the preparation of his College lectures, on which he entered with much enthusiasm, his mind was fully occujjied and the work was congenial. He had formed a number of valued friendships, and he had a variety of interests, scientific and literary, attaching him to the University. Nevertheless, when the chair of Natural Philosophy in Marischal College, Aberdeen, fell vacant, Maxwell became a candidate. This step was probably taken in deference to his father's wishes, as the long summer vacation of the Scottish College would enable him to reside with his father at Glenlair for half the year continuously. He obtained the professorship, but unhappily the kind intentions which prompted him to apply for it were frustrated by the death of his father, which took place in April, 1856. It is doubtful whether the change from the Trinity lectureship to the Aberdeen professorship was altogether prudent. The advantages were the possession of a laboratory and the long uninterrupted summer vacation. But the labour of drilling classes composed chiefly of comparatively young and untrained lads, in the elements of mechanics and physics, was not the work for which Maxwell was specially fitted. On the other hand, in a large college like Trinity there could not fail to have been among its undergraduate members, some of the most promising young mathematicians of the University, capable of appreciating his original genius and immense knowledge, by instructing whom he would himself have derived ad- vantage. In 1856 Maxwell entered upon his duties as Professor of Natural Philosophy at Marischal College, and two years afterwards he married Katharine Mary Dewar, daughter of the Principal of the College. He in consequence ceased to be a Fellow of Trinity College, but was afterwards elected an honorary Fellow, at the same time as Professor Cayley. During the years 1856 — 60 he was still actively employed upon the subject of colour sensation, to which he contributed a new method of measurement in the ingenious instru- ment known as the colour-box. The most serious demands upon his powers and upon his time were made by his investigations on the Stability of Saturn's Rings. This was the subject chosen by the Examiners for the Adams Prize Essay to be adjudged in 1857, and was advertised in the following terms : — "The Problem may be treated on the supposition that the system of Rings is exactly or very approximately concentric with Saturn and symmetrically disposed about the plane of his equator and diflferent hypotheses may be made respecting the physical constitution of the Rings. It may be supposed (1) that they are rigid; (2) that they are fluid and in part aeriform ; (8) that they consist of masses of matter not materially coherent. The question will be considered to be answered by ascertaining on these hypotheses severally whether the conditions of mechanical stability are satisfied by the mutual attractions and motions of the Planet and the Rings." PREFACE. XV "It is desirable that an attempt should also be made to determine on which of the above hypotheses the appearances both of the bright rings and the recently discovered dark ring may be most satisfactorily explained; and to indicate any causes to which a change of form such as is supposed from a comparison of modern with the earlier observations to have taken place, may be attributed." It is sufficient to mention here that Maxwell bestowed an immense amount of labour in working out the theory as proposed, and that he arrived at the conclusion that "the only system of rings which can exist is one composed of an indefinite number of unconnected particles revolving round the planet with different velocities according to their respective distances. These particles may be arranged in a series of narrow rings, or they may move about through each other irregularly. In the first case the destruction of the system will be very slow, in the second case it will be more rapid, but there may be a tendency towards an arrangement in narrow rings which may retard the process." Part of the work, dealing with the oscillatory waves set up in a ring of satellites, was illustrated by an ingenious mechanical contrivance which was greatly admired when exhibited before the Royal Society of Edinburgh. This essay, besides securing the prize, obtained for its author great credit among scientific men. It was characterized by Sir George Airy as one of the most remarkable applications of Mathematics to Physics that he had ever seen. The suggestion has been made that it was the irregular motions of the particles which compose the Rings of Saturn resulting on the whole in apparent regularity and uni- formity, which led Maxwell to the investigation of the Kinetic Theory of Gases, his first contribution to which was read to the British Association in 1859. This is not unlikely, but it must also be borne in mind that Bernoulli's Theory had recently been revived by Herapath, Joule and Clausius whose writings may have drawn Maxwell's attention to the subject. In 1860 King's College and Marischal College were joined together as one institution, now known as the University of Aberdeen. The new chair of Natural Philosophy thus created was filled up by the appointment of David Thomson, formerly Professor at King's College and Maxwell's senior. Professor Thomson, though not comparable to Maxwell as a physicist, was nevertheless a remarkable man. He was distinguished by singular force of character and great administrative faculty and he had been prominent in bringing about the fusion of the Colleges. He was also an admirable lecturer and teacher and had done much to raise the standard of scientific education in the north of Scotland. Thus the choice made by the Commissioners, though almost inevitable, had the effect of making it appear that Maxwell failed as a teacher. There seems however to be no evidence to support such an inference. On the contrary, if we may judge from the number of voluntary students attending his classes in his last College session, he would seem to have been as popular as a professor as he was personally estimable. XVI PREFACE. This is also borne out by the fact that he was soon afterwards elected Professor of Natural Philosophy and Astronomy in King's College, London. The new appointment had the advantage of bringing him much more into contact with men in his own department of science, especially with Faraday, with whose electrical work his own was so intimately connected. In 1862—63 he took a prominent part in the experiments organised by a Committee of the British Association for the determination of electrical resistance in absolute measure and for placing electrical measurements on a satisfactory basis. In the experiments which were conducted in the laboratory of King's College upon a plan due to Sir W. Thomson, two long series of measurements were taken in successive years. In the first year, the working members were Maxwell, Balfour Stewart and Fleeming Jenkm ; in the second, Charles Hockin took the place of Balfour Stewart. The work of this Committee was communicated in the form of reports to the British Association and was afterwards republished in one volume by Fleeming Jenkin. Maxwell was a professor in King's College from 1860 to 1865, and this period of his life is distinguished by the production of his most important papers. The second memoir on Colours made its appearance in 1860. In the same year his first papers on the Kinetic Theory of Gases were published. In 1861 came his papers on Physical Lines of Force and in 1864 his greatest memoir on Electricity,— a Dynamical Theory of the Electro- magnetic Field. He must have been occupied with the Dynamical Theory of Gases in 1865, as two important papers appeared in the following year, first the Bakerian lecture on the Viscosity of Gases, and next the memoir on the Dynamical Theory of Gases. The mental strain involved in the production of so much valuable work, combined with the duties of his professorship which required his attention during nine months of the year, seems to have influenced him in a resolution which in 1865 he at length adopted of resigning his chair and retiring to his country seat. Shortly after this he had a severe illness. On his recovery he continued his work on the Dynamical Theory of Gases, to which reference has just been made. For the next few years he led a quiet and secluded life at Glenlair, varied by annual visits to London, attendances at the British Association meetings and by a tour in Italy in 1867. He was also Moderator or Examiner in the Mathematical Tripos at Cambridge on several occasions, oflfices which entailed a few weeks' residence at the University in winter. His chief employment during those years was the preparation of his now celebrated treatise on Electricity and Magnetism which, however, was not published till 1873. He also wrote a treatise on Heat which was published in 1871. In 1871 Maxwell was, with some reluctance, induced to quit his retreat in the country and to enter upon a new career. The University of Cambridge had recently resolved to found a professorship of physical science, especially for the cultivation and teaching of the subjects of Heat, Electricity and Magnetism. In furtherance of this object her Chancellor, the Duke of Devonshire, had most generously undertaken to build a laboratory and furnish it with the necessary apparatus. Maxwell was invited to fill the PREFACE. XVll new chair thus formed and to superintend the erection of the laboratory. In October, 1871, he delivered his inaugural lecture. The Cavendish Laboratory, so called after its founder, the present venerable chief of the family which produced the great physicist of the same name, was not completed for practical work until 1874. In June of that year it was formally presented to the University by the Chancellor. The building itself and the fittings of the several rooms were admirably contrived mainly by Maxwell himself, but the stock of apparatus was smaller than accorded with the generous intentions of the Chancellor. This defect must be attributed to the anxiety of the Professor to procure only instruments by the best makers and with such improvements as he could himself suggest. Such a defect therefore required time for its removal and afterwards in great measure disappeared, apparatus being constantly added to the stock as occasion demanded. One of the chief tasks which Maxwell undertook was that of superintending and directing the energies of such young Bachelors of Arts as became his pupils after having acquired good positions in the University examinations. Several pupils, who have since acquired distinction, carried out valuable experiments under the guidance of the Professor. It must be admitted, however, that the numbers were at first small, but perhaps this was only to be expected from the traditions of so many years. The Professor was singularly kind and helpful to these pupils. He would hold long conversations with them, opening up to them the stores of his mind, giving them hints as to what they might try and what avoid, and was always ready with some ingenious remedy for the experimental troubles which beset them. These conversations, always delightful and instructive, were, according to the account of one of his pupils, a liberal education in themselves, and were repaid in the minds of the pupils by a grateful affection rarely accorded to any teacher. Besides discharging the duties of his chair. Maxwell took an active part in conducting the general business of the University and more particularly in regulating the courses of study in Mathematics and Physics. For some years previous to 1866 when Maxwell returned to Cambridge as Moderator in the Mathematical Tripos, the studies in the University had lost touch with the gi-eat scientific movements going on outside her walls. It was said that some of the subjects most in vogue had but little interest for the present generation, and loud complaints began to be heard that while such branches of knowledge as Heat, Electricity and Magnetism, were left out of the Tripos examination, the candidates were wasting theii' time and energy upon mathematical trifles barren of scientific interest and of practical results. Into the movement for reform Maxwell entered warmly. By his questions in 1866 and subsequent years he infused new life into the examination ; he took an active part in drafting the new scheme introduced in 187.3 ; but most of all by his writings he exerted a powerful influence on the younger members of the University, and was largely instrumental in bringing about the change which has been now effected. VOL. I. C Xviii PREFACE. In the first few years at Cambridge Maxwell was busy in giving the final touches to his great work on Electricity and Magnetism and in passing it through the press. This work was published in 1873, and it seems to have occupied the most of his attention for the two previous years, as the few papers published by him during that period relate chiefly to subjects forming part of the contents. After this publication his contributions to scientific journals became more numerous, those on the Dynamical Theory of Gases being perhaps the most important. He also wrote a great many short articles and reviews which made their appearance in Nature and the Encyclopcedia Britannica. Some of these essays are charming expositions of scientific subjects, some are general criticisms of the works of contemporary writers and others are brief and appreciative biographies of fellow workers in the same fields of research. An undertaking in which he was long engaged and which, though it proved exceedingly interesting, entailed much labour, was the editing of the "Electrical Kesearches" of the Hon. Henry Cavendish. This work, published in 1879, has had the effect of increasing the reputation of Cavendish, disclosing as it does the unsuspected advances which that acute physicist had made in the Theory of Electricity, especially in the measurement of electrical quantities. The work is enriched by a variety of valuable notes in which Cavendish's views and results are examined by the light of modern theory and methods. Especially valuable are the methods applied to the determination of the electrical capacities of con- ductors and condensers, a subject in which Cavendish himself shewed considerable skill both of a mathematical and experimental character. The importance of the task undertaken by Maxwell in connection with Cavendish's papers will be understood from the following extract from his introduction to them. "It is somewhat difficult to account for the fact that though Cavendish had prepared a complete description of his experiments on the charges of bodies, and had even taken the trouble to write out a fair copy, and though all this seems to have been done before 1774 and he continued to make experiments in Electricity till 1781 and lived on till 1810, he kept his manuscript by him and never published it.'' "Cavendish cared more for investigation than for publication. He would under- take the most laborious researches in order to clear up a difficulty which no one but himself could appreciate or was even aware of, and we cannot doubt that the result of his enquiries, when successful, gave him a certain degree of satisfaction. But it did not excite in him that desire to communicate the discovery to others which in the case of ordinary men of science, generally ensures the publication of their results. How completely these researches of Cavendish remained unknown to other men of science is shewn by the external history of electricity." It will probably be thought a matter of some difficulty to place oneself in the position of a physicist of a century ago and to ascertain the exact bearing of his experiments. But Maxwell entered upon this undertaking with the utmost enthusiasm and PREFACE. XIX succeeded in completely identifying himself with Cavendish's methods. He shewed that Cavendish had really anticipated several of the discoveries in electrical science which have been made since his time. Cavendish was the first to form the conception of and to measure Electrostatic Capacity and Specific Inductive Capacity; he also anticipated Ohm's law. The Cavendish papers were no sooner disposed of than Maxwell set about preparing a new edition of his work on Electricity and Magnetism ; but unhappily in the summer term of 1879 his health gave way. Hopes were however entertained that when he returned to the bracing air of his country home he would soon recover. But he lingered through the summer months with no signs of improvement and his spii'its gradually sank. He was finally informed by his old fellow-student, Professor Sanders, that he could not live more than a few weeks. As a last resort he was brought back to Cambridge in October that he might be under the charge of his favourite physician, Dr Paget*. Nothing however could be done for his malady, and, after a painful illness, he died on the 5th of November, 1879, in his 49th year. Maxwell was thus cut off in the prime of his powers, and at a time when the depart- ments of science, which he had contributed so much to develop, were being every day extended by fresh discoveries. His death was deplored as an irreparable loss to science and to the University, in which his amiable disposition was as universally esteemed as his genius was admu'ed. It is not intended in this preface to enter at length into a discussion of the relation which Maxwell's work bears historically to that of his predecessors, or to attempt to estimate the effect which it has had on the scientific thought' of the present day. In some of his papers he has given more than usually copious references to the works of those by whom he had been influenced; and in his later papers, especially those of a more popular nature which appeared in the Encyclopcedia Britannica, he has given full historical outlines of some of the most prominent fields in which he laboured. Nor does it appear to the present editor that the time has yet arrived when the quickening influence of Maxwell's mind on modern scientific thought can be duly estimated. He therefore proposes to himself the duty of recalling briefly, according to subjects, the most important speculations in which Maxwell engaged. His works have been arranged as far as possible in chronological order but they fall naturally under a few leading heads; and perhaps we shall not be far wrong if we place first in importance his work in Electricity. His first paper on this subject bearing the title "On Faraday's Lines of Force" was read before the Cambridge Philosophical Society on Dec. 11th, 1855. He had been previously attracted by Faraday's method of expressing electrical laws, and he here set before himself the task of shewing that the ideas which had guided Faraday's researches were not incon- sistent with the mathematical formulae in which Poisson and others had cast the laws of * Now Sir George Edward Paget, K.C.B. c2 XX PREFACE. Electricity. His object, he says, is to find a physical analogy which shall help the mind to grasp the results of previous investigations "without being committed to any theory founded on the physical science from which that conception is borrowed, so that it is neither drawn aside from the subject in the pursuit of anal)rtical subtleties nor carried beyond the truth by a favorite hypothesis." The laws of electricity are therefore compared with the properties of an incompressible fluid the motion of which is retarded by a force proportional to the velocity, and the fluid is supposed to possess no inertia. He shews the analogy which the lines of flow of such a fluid would have with the lines of force, and deduces not merely the laws of Statical Electricity in a single medium but also a method of representing what takes place when the action passes from one dielectric into another. In the latter part of the paper he proceeds to consider the phenomena of Electro- magnetism and shews how the laws discovered by Ampere lead to conclusions identical with those of Faraday. In this paper three expressions are introduced which he identifies with the components of Faraday's electrotonic state, though the author admits that he has not been able to frame a physical theory which would give a clear mental picture of the various connections expressed by the equations. Altogether this paper is most important for the light which it throws on the principles which guided Maxwell at the outset of his electrical work. The idea of the electrotonic state had already taken a firm hold of his mind though as yet he had formed no physical explanation of it. In the paper "On Physical Lines of Force" printed in the Philosophical Magazine, Vol. XXI. he resumes his speculations. He explains that in his former paper he had found the geometrical significance of the Electrotonic state but that he now proposes "to examine magnetic phenomena from a mechanical point of view." Accordingly he propounds his remarkable speculation as to the magnetic field being occupied by molecular vortices, the axes of which coincide with the lines of force. The cells within which these vortices rotate are supposed to be separated by layers of particles which serve the double purpose of transmitting motion from one cell to another and by their own motions constituting an electric current. This theory, the parent of several working models which have been devised to represent the motions of the dielectric, is remarkable for the detail with which it is worked out and made to explain the various laws not only of magnetic and electromagnetic action, but also the various forms of electrostatic action. As Maxwell subsequently gave a more general theory of the Electromagnetic Field, it may be inferred that he did not desire it to be supposed that he adhered to the views set forth in this paper in every particular- but there is no doubt that in some of its main features, especially the existence of rotation round the lines of magnetic force, it expressed his permanent convictions. In his treatise on "Electricity and Magnetism," Vol. Ii. p. 416, (2nd edition 427) after quoting from Sir W. Thomson on the explanation of the magnetic rotation of the plane of the polarisation of light, he goes on to say of the present paper, PREFACE. XXI " A theory of molecular vortices which I worked out at considerable length was published in the Phil. Mag. for March, April and May, 1861, Jan. and Feb. 1862." "I think we have good evidence for the opinion that some phenomenon of rotation is going on in the magnetic field, that this rotation is performed by a great mimber of very small portions of matter, each rotating on its own axis, that axis being parallel to the direction of the magnetic force, and that the rotations of these various vortices are made to depend on one another by means of some mechanism between them." "The attempt which I then made to imagine a working model of this mechanism must be taken for no more than it really is, a demonstration that mechanism may be imagined capable of producing a connection mechanically equivalent to the actual connection of the parts of the Electromagnetic Field." This paper is also important as containing the first hint of the Electromagnetic Theory of Light which was to be more fully developed afterwards in his third great memoir " On the Dynamical Theory of the Electromagnetic Field." This memoir, which was presented to the Royal Society on the 27th October, 1864, contains Maxwell's mature thoughts on a subject which had so long occupied his mind. It was afterwards reproduced in his Treatise with trifling modifications in the treatment of its parts, but without substantial changes in its main features. In this paper Maxwell reverses the mode of treating electrical phenomena adopted by previous mathematical writers; for while they had sought to build up the laws of the subject by starting from the principles discovered by Ampere, and deducing the induction of currents from the conservation of energy. Maxwell adopts the method of first arriving at the laws of induction and then deducing the mechanical attractions and repulsions. After recalling the general phenomena of the mutual action of currents and magnets and the induction produced in a circuit by any variation of the strength of the field in which it lies, the propagation of light through a luminiferous medium, the properties of dielectrics and other phenomena which point to a medium capable of transmitting force and motion, he proceeds. — " Thus then we are led to the conception of a complicated mechanism capable of a vast variety of motions but at the same time so connected that the motion of one part depends, according to definite relations, on the motion of other parts, these motions being communicated by forces arising from the relative displacement of theii- connected parts, in virtue of then- elasticity. Such a mechanism must be subject to the laws of Dynamics." On applying dynamical principles to such a connected system he attains certain general propositions which, on being compared with the laws of induced currents, enable him to identify certain features of the mechanism with properties of currents. The induction of currents and their electromagnetic attraction are thus explained and connected. XXll PREFACE. In a subsequent part of the memoir he proceeds to establish from these premises the general equations of the Field and obtains the usual formula? for the mechanical force on cuiTsnts, magnets and bodies possessing an electrostatic charge. He also returns to and elaborates more fully the electromagnetic Theory of Light. His equations shew that dielectrics can transmit only transverse vibrations, the speed of propagation of which in air as deduced from electrical data comes out practically identical with the known velocity of light. For other dielectrics the index of refraction is equal to the square root of the product of the specific inductive capacity by the coefficient of magnetic induction, which last factor is for most bodies practically unity. Various comparisons have been made with the view of testing this deduction. In the case of paraffin wax and some of the hydrocarbons, theory and experiment agree, but thi^ is not the case with glass and some other substances. Maxwell has also applied his theory to media which are not perfect insulators, and finds an expression for the loss of light in passing through a stratum of given thickness. He remarks in confirmation of his result that most good conductors are opaque while insulators are transparent, but he also adds that electrolytes which transmit a current freely are often transparent, while a piece of gold leaf whose resistance was determined by Mr Hockin allowed far too great an amount of light to pass. He observes however that it is possible "there is less loss of energy when the electromotive forces are reversed with the rapidity of light than when they act for sensible times as in our experiments." A similar explanation may be given of the discordance between the calculated and observed values of the specific inductive capacity. Prof J. J. Thomson in the Proceedings of the Royal Society, Vol. 46, has described an experiment by which he has obtained the specific inductive capacities of various dielectrics when acted on by alternating electric forces whose frequency is 25,000,000 per second. He finds that under these conditions the specific inductive capacity of glass is very nearly the same as the square of the refractive index, and very much less than the value for slow rates of reversals. In illustration of these remarks may be quoted the observations of Prof Hertz who has shewn that vulcanite and pitch are transparent for waves, whose periods of vibration are about three hundred millionths of a second. The investigations of Hertz have shewn that electro-dynamic radiations are transmitted in waves with a velocity, which, if not equal to, is comparable with that of light, and have thus given conclusive proof that a satisfactory theory of Electricity must take into account in some form or other the action of the dielectric. But this does not prove that Maxwell's theory is to be accepted in every particular. A peculiarity of his theory is, as he himself points out in his treatise, that the variation of the electric displacement is to be treated as part of the cun-ent as well as the curi-ent of conduction, and that it is the total amount due to the sum of these which flows as if electricity were an incompressible fluid, and which determines external electrodynamic actions. In this respect it differs from the theory of Helmholtz which also takes into account the action of the dielectric. Professor J. J. Thomson in his Review of Electric Theories has entered into a full discussion of the points at issue PREFACE. XXlll between the two above mentioned theories, and the reader is referred to his paper for further information*. Maxwell in the memoir before us has also applied his theory to the passage of light through crystals, and gets rid at once of the wave of normal vibrations which has hitherto proved the stumbling block in other theories of light. The electromagnetic Theory of Light has received numerous developments at the hands of Lord Rayleigh, Mr Glazebrook, Professor J. J. Thomson and others. These volumes also contain various shorter papers on Electrical Science, though perhaps the most complete record of Maxwell's work in this department is to be found in his Treatise on Electricity and Magnetism in which they were afterwards embodied. Another series of papers of hardly less importance than those on Electricity are the various memoirs on the Dynamical Theory of Gases. The idea that the properties of matter might be explained by the motions and impacts of their ultimate atoms is as old as the time of the Greeks, and Maxwell has given in his paper on "Atoms" a full sketch of the ancient controversies to which it gave rise. The mathematical difficulties of the speculation however were so great that it made little real progress till it was taken up by Clausius and shortly afterwards by Maxwell. The first paper by Maxwell on the subject is entitled "Illustrations of the Dynamical Theory of Gases" and was published in the Philosophical Magazine for January and July, 1860, having been read at a meeting of the British Association of the previous year. Although the methods developed in this paper were afterwards abandoned for others, the paper itself is most interesting, as it indicates clearly the problems in the theory which Maxwell proposed to himself for solution, and so far contains the germs of much that was treated of in his next memoir. It is also epoch-making, inasmuch as it for the first time enumerates various propositions which are characteristic of Maxwell's work in this subject. It contains the first statement of the distribution of velo- cities according to the law of errors. It also foreshadows the theorem that when two gases are in thermal equilibrium the mean kinetic energy of the molecules of each system is the same; and for the first time the question of the viscosity of gases is treated dynamically. In his great memoir "On the Dynamical Theory of Gases" published in the Philo- sophical Transactions of the Royal Society and read before the Society in May, 1866, he returns to this subject and lays down for the first time the general dynamical methods appropriate for its treatment. Though to some extent the same ground is traversed as in his former paper, the methods are widely different. He here abandons his former hypothesis that the molecules are hard elastic spheres, and supposes them to repel each other with forces varying inversely as the fifth power of the distance. His chief reason for assuming this law of action appears to be that it simplifies considerably the calculation of the collisions between the molecules, and it leads to the conclusion that the coefficient of viscosity is directly proportional to the absolute temperature. He himself undertook an experimental enquiry for the purpose of verifying this conclusion, and, in his paper on the Viscosity of Gases, he satisfied himself of its correctness. A re-examination of the numerical * British Association Report, 1885. XXIV PREFACE. reductions made in the course of his work discloses however an inaccuracy which materially affects the values of the coefficient of viscosity obtained. Subsequent experiments also seem to shew that the concise relation he endeavoured to establish is by no means so near the truth as he supposed, and it is more than doubtful whether the action between two molecules can be represented by any law of so simple a character. In the same memoir he gives a fresh demonstration of the law of distribution of velocities, but though the method is of permanent value, it labours under the defect of assuming that the distribution of velocities in the neighbourhood of a point is the same in every direction, whatever actions may be taking place within the gas. This flaw in the argument, first pointed out by Boltzmann, seems to have been recognised by Maxwell, who in his next paper " On the Stresses in Rarefied Gases arising from inequalities of Temperature," published in the Philosophical Transactions for 1879, Part i., adopts a form of the distribution function of a somewhat different shape. The object of this paper was to arrive at a theory of the effects observed in Crookes's Radiometer. The results of the investigation are stated by Maxwell in the introduction to the paper, from which it would appear that the observed motion cannot be explained on the Dynamical Theory, unless it be supposed that the gas in contact with a solid can slide along the surface with a finite velocity between places whose temperatures are different. In an appendix to the paper he shews that on certain assumptions regarding the nature of the contact of the solid and gas, there will be, when the pressure is constant, a flow of gas along the surface from the colder to the hotter parts. The last of his longer papers on this subject is one on Boltzmann's Theorem. Throughout these volumes will be found numerous shorter essays on kindred subjects, published chiefly in Nature and in the EncyclopcBdia Britannica. Some of these contain more or less popular expositions of this subject which Maxwell had himself in gi-eat part created, while others deal with the work of other writers in the same field. They are profoundly suggestive in almost every page, and abound in acute criticisms of speculations which he could not accept. They are always interesting; for although the larger papers are sometimes difficult to follow, Maxwell's more popular writings are characterized by extreme lucidity and simplicity of style. The first of Maxwell's papers on Colour Perception is taken from the Transactions of the Royal Scottish Society of Ai^ts and is in the form of a letter to Dr G. Wilson dated Jan. 4, 1855. It was followed directly afterwards by a communication to the Royal Society of Edinburgh, and the subject occupied his attention for some years. The most important of his subsequent work is to be found in the papers entitled "An account of Experiments on the Perception of Colour" published in the Philosophical Magazine, Vol. xiv. and "On the Theory of Compound Colours and its relation to the colours of the spectrum" in the Philosophical Transactions for the year 1860. We may also refer to two lectures delivered at the Royal Institution, in which he recapitulates and enforces his main positions in his usual luminous style. Maxwell from the first adopts Young's Theory of Colour Sensation, according to which all colours may ultimately be reduced to three, a red, a green and PREFACE. XXV a violet. This theory had been revived by Helmholtz who endeavoured to find for it a physiological basis. Maxwell however devoted himself chiefly to the invention of accurate methods for combining and recording mixtures of colours. His first method of obtaining mixtures, that of the Colour Top, is an adaptation of one formerly employed, but in Maxwell's hands it became an instrument capable of giving precise numerical results by means which he added of varying and measuring the amounts of colour which were blended in the eye. In the representation of colours diagrammatically he followed Young in employing an equilateral triangle at the angles of which the fundamental colours were placed. All colours, white included which may be obtained by mixing the fundamental colours in any proportions will then be represented by points lying within the triangle. Points without the triangle represent colours which must be mixed with one of the funda- mental tints to produce a mixture of the other two, or with which two of them must be mixed to produce the third. In his later papers, notably in that printed in the Philosophical Transactions, he adopts the method of the Colour Box, by which different parts of the spectrum may be mixed in different proportions and matched with white, the intensity of which has been suitably diminished. In this way a series of colour equations are obtained which can be used to evaluate any colour in terms of the three fundamental colours. These observations on which Maxwell expended great care and labour, constitute by far the most important data regarding the combinations of colour sensations which have been yet obtained, and are of permanent value whatever theory may ultimately be adopted of the physiology of the perception of colour. In connection with these researches into the sensations of the normal eye, may be mentioned the subject of colour-blindness, which also engaged Maxwell's attention, and is discussed at considerable length in several of his papers. Geometrical Optics was another subject in which Maxwell took much interest. At an early period of his career he commenced a treatise on Optics, which however was never completed. His first paper "On the general laws of optical instruments," appeared in 1858, but a brief account of the first part of it had been previously communicated to the Cambridge Philosophical Society. He therein lays down the conditions which a perfect optical instrument must fulfil, and shews that if an instrument produce perfect images of an object, i.e. images free from astigmatism, curvature and distortion, for two different positions of the object, it will give perfect images at all distances. On this result as a basis, he finds the relations between the foci of the incident and emergent pencils, the magnifying power and other characteristic quantities. The subject of refraction through optical combinations was afterwards treated by him in a different manner, in three papers communicated to the London Mathematical Society. In the first (1873), "On the focal lines of a refracted pencil," he applies Hamilton's characteristic function to determine the focal lines of a thin pencil refracted from one isotropic medium into another at any surface of separation. In the second (1874), "On VOL. I. <* XXVI PREFACE. Hamilton's characteristic function for a narrow beam of light/' he considers the more general question of the passage of a ray from one isotropic medium into another, the two media being separated by a third which may be of a heterogeneous character. He finds the most general form of Hamilton's characteristic function from one point to another, the first being in the medium in which the pencil is incident and the second in the medium in which it is emergent, and both points near the principal ray of the pencil. This result is then applied in two particular cases, viz. to determine the emergent pencil (1) from a spectroscope, (2) from an optical instrument symmetrical about its axis. In the third paper (1876) he resumes the last-mentioned application, discussing this case more fully under a somewhat simplified analysis. It may be remarked that all these papers are connected by the same idea, which was — first to study the optical effects of the entire instrument without examining the mechanism by which these effects are produced, and then, as in the paper in 18-58, to supply whatever data may be necessary by experiments upon the instrument itself. Connected to some extent with the above papers is an investigation which was published in 1868 "On the cyclide." As the name imports, this paper deals chiefly with the geometrical properties of the surface named, but other matters are touched on, such as its conjugate isothermal functions. Primarily however the investigation is on the orthogonal surfaces to a system of rays passing accurately through two lines. In a footnote to this paper Maxwell describes the stereoscope which he invented and which is now in the Cavendish Laboratory. In 1868 was also published a short but important article entitled " On the best arrange- ment for producing a pure spectrum on a screen." The various papers relating to the stresses experienced by a system of pieces joined together so as to form a frame and acted on by forces form an important group connected with one another. The first in order was " On reciprocal figures and diagrams of forces," published in 1864. It was immediately followed by a paper on a kindred subject, " On the calculation of the equilibrium and stiffness of frames." In the first of these Maxwell demonstrates certain reciprocal properties in the geometry of two polygons which are related to one another in a particular way, and establishes his well-known theorem in Graphical Statics on the stresses in frames. In the second he employs the principle of work to problems connected with the stresses in frames and structures and with the deflections arising from extensions in any of the connecting pieces. A third paper "On the equilibrium of a spherical envelope," published in 18Q1, may here be referred to. The author therein considers the stresses set up in the envelope by a system of forces applied at its surface, and ultimately solves the problem for two normal forces applied at any two points. The solution, in which he makes use of the principle of inversion as it is appUed in various electrical questions, turns ultimately on the deter- mination of a certain function first introduced by Sir George Airy, and called by Maxwell PREFACE. XXVU Airy's Function of Stress. The methods which in this paper were attended with so much success, seem to have suggested to Maxwell a reconsideration of his former work, with the view of extending the character of the reciprocity therein established. Accordingly in 1870 there appeared his fourth contribution to the subject, "On reciprocal figures, frames and diagrams of forces." This important memoir was published in the Transactions of the Royal Society of Edinburgh, and its author received for it the Keith Prize. He begins with a remarkably beautiful construction for drawing plane reciprocal diagrams, and then proceeds to discuss the geometry and the degrees of freedom and constraint of polyhedral frames, his object being to lead up to the limiting case when the faces of the polyhedron become infinitely small and form parts of a continuous surface. In the course of this work he obtains certain results of a general character relating to inextensible surfaces and certain others of practical utility relating to loaded frames. He then attacks the general problem of representing graphically the internal stress of a body and by an extension of the meaning of "Diagram of Stress," he gives a construction for finding a diagram which has mechanical as well as geometrical reciprocal properties with the figure supposed to be under stress. It is impossible with brevity to give an account of this reciprocity, the development of which in Maxwell's hands forms a very beautiful example of analysis. It will be sufficient to state that under restricted conditions this diagram of stress leads to a solution for the components of stress in terms of a single function analogotis to Airy's Function of Stress. In the remaining parts of the memoir there is a discussion of the equations of stress, and it is shewn that the general solution may be expressed in terms of three functions analogous to Airy's single function in two dimensions. These results are then applied to special cases, and in particular the stresses in a horizontal beam with a uniform load on its upper surface are fully investigated. On the subjects in which Maxwell's investigations were the most numerous it has been thought necessary, in the observations which have been made, to sketch out briefly the connections of the various papers on each subject with one another. It is not how- ever intended to enter into an account of the contents of his other contributions to science, and this is the less necessary as the reader may readily obtain the information he may require in Maxwell's own language. It was usually his habit to explain by way of introduction to any paper his exact position with regard to the subject matter and to give a brief account of the nature of the work he was contributing. There are however several memoirs which though unconnected with others are exceedingly interesting in them- selves. Of these the essay on Saturn's Rings will probably be thought the most important as containing the solution of a difficult cosmical problem ; there are also various papers on Dynamics, Hydromechanics and subjects of pure mathematics, which are most useful con- tributions on the subjects of which they treat. The remaining miscellaneous papers may be classified under the following heads: (a) Lectures and Addresses, (6) Essays or Short Treatises, (c) Biographical Sketches, (d) Criticisms and Reviews. d 2 XXviii PREFACE. Class (a) comprises his addresses to the British Association, to the London Mathematical Society, the Rede Lecture at Cambridge, his address at the opening of the Cavendish Laboratory and his Lectures at the Royal Institution and to the Chemical Society. Class (6) includes all but one of the articles which he contributed to the Encyclo- pcedia Britannica and several others of a kindred character to Nature. Class (c) contains such articles as "Faraday" in the Encyclopcedia Britannica and " Helmholtz '' in Nature. Class (d) is chiefly occupied with the reviews of scientific books as they were pub- lished. These appeared in Nature and the most important have been reprinted in these In some of these writings, particularly those in class (6), the author allowed himself a greater latitude in the use of mathematical symbols and processes than in others, as for instance in the article " Capillary Attraction," which is in fact a treatise on that subject treated mathematically. The lectures were upon one or other of the three departments of Physics with which he had mainly occupied himself; — Colour Perception, Action through a Medium, Molecular Physics; and on this account they are the more valuable. In the whole series of these more popular sketches we find the same clear, graceful delineation of principles, the same beauty in arrangement of subject, the same force and precision in expounding proofs and illustrations. The style is simple and singularly free from any kind of haze or obscurity, rising occasionally, as in his lectures, to a strain of subdued eloquence when the emotional aspects of the subject overcome the purely speculative. The books which were written or edited by Maxwell and published in his lifetime but which are not included in this collection were the "Theory of Heat" (1st edition, 1871); " Electricity and Magnetism " (1st edition, 1873) ; " The Electrical Researches of the Hon- ourable Henry Cavendish, F.R.S., written between 1771 and 1781, edited from the original manuscripts in the possession of the Duke of Devonshire, K.G." (1879). To these may be added a graceful little introductory treatise on Dynamics entitled "Matter and Motion" (published in 1876 by the Society for promoting Christian Knowledge). Maxwell also contributed part of the British Association Report on Electrical Units which was afterwards published in book form by Fleeming Jenkin. The "Theory of Heat" appeared in the Text Books of Science series published by Longmans, Green and Co., and was at once -hailed as a beautiful exposition of a subject, part of which, and that the most interesting part, the mechanical theory, had as yet but commenced the existence which it owed to the genius and labours of Rankine, Thomson and Clausius. There is a certain charm in Maxwell's treatise, due to the freshness and originality of its expositions which has rendered it a great favourite with students of Heat. After his death an "Elementary Treatise on Electricity," the greater part of which he had written, was completed by Professor 'Garnett and published in 1881. The aim of this PREFACE. XXIX treatise and its position relatively to his larger work may be gathered from the following extract from Maxwell's preface. " In this smaller book I have endeavoured to present, in as compact a form as I can, those phenomena which appear to throw light on the theory of electricity and to use them, each in its place, for the development of electrical ideas in the mind of the reader." "In the larger treatise I sometimes made use of methods which I do not think the best in themselves, but without which the student cannot follow the investigations of the founders of the Mathematical Theory of Electricity. I have since become more convinced of the superiority of methods akin to those of Faraday, and have therefore adopted them from the first." Of the "Electricity and Magnetism" it is difficult to predict the future, but there is no doubt that since its publication it has given direction and colour to the study of Electrical Science. It was the master's last word upon a subject to which he had devoted several years of his life, and most of what he wrote found its proper place in the treatise. Several of the chapters, notably those on Electromagnetism, are practically reproductions of his memoii's in a modified or improved form. The treatise is also remarkable for the handling of the mathematical details no less than for the exposition of physical principles, and is enriched incidentally by chapters of much originality on mathematical subjects touched on in the course of the work. Among these may be mentioned the dissertations on Spherical Harmonics and Lagrange's Equations in Dynamics. The origin and growth of Maxwell's ideas and conceptions of electrical action, cul- minating in his treatise where all these ideas are arranged in due . connection, form an interesting chapter not only in the history of an individual mind but in the history of electrical science. The importance of Faraday's discoveries and speculations can hardly be overrated in their influence on Maxwell, who tells us that before he began the study of electricity he resolved to read none of the mathematics of the subject till he had first mastered the "Experimental Researches." He was also at first under deep obligations to the ideas contained in the exceedingly important papers of Sir W. Thomson on the analogy between Heat-Conduction and Statical Electricity and on the Mathematical Theory of Electricity in Equilibrium. In his subsequent efforts we must perceive in Maxwell, possessed of Faraday's views and embued with his spirit, a vigorous intellect bringing to bear on a subject still full of obscurity the steady light of patient thought and expending upon it all the resources of a never failing ingenuity. Royal Naval College, Greenwich, August, 1890. TABLE OF CONTENTS. I. On the Description of Oval Curves and those having a plurality of Foci; with remarks by Professor Forbes II. On the Theory of Rolling Curves III. On the Equilibrium of Elastic Solids . Solutions of Problems ..... IV. On the Transformation of Surfaces by Bending . V. On a particular case of the descent of a heavy body in a resisting medium VI. On the Theory of Colours in relation to Golour-Blindness VII. Experiments on Colour as perceived by the Eye, with remurks on Colour-Blindness "~^^ VIII. On Faraday's Lines of Force ..... ... IX. Description of a New Form of the Platometer, an Instrument for measuring the areas of Plane Figures drawn on paper . . ... X. On the elementary theory of Optical Instruments ....... XI. On a method of drawing the Theoretical Fonns of Faraday's Lines of Force without calcidation ... ....... XII. On the unequal sensibility of the Foramen Centrale to Light of different Colours XIII. On the Theory of Compound Colours with reference to mixtures of Blue and Yellow Light ............. XIV. On an instrument to illustrate Poinsot's Theory of Rotation .... XV. On a Dynamical Top, for exhibiting the phenomena of the motions of a body of invariable form about a fixed point, with some suggestimis as to the Earth's motion ..... . . XVI. Account of Experiments on the Perception of Colour XVII. On the general laws of Optical Instruments r^ XVIII. On Theories of the Constitution of Saturn's Rings """ XIX. On the stability of the motion of Saturn's Rings ■^ XX. Illustrations of the Dynamical Theory of Gases . XXI. On the Theory of Compound Colours and the Relations of the Spectrum ....... XXII. On the Theory of Three Primary Colours . ■^ XXIII. On Physical Lines of Force .... XXIV. On Reciprocal Figures and Diagrams of Forces . ^^J XXV. A Dynamical Theory of the Electromagnetic Field XXVI. On the Calculation of the Equilibrium and Stiffness of Frames Colours of the PAGE 1 4 30 74 80 115 119 126 ^ 1.5.5 230 238 241 242 ' 243 ' 246 248 263 271 286 288 377 410 445 451 514 526 598 ERRATA. Page 40. In the first of equations (12), second group of terms, read d'i £^2 (^2 instead of d^ cP cP with corresponding changes in the other two equations. Page 153, five lines from bottom of page, read 127 instead of 276. Page 591, four lines from bottom of page the equation should be (PM '= m_, is the tractory of the circle, the equation of which is ^ = cos"' k/^~^- '^^® curve whose equation is f = ^ — :; seems to be among spirals what the catenary is among curves whose equations are between rec- tangular co-ordinates ; for, if we represent the vertical direction by the radius vector, the tangent of the angle which the curve makes with this line is proportional to the length of the curve reckoned from the origin ; the point at the distance a from a straight line rolled on this curve generates a circle, and when rolled on the catenary produces a straight line ; the involute of this curve is the tractory of the circle, and that of the catenary is the tractory of the straight line, and the tractory of the circle rolled on that of the straight line traces the straight line ; if this curve is rolled on the catenary, it produces the straight line touching the catenary at its vertex; the method of drawing VOL. I. 3 18 THE THEORY OF ROLLING CURVES. tangents is the same as in the catenary, namely, by describing a circle whose radius is a on the production of the radius vector, and drawing a tangent to the circle from the given point. In the next case the rolled curve is the same as the fixed curve. It is evident that the traced curve will be similar to the locus of the intersection of the tangent with the perpendicular from the pole ; the magnitude, however, of the traced curve wUl be double that of the other curve ; therefore, if we call r„ = <^„^„ the equation to the fixed curve, r^ = ^^Qi that of the traced curve, we have n = 2^„, ^^ = ^„_cos^'^ = ^„-J+sin-^^, also, Pl = P2 Similarly, r, = 2p, = 2r,^ = i ^l 4r, (^Y , 6, = 6,- 2 cos"^ ^ . Similarly, r,. = 2p,.., = 2r„_, ^ &c. = 2V„ (^Y , and P^ — P" Let 6^ become 6,,' ; 9,, 6,' and ^ , P\ . Let 6^' -6, = a, ' ' « = ^n' - 0,, = e:~e,-n cos- ^ + n COS-' ^' ; .: cos-'^-cos-'^' = - + t^' fn r„ n n THE THEOEY OF ROLLING CUEVES. 19 Now, cos"^ £^ is the complement of the angle at which the curve cuts the ' n radius vector, and cos"' ^ -cos"' -^ is the variation of this angle when 0„ varies by an angle equal to a. Let this variation = ^ ; then if 6, - 6^ = /8, n n Now, if n increases, j> wUl diminish ; and if n becomes infinite, <^ = ^^ + ^5" = when a and ^ are finite. Therefore, when n is infinite, ^ vanishes ; therefore the curve cuts the radius vector at a constant angle ; therefore the curve is the logarithmic spiral. Therefore, if any curve be rolled on itself, and the operation repeated an infinite number of times, the resulting curve is the logarithmic spiral. Hence we may find, analytically, the curve which, being roUed on itself, traces itself. For the curve which has this property, if roUed on itself, and the operation repeated an infinite number of times,, wiU still trace itself. But, by this proposition, the resulting curve is the logarithmic spiral ; therefore the curve required is the logarithmic spiral. As an example of a curve rolling on itself, we will take the curve whose equation is / 6V r„ = 2"a(cos— . n Hare -f^=2'.(sm§)(cosf J^c I 6\^ 2=V (cos-j 2"a(cos — ) / /,\„+i + sm n 3—2 20 THE THEORY OF ROLLING CURVES. ft ft Now e,-0,= - COS-^ I? = - C03-^ COS ^ = ^ , n + l' substituting this value of ft, in the expression for r„ (ft x^^^ similarly, if the operation be repeated m times, the resulting curve is g \n+m 2"+'"a(cos- " When «=1, the curve is r = 2a cos ^, the equation to a circle, the pole being in the circumference. When n = 2, it is the equation to the cardioid r = 4a ( cos ^ In order to obtain the cardioid from the circle, we roll the circle upon itself, and thus obtain it by one operation ; but there is an operation which, being performed on a circle, and again on the resulting curve, will prodvice a cardioid, and the intermediate curve between the circle and cardioid is r = 2^a(cos — 1 . As the operation of rolling a curVe on itself is represented by changing n into {n+l) in the equation, so this operation may be represented by changing n into (» + ■!). Similarly there may be many other fractional operations performed upon the curves comprehended under the equation r = 2"a (cos - \ ^. We may also find the curve, which, being rolled on itself, will produce a given curve, by making n= —I, THE THEORY OE ROLLING CURVES. 21 We may likewise prove by the same method as before, that the result of performing this inverse operation an infinite number of times is the logarithmic spiral. As an example of the inverse method, let the traced line be straight, let its equation be r, = 2a sec ^«, then £_j ^ ^ ^ 2a ^ _2a_ ^-1 n ^0 2jp_,' .-. p\, = ar^,, therefore suppressing the suffix, " + de^ fdrV r' , dO 1 '' dr Jr r ^ 1 .-. 6= cos M \r 0' 2a 1 - COS d '■ the polar equation of the parabola vrhose parameter is 4a. The last case which we shall here consider affords the means of constructing two wheels whose centres are fixed, and which shall roll on each other, so that the angle described by the first shall be a giten function of the angle described by the seconds Let 0^ = ^d-a then r^ + r^ = a, and -m = — ', dd^ a — r^ ' Let us take as an example, the pair of wheels which will represent the angular motion of a comet in a parabola. 22 THE THEORY OE ROLLING CURVES. Here ^. = tan-. . =-— ^ = ^-^^ 2 cos — r, 1 ' ' a 2 + cos ^1 ' therefore the first wheel is an elhpse, whose major axis is equal to f of the distance between the centres of the wheels, and in which the distance between the foci is half the major axis. Now since ^i = 2 tan"' Q^ and r-j = a — n, ^=2- ?!:_2 a which is the equation to the wheel which revolves with constant angular velocity. Before proceeding to give a list of examples of rolling curves, we shall state a theorem which is almost self-evident after what has been shewn pre- viously. Let there be three curves, A, B, and O. Let the curve A, when roUed on itself, produce the curve B, and when rolled on a straight line let it produce the curve C, then, if the dimensions of C be doubled, and B be rolled on it, it will trace a straight line. A Collection of Examples of Boiling Curves, First. Examples of a curve rolling on a straight line. Ex. 1, When the rolling curve is a circle whose tracing-point is in the circumference, the curve traced is a cycloid, and when the point is not in the circumference, the cycloid becomes a trochoid. Ex. 2. When the rolling curve is the involute of the circle whose radius is 2a, the traced curve is a parabola whose parameter is 4a. THE THEOEY OE ROLLING CUBVES. 23 Ex. 3. When the rolled curve is the parabola whose parameter is 4a, the traced curve is a catenary whose parameter is a, and whose vertex is distant a from the straight line. Ex. 4. When the rolled curve is a logarithmic spiral, the pole traces a straight line which cuts the fixed line at the same angle as the spiral cuts the radius vector. Ex. 5. When the rolled curve is the hyperbolic spiral, the traced curve is the tractory of the straight line. Ex. 6. When the rolled curve is the polar catenary '-/ the traced curve is a circle whose radius is a, and which touches the straight line. Ex. 7. When the equation of the rolled curve is the traced curve is the hyperbola whose equation is Second. In the examples of a straight Hne roUing on a curve^ we shall use the letters A, B, and C to denote the three curves treated of in page 22. Ex. 1. When the curve ^ is a circle whose radius is a, then the curve B is the involute of that circle, and the curve C is the spiral of Archimedes, r = ad. Ex. 2. When the curve ^ is a catenary whose equation is the curve B is the tractory of the straight line, whose equation is ■M = alog ■ +'Ja^-x', a + \/a' — 3? and C is a straight line at a distance a froin the vertex of the catenary. 24 THE THEORY OF ROLLING CURVISi Ex. 3. When the curve A is the polar catenary the curve B is the tractory of the circle ^ = cos" a N r" ' . a and the curve C is a circle of which the radius is - . TJiird. Examples of one curve rolling on another, and tracing a straight line. Ex. 1. The curve whose equation is ^ = ^r-» + &c. + Kr'^ + Lr-^ + il^log r + Nr + kc. + Zr\ when roUed on the curve whose equation is V = -^ ^a;^-" + &c. + 2irx-^ - X log a; + ifa; + ^ iVaf + &c. + — — Zx"+S traces the axis of y. Ex. 2. The circle whose equation is r = acos^ rolled on the circle whose radius is a traces a diameter of the circle. Ex. 3. The curve whose equation is t- /^ 6= , / 1 — versm - , rolled on the circle whose radius is a, traces the tangent to the circle. Ex. 4. If the fixed curve be a parabola whose parameter is 4a, and if "we roU on it the spiral of Archimedes r=^a9, the pole will trace the axis of the parabola. Ex. 5. If we roll an equal parabola on it, the focus will trace the directrix of the first parabola. Ex. 6. If we roll on it the curve '^ = jQi the pole wUl trace the tangent at the vertex of the parabola. THE THEOEY OE ROLLING CURVES. 25 Ex. 7. If we roll tlie curve whose equation is r = acos [t^ + — = 1 on the ellipse whose equation is the pole will trace the axis h. Ex. 8. If we roll the curve whose equation is T = - (e* — e ^ on the hyperbola whose equation is ¥ a? ' the pole will trace the axis b. Ex. 9. If we roll the lituus, whose equation is /y*^ — 30' on the hyperbola whose equation is xy = a', the pole will trace the asymptote. Ex. 10. The cardioid whose equation is r = a (1 + COS 0), roUed on the cycloid whose equation is y — a versin"^ - + v 2a.x - x^ traces the base of the cycloid. Ex. 11. The curve whose equation is = versm"' - + 2 / 1, a \J r rolled on the cycloid, traces the tangent at the vertex. VOL, I. 26 THE THEOEY OF ROLLING CURVES. Ex, 12. The straight line whose equation is r = a sec d, rolled on a catenary whose parameter is a, traces a line whose distance from the vertex is a. Ex. 13. The part of the polar catenary whose equation is /I / , 2a. rolled on the catenary, traces the tangent at the vertex. Ex. 14. The other part of the polar catenary whose equation is rolled on the catenary, traces a line whose distance from the vertex is equal to 2a. Ex. 15. The tractory of the circle whose diameter is a, rolled on the tractory of the straight line whose constant tangent is a, produces the straight line. Ex. 16. The hyperbolic spiral whose equation is a rolled on the logarithmic curve whose equation is y = a\og-, traces the axis of y or the asymptote. Ex. 17. The mvolute of the circle whose radius is a, rolled on an orthogonal trajectory of the catenary whose equation is traces the axis of y. Ex. 18. The curve whose equation is "i^'W^^'- THE THEORY OF EOLLING CURVES. 27 rolled on the witch, whose equation is traces the asymptote. Ex. 19. The curve whose equation is r = a tan 6, rolled on the curve whose equation is traces the axis of y. Ex. 20. The curve whose equation is 2r V — , ■^^ rolled on the curve whose equation is T = : a tan 6, traces the axis of y. Ex. '. 21. The curve whose equation is r = a (sec 6 - tan ei rolled on the curve whose equation is 2/ = «log(j+l), traces the axis of y. Fourth. Examples of pairs of roUing curves which have their poles at a fixed distance = a. T (The straight Hne whose equation is ^=sec"^-. Ex. 1. \ ^ \.The polar catenary whose equation is 6= ±^ 1 +—^ . Ex. 2. Two equal ellipses or hyperbolas centered at the foci. Ex. -3. Two equal logarithmic spirals. (Circle whose equation is r = 2a cos 6. r^ r Curve whose equation is ^ = ^2--l +versin"^-. 4—2 28 THE THEORY OE ROLLING CURVES. [Cardioid whose equation is r=2a{l+ cos d). Ex. 5. •< , /3 • -1 *■ 1 ^ [Curve whose equation is ^ = sin '^'^^'^S j^—^+a' Ex. 6. Ex. 7. Ex. 8. Ex. 9. Ex. 10. f Conchoid, ^ = « ( sec^- 1) [Curve, ^ = J^-^+^^'' [Spiral of Archimedes, r = a9. [Curve. 6' = ^ + log^. ^,r a I Hyperbolic spiral, ''^~0 Curve, r = a ° a a 1 (Ellipse whose equation la ''"^oT 3" Curve, r = a(l-| 2 {2-0'))' [Involute of circle, 6= /——I sec ^- . i^ifi/^. Examples of curves rolling on themselves. Ex. 1. When the curve which rolls on itself is a circle, equation r = a cos 9, the traced ciurve is a cardioid, equation r = a(l + cos^). Ex. 2. When it is the curve whose equation is r = 2"a ( cos - 1 , the equation of the traced curve is / '^"^^ r = 2"+'a cos n + 1 Ex. 3. When it is the involute of the circle, the traced curve is the spiral of Archimedes. THE THEORY OF ROLLING CURVES, 29 Ex. 4. When it is a parabola, the focus traces the directrix, and the vertex traces the cissoid. Ex, 5. When it is the hyperbolic spiral, the traced curve is the tractory of the circle. Ex. 6. When it is the polar catenary, the equation of the traced curve is /) /2a - • -1 ^ o = ^ / 1 — versm - . S] r a Ex. 7. When it is the curve whose equation is the equation of the traced curve is rt=a (e* — e~*). This paper commenced with an outline of the nature and history of the problem of rolling curves, and it was shewn that the subject had been discussed previously, by several geometers, amongst whom were De la Hire and Nicole in the Memoir es de I'Academie, Euler, Professor Willis, in his Principles of Mechanism, and the Rev. H. Holditch in the Cambridge Philosophical Transactions. None of these authors, however, except the two last, had made any application of their methods ; and the principal object of the present communication was to find how far the general equations could be simplified in particular cases, and to apply the results to practice. Several problems were then worked out, of which some were applicable to the generation of curves, and some to wheelwork j while others were interesting as shewing the relations which exist between different curves; and, finally, a collection of examples was added, as an illus' tration of the fertility of the methods employed. [From the Transactions of the Royal Society of Edinburgh, Vol. XX. Part i.] III. — On the Equilibrium of Elastic Solids. There are few parts of mechanics in which theory has differed more from experunent than in the theory of elastic soHds. Mathematicians, setting out from very plausible assumptions with respect to the constitution of bodies, and the laws of molecular action, came to conclusions which were shewn to be erroneous by the observations of experimental philoso- phers. The experiments of CErsted proved to be at variance with the mathe- matical theories of Navier, Poisson, and Lame and Clapeyron, and apparently deprived this practically important branch of mechanics of all assistance from mathematics. The assumption on which these theories were founded may be stated thus : — Solid bodies are composed of distinct molecules, which are hept at a certain distance from each other by the opposing principles of attraction and heat. When the distance between two molecules is changed, they act on each other with a force whose direction is in the line joining the centres of the molecules, and whose magnitude is equal to the change of distance multiplied into a function of the distance ivhich vanishes when that distance becomes sensible. The equations of elasticity deduced from this assumption contain only one coefficient, which varies with the nature of the substance. The insufficiency of one coefficient may be proved from the existence of bodies of different degrees of solidity. No effort is required to retain a liquid in any form, if its volume remain unchanged; but when the form of a solid is changed, a force is called into action which tends to restore its former figure ; and this constitutes the differ- THE EQUILIBRIUM OF ELASTIC SOLIDS, 31 ence between elastic solids and fluids. Both tend to recover their volume, but fluids do not tend to recover their shape. Now, since there are in nature bodies which are in every intermediate state from perfect soHdity to perfect liquidity, these two elastic powers cannot exist in every body in the same proportion, and therefore all theories which assign to them an invariable ratio must be erroneous. I have therefore substituted for the assumption of Navier the following axioms as the results of experiments. If three pressures in three rectangular axes be applied at a point in an elastic soHd, — 1. The sum of the three lyressures is proportional to the sum of the com- pressions which they produce. 2. The difference between two of the pressures is proportional to the differ- ence of the compressions which they produce. The equations deduced from these axioms contain two coefficients, and differ from those of Navier only in not assuming any invariable ratio between the cubical and linear elasticity. They are the same as those obtained by Professor Stokes from his equations of fluid motion, and they agree with all the laws of elasticity which have been deduced from experiments. In this paper pressures are expressed by the number of units of weight to the unit of surface ; if in English measure, in pounds to the square inch, or in atmospheres of 15 pounds to the square inch. Compression is the proportional change of any dimension of the solid caused by pressure, and is expressed by the quotient of the change of dimension divided by the dimension compressed'"'. Pressure wiU be understood to include tension, and compression dilatation ; pressure and compression being reckoned positive. Elasticity is the force which opposes pressure, and the equations of elasticity are those which express the relation of pressure to compression f. Of those who have treated of elastic solids, some have confined themselves to the investigation of the laws of the bending and twisting of rods, without * The laws of pressure and compression may be found in the Memoir of Lame and Clapeyron. See note A. t See note B. 32 THE EQUILIBRIUM OF ELASTIC SOLIDS. considering the relation of the coefficients which occur in these two cases; while others have treated of the general problem of a solid body exposed to any forces. The investigations of Leibnitz, Bernoulli, Euler, Varignon, Young, La Hire, and Lagrange, are confined to the equilibrium of bent rods; but those of Navier, Poisson, Lamd and Clapeyron, Cauchy, Stokes, and Wertheim, are principally directed to the formation and application of the general equations. The investigations of Navier are contained in the seventh volume of the Memoirs of the Institute, page 373 ; and in the Annates de Ghimie et de Physique, 2° Sdrie, XV. 264, and xxxviiL 435 ; L' Application de la Mecanique, Tom. I. Those of Poisson in Mem. de I'Institut, viii. 429 ; Annates de Chimie, 2® Serie, xxxvl 334 ; xxxvii. 337 ; xxxvin. 338 ; XLii. Journal de I'Ecole Polytechnique, cahier xx., with an abstract in Annates de Chimie for 1829. The memoir of MM. Lamd and Clapeyron is contained in Crelle's Mathe- matical Journal, Vol. vii. ; and some observations on elasticity are to be found in Lamp's Cours de Physique. M. Cauchy's investigations are contained in his Exercices d'Analyse, Vol. iii. p. 180, published in 1828. Instead of supposing each pressure proportional to the linear compression which it produces, he supposes it to consist of two parts, one of which is pro- portional to the Hnear compression in the direction of the pressure, while the other is proportional to the diminution of volume. As this hypothesis admits two coefficients, it differs from that of this paper only in the values of the coefficients selected. They are denoted by K and Jc, and K^fx — ^m, h = m. The theory of Professor Stokes is contained in Vol. viii. Part 3, of the Cambridge^ Philosophical Transactions, and was read April 14, 1845. He states his general principles thus : — " The capability which solids possess of being put into a state of isochronous vibration, shews that the pressures called into action by small displacements depend on homogeneous functions of those displacements of one dimension. I shall suppose, moreover, according to the general principle of the superposition of small quantities, that the pressures due to different displacements are superimposed, and, consequently, that the pressures are linear functions of the displacements." THE EQUILIBRIUM OE ELASTIC SOLIDS. 33 Having assumed the proportionality of pressure to compression, he proceeds to define his coefiicients. — "Let —^8 be the pressures corresponding to a uniform linear dilatation 8 when the solid is in equilibrium, and suppose that it becomes mAh, in consequence of the heat developed when the solid is in a state of rapid vibration. Suppose, also, that a displacement of shifting parallel to the plane xy, for which Sx — kx, 8y= —hy, and Sz = 0, calls into action a pressure —Bh on a plane perpendicular to the axis of x, and a pressure Bk on a plane perpendicular to the axis of y ; the pressure on these planes being equal and of contrary signs ; that on a plane perpendicular to z being zero, and the tan- gential forces on those planes being zero." The coefficients A and B, thus defined, when expressed as in this paper, are A = 3/x., B= - . Professor Stokes does not enter into the solution of his equations, but gives their results in some particular cases. 1. A body exposed to a uniform pressure on its whole surface. 2. A rod extended in the direction of its length. 3. A cyhnder twisted by a statical couple. He then points out the method of finding A and B from the last two cases. While explaining why the equations of motion of the luminiferous ether are the same as those of incompressible elastic solids, he has mentioned the property of plasticity or the tendency which a constrained body has to reheve itself from a state of constraint, by its molecules assuming new positions of equi- librium. This property is opposed to linear elasticity; and these two properties exist in all bodies, but in variable ratio. M. Wertheun, in Annales de Chimie, 3« S^rie, xxiii., has given the results of some experiments on caoutchouc, from which he finds that K^h, or /A = |m; and concludes that lc = K m. all substances. In his equations, /a is therefore made equal to |^m. The accounts of experimental researches on the values of the coefficients are so numerous that I can mention only a few. Canton, Perkins, (Ersted, Aime, Colladon and Sturm, and Regnault, have determined the cubical compressibilities of substances; Coulomb, Duleau, and Giulio, have calculated the linear elasticity from the torsion of wires; and a great many observations have been made on the elongation and bending of beams. VOL. I. 34 THE EQUILIBRIUM OF ELASTIC SOLIDS. I have found no account of any experiments on tlie relation between the doubly refracting power communicated to glass and other elastic solids by com- pression, and the pressure which produces it'" ; but the phenomena of bent glass seem to prove, that, in homogeneous singly-refracting substances exposed to pressures, the principal axes of pressure coincide with the principal axes of double refraction ; and that the difference of pressures in any two axes is proportional to the difference of the velocities of the oppositely polarised rays whose directions are parallel to the third axis. On this principle I have calculated the phenomena seen by polarised light in the cases where the solid is, bounded by parallel planes. In the following pages I have endeavoured to apply a theory identical with that of Stokes to the solution of problems which have been selected on account of the possibility of fulfilling the conditions. I have not attempted to extend the theory to the case of imperfectly elastic bodies, or to the laws of permanent bending and breaking. The solids here considered are supposed not to be compressed beyond the limits of perfect elasticity. The equations employed in the transformation of co-ordinates may be found in Gregory's Solid Geometry. I have denoted the displacements by Sa:;, Sy, Sz. They are generally denoted by a, ^, y ; but as I had employed these letters to denote the principal axes at any point, and as this had been done throughout the paper, I did not alter a notation which to me appears natural and intelligible. The laws of elasticity express the relation between the changes of the dimensions of a body and the forces which produce them. These forces are called Pressures, and their effects Compressions. Pressures are estimated in pounds on the square inch, and compressions in fractions of the dimensions compressed. Let the position of material points in space be expressed by their co-ordinates X, y, and z, then any change in a system of such points is expressed by giving to these co-ordinates the variations Sec, hy, Zz, these variations being functions of X, y, z. * See note C. THE EQUILIBRIUM OF ELASTIC SOLIDS. 35 The quantities Sx, Sy, Sz, represent the absolute motion of each point in the directions of the three co-ordinates ; but as compression depends not on absolute, but on relative displacement, we have to consider only the nine quantities — dSx dSx dBx dx ' dy ' dz ' dSy dBy dSy dx ' dy ' dz ' dBz dSz dSz dx ' dy ' dz ' Since the number of these quantities is nine, if nine other independent quantities of the same kind can be found, the one set may be found in terms of the other. The quantities which we shall assume for this purpose are — 1. Three compressions, — , -^ , — , in the directions of three principal a p y axes a, /3, y. 2. The nine direction-cosines of these axes, with the six connecting eqxia- tions, leaving three independent quantities. (See Gregory's Solid Geometry.) 3. The small angles of rotation of this system of axes about the axes of X, y, z. The cosines of the angles which the axes of x, y, z make with those of a, yS, y are cos (aOa;) = a^, cos (ySOa;) = \, cos (yOa;) = Cj, cos (aOy) = tta, cos (/SOy) — 62, cos (yO?/) = c^, cos (aOz) =a^, cos (/30z) =?>3, cos (yOz) = C3. These direction-cosines are connected by the six equations, < + ^i + Ci = 1> «A + &1&2 + CA = 0, ai + 6/ + ci=l, a,a, + W + cfi^ = 0. The rotation of the system of axes a, /8, y, round the axis of X, from y to z, =8^i, y, from z to x, =89^, z, from X to y, =863; 5—2 36 THE EQUILIBRIUM OF ELASTIC SOLIDS. By resolving the displacements 8a, Sfi, Sy, 80^, Sd^, 86^, in the directions of the axes x, y, z, the displacements in these axes are found to be 8x == ttiSa + 6i8/3 + CiSy — Z6^z + hd^y, Sy = a^Sa + b^8/3 + cJ8-y — Sd^x + S9^z, Sz = a^Sa + h^S/S + c^Sy — hO^y + hO^x. Sa 8y8 = y8§, and 87 = 7^ p y But 8a = a a and a = a,x + a^ + afi, fi = h^x + b2y + b^z, and y — c^x + c.,y + c^z. Substituting these values of Sa, 8/3, and 8y in the expressions for Bx, Sy, 8z, and differentiating with respect to x, y, and z, in each equation, we obtain the equations dhx ha „ . 8/3 J 2 , 8y ax a p y dSy Sa , SB, ^ Sy , dy a ' fi ' y ' d8z Sa 8/3 , . Sy 2 dz ,(1). c^SiC 8a 8/3 , , Sy „/> c^Stc Sa , Si8 , , Sy -, . (iSw Sa SjS,, 8y -^ = — 1*2«3 + -^ &,&3 + -^ c,C3 + 8^, dx a Sa a p y cZSz Sa 8;S,, Sy -^ = — a3a, + -^ 63&1 + -^ c,c, + 86, dSz dy a oa a /3 8 ■he. Equations of compression. (2). Equations of the equilibrium of an element of the solid. The forces which may act on a particle of the solid are :— 1. Three attractions in the direction of the axes, represented by X, Y, Z. 2. Six pressures on the six faces. THE EQUILIBEIUM OF ELASTIC SOLIDS. 37 3. Two tangential actions on each face. Let the six faces of the small parallelepiped be denoted by x^, y^, z^, x^, y.^, and Zaj then the forces acting on x^ are : — 1. A normal pressure p.^ acting in the direction of x on the area dydz. 2. A tangential force q^ acting in the direction of y on the same area. 3. A tangential force q^ acting in the direction of z on the same area, and so on for the other five faces, thus : — Forces which act in the direction of the axes of X y z On the face x^ —p-fiiydz — q^dydz — q^dydz X, {Pi+ J^dx)dydz {(1^+ J^dx)dydx {q.^+-£dx)dydz Vi — q^dzdx —p^zdx — q^dzdx 2/2 {q\+ Jydy)dzdx {p^ + 77 (^y) dzdx {(1^+ ^ydy)d^dx Zi — qdxdy — q^dxdy -p,dxdy z. fe+ -j^dz)dxdy' {q' + ^dz)dxdy {Pz + -^dz)dxdy Attractions, pXdxdydz p Ydxdydz pZdxdydz Taking the moments of these forces round the axes of the particle, we find qi=qi, q^=q..> ql= A y, which wUl become the principal axes of compression, and the com- .„ , Sa hB Sy pressions will be — , -7^ , -^ . a P y The fundamental assumption from which the following equations are deduced is an extension of Hooke's law, and consists of two parts. I. The sum of the compressions is proportional to the sum of the pressures. II. The difference of the compressions is proportional to the difference of the pressures. These laws are expressed by the following equations I. (P, + P, + P,).3^(^ + | + ^ (4). II. Sa Sy8\ ■> a /3 h y {P,-P,) = m (P.-P.)=«.(f-^) Equations of Elasticity. ,(5). The quantity ^ is the coefficient of cubical elasticity, and m that of linear elasticity. THE EQUILIBRIUM OF ELASTIC SOLIDS. 39 By solving these equations, the values of the pressures Pj, P„, Pg, and the compressions — > ~§ > ™^y ^^ found. (6). a V97-3i)*^' + ^- + ^'> + S^' a Sy p \9jU, 3m/ ^ ^ m (7). From these values of the pressures in the axes a, /S, y, may be obtained the equations for the axes x, y, z, by resolutions of pressures and compressions'"". For and q — aaP^ + hhP^ + ccP.^ ; dSx . ^ , /dSx dSy dSz\ 2 i — \r 3 I \jlx dy dz ) dx , , fdhx dhii dSz\ dSy , , /dSx dSy d8z\ dSz i>3 = (/^-im)^^- + -^ + ^j+m^ in Idhy dSz ^'^2 \d^'^ dj m fdZz dhx (8). ^^"2 \dx ^ dz m /dSx dSy ^'^2 \dy'^~d^J _ * See tlie Memoir of Lame and Clapeyron, and note A. .(9). 40 THE EQUILIBRIUM OE ELASTIC SOLIDS. dSx dx dSy dy cZ8z dz (^-3^)^^^+^^+^^) + ^^^ = (^-i)^^^+^^+^^) + ^^^ t^-^^^^P^^P^^^l^' dSx i = o, are the compression and pressure in the direction of the axis at any point. -T^ = -T— , ^2 =p, are the compression and pressure in the direction of the radius. P3 = q, are the compression and pressure in the direction of the tangent. dZx dx dBx dx ' dSz dz dSrO drd 8r r Equations (9) become, when expressed in terms of these co-ordinates- _m d80 ^'^2'^'dr ^3 = m dS0 2 dx m dSx ^'~2 '^ dx .(14). 2 ' dr The length of the cylinder is b, and the two radii a^^ and a^ in every case. VOL. I. 42 THE EQUILIBRIUM OF ELASTIC SOLIDS. Case I. The first equation is applicable to the case of a hollow cylinder, of which the outer surface is fixed, while the inner surface is made to turn through a small angle 8^, by a couple whose moment is M. The twisting force M is resisted only by the elasticity of the solid, and therefore the whole resistance, in every concentric cylindric surface, must be equal to M. The resistance at any point, multiplied into the radius at which it acts, is expressed by 2 dr rq, = - r- Therefore for the whole cylindric surface -7— imn^% — M. dr Whence W = -^J\--, 2Tnno \ai a.^ 2 ' The optical effect of the pressure of any point is expressed by I = ^qJ) = ^.—- (15). Therefore, if the solid be viewed by polarized light (transmitted parallel to the axis), the difference of retardation of the oppositely polarized rays at any point in the solid wdl be inversely proportional to the square of the distance from the axis of the cylinder, and the planes of polarization of these rays will, be inclined 45° to the radius at that point. The general appearance is therefore a system of coloured rings arranged oppositely to the rings in uniaxal crystals, the tints ascending in the scale as they approach the centre, and the distance between the rings decreasing towards the centre. The whole system is crossed by two dark bands inclined 45° to the plane of primitive polarization, when the plane of the analysing plate is perpen- dicular to that of the first polarizing plate. THE EQUILIBRIUM OP ELASTIC SOLIDS. 43 A jelly of isinglass poured when hot between two concentric cylinders forms, when cold, a convenient solid for this experiment ; and the diameters of the rings may be varied at pleasure by changing the force of torsion applied to the interior cyHnder. By continuing the force of torsion while the jelly is allowed to dry, a hard plate of isinglass is obtained, which still acts in the same way on polarized light, even when the force of torsion is removed. It seems that this action cannot be accounted for by supposing the interior parts kept in a state of constraint by the exterior parts, as in unannealed and heated glass ; for the optical properties of the plate of isinglass are such as would indicate a strain preserving in every part of the plate the direction of the original strain, so that the strain on one part of the plate cannot be main- tained by an opposite strain on another part. Two other uncrystalhsed substances have the power of retaining the polariz- ing structure developed by compression. The first is a mixture of wax and resin pressed into a thin plate between two plates of glass, as described by Sir David Brewster, in the Philosophical Transactions for 1815 and 1830. When a compressed plate of this substance is examined with polarized light, it is observed to have no action on light at a perpendicular incidence ; but when inclined, it shews the segments of coloured rings. This property does not belong to the plate as a whole, but is possessed by every part of it. It is therefore similar to a plate cut from a uniaxal crystal perpendicular to the axis. I find that its action on light is like that of a positive crystal, while that of a plate of isinglass similarly treated would be negative. The other substance which possesses similar properties is gutta percha. This substance in its ordinary state, when cold, is not transparent even in thin films; but if a thin film be drawn out gradually, it may be extended to more than double its length. It then possesses a powerful double refraction, which it retains so strongly that it has been used for polarizing light"'. As one of its refractive indices is nearly the same as that of Canada balsam, while the other is very different, the common surface of the gutta percha and Canada balsam will transmit one set of rays much more readily than the other, so that a film of extended gutta percha placed between two layers of Canada balsam acts like * By Dr Wright, I believe. 6—2 44 THE EQUILIBRIUM OF ELASTIC SOLIDS. a plate of nitre treated in the same way. That these films are in a state of constraint may be proved by heating them slightly, when they recover their original dimensions. As all these permanently compressed substances have passed their limit of perfect elasticity, they do not belong to the class of elastic solids treated of in this paper ; and as I cannot explain the method by which an uncrystallised body maintains itself in a state of constraint, I go on to the next case of twisting, which has more practical importance than any other. This is the case of a cylinder fixed at one end, and twisted at the other by a couple whose moment is M. Case II. In this case let hO be the angle of torsion at any point, then the resistance to torsion in any circular section of the cylinder is equal to the twisting force M. The resistance at any point in the circular section is given by the second Equation of (14). m dZ9 ^- 2 dx This force acts at the distance r from the axis ; therefore its resistance to torsion will be q.^, and the resistance in a circular annulus will be q^lTtrdr = rmrr^ -i— dr dx and the whole resistance for the hollow cylinder will be expressed by ,^ miT dS0 , , ,, , , ^^T^(»^-<) (16). m = 4lf 2^ 720 if / 6 \ "^ = -^ n(^?3^^) (17)- In this equation, m is the coefficient of linear elasticity; a, and a, are the radii of the exterior and interior surfaces of the hollow cylinder in inches ; M is the moment of torsion produced by a weight acting on a lever, and is expressed THE EQUILIBRIUM OE ELASTIC SOLIDS. 45 by the product of the number of pounds in the weight into the number of inches in the lever ; b is the distance of two points on the cyhnder whose angular motion is measured by means of indices, or more accurately by small mirrors attached to the cyhnder ; n is the difference of the angle of rotation of the two indices in degrees. This is the most accurate method for the determination of m independently of fjb, and it seems to answer best with thick cyKnders which cannot be used with the balance of torsion, as the oscillations are too short, and produce a vibration of the whole apparatus. Case III. A hoUow cyhnder exposed to normal pressures only. When the pressures parallel to the axis, radius, and tangent are substituted for p^, p^, and ps, Equations (10) become dSx dSr dr ^Sir^)^8r^n _ M 1 (20). By multiplying Equation (20) by r, differentiating with respect to r, and comparing this value of -i— with that of Equation (19), p-q _(^_ _1\ (d^ <^ ,d(i\ _'^dq rm ~ \9/A 3m/ \dr dr dr) m dr ' The equation of the equilibrium of an element of the solid is obtained by considering the forces which act on it in the direction of the radius. By equating the forces which press it outwards with those pressing it inwards, we find the equation of the equihbrium of the element, iz£ = ^ (21). r dr 46 THE EQUILIBRIUM OE ELASTIC SOLIDS. By comparing tMs equation with, the last, we find \9ju, SmJ dr \9fJi 3mJ \dr dr/ Integrating, (i-s4)''+(^+3i)(^+«)=<'- Since 0, the longitudinal pressure, is supposed constant, we may assume c. = 12 ={P + ^)- 9fi 3m Therefore q-p = c^-2p, therefore by (21), dp 2p _ C2 dr r r ' a linear equation, which gives 1 c. The coefficients c^ and c, must be found from the conditions of the surface of the solid. If the pressure on the exterior cylindric surface whose radius is a^ be denoted by h^, and that on the interior surface whose radius is a. by h.,, then p = h^ when r = a^ and p = h^ when r=^a^ and the general value of p is P = ^-\ V---^ -1 \ (22). dp o^iW K—K 1 /„,N a^h^ — aA a^a^ h, — K , ^ / = 6.,(^_g)=_26^^A:A (24). This last equation gives the optical effect of the pressure at any point. The law of the magnitude of this quantity is the inverse square of the radius, as in THE EQUILIBRIUM OP ELASTIC SOLIDS. 47 Case I. ; but tlie direction of the principal axes is different, as in tHs case they are parallel and perpendicular to the radius. The dark bands seen by polarized light wiU therefore be parallel and perpendicular to the plane of polarization, in- stead of being inclined at an angle of 45°, as in Case I. By substituting in Equations (18) and (20), the values of p and q given in (22) and (23), we find that when r = a.^, Sx ^ /l\ / 2 (^i%-(^^% \ . A / ^''^1 ~ ^'^^ = (^TT- + ;r- V 2 (Ml' - KO -^—. (^-^ \9/i 3m/ ^'' ^ "■' al-a^\^iL 3m, ,„i Sr 1 / ^a^h,—cCh\ 1 /a,% + 3a„% — 4:alL When r = a„ — .= —0 + 2^4 T^ + F" -^^ ^^4 —-o r 9ju, \ «! — a/ / 3m \ a/ — a/ .(25). /_1 l_\ , 1 /2a,' «/-|-3aA , a' /2 ~^\9^ 3^;''" 'ai^-a/\9/i "*■ 3m / '«,= -«/ ^9/x"'" 3m .(26). From these equations it appears that the longitudinal compression of cylin- dric tubes is proportional to the longitudinal pressure referred to unit of surface when the lateral pressures are constant, so that for a given pressure the com- pression is inversely as the sectional area of the tube. These equations may be simplified in the following cases : — 1. When the external and internal pressures are equal, or h, — \. 2. When the external pressure is to the internal pressure as the square of the interior diameter is to that of the exterior diameter, or when a^\ = a^K. 3. When the cylinder is soHd, or when a2 = 0. 4. When the solid becomes an indefinitely extended plate with a cylindric hole in it, or when a^ becomes infinite. 5. When pressure is applied only at the plane surfaces of the solid cylinder, and the cylindric surface is prevented from expanding by being inclosed m u 1 ^^ strong case, or when — = 0. 6. When pressure is applied to the cylindric surface, and the ends are retained at an invariable distance, or when — ^ = 0. 48 THE EQUILIBRIUM OF ELASTIC SOLIDS. 1. When h^ = h,, the equations of compression become (27). 3m (h -n) 3to 9/A ^ 3my ' \9/i, ' 3m, When h^ = h, = o, then Sec _ Sr _ ^1 X r 3/x " The compression of a cylindrical vessel exposed on all sides to the same hydrostatic pressure is therefore independent of m, and it may be shewn that the same is true for a vessel of any shape. 2. When a^\ = a^}i^, 8a!_ /_!_ _2_ X \9/A 3m, 7 = 9;i(^) + 3^(3^'-^) (28). \9/A 3m/ m j In this case, when o = 0, the compressions are independent of /a. 3. In a solid cylinder, ^2 = 0, The expressions for — and — are the same as those in the first case, when ^ X r • ' hi = h^. When the longitudinal pressure o vanishes, 8a; x~^^'[9iM 3m)' Sr^, /2 1_\ r ' \9/i 3m/ THE EQUILIBEIUM OF ELASTIC SOLIDS. 49 When the cyUnder is pressed on the plane sides only, Sx ^ (,9/* "^ 3m/ ' r \9ja 3m/ X Sr 4. When the solid is infinite, or when aj is infinite, r" 2(0 r 9ju, ^ ' 3m ^ ' 5. When 8r = in a soHd cylinder, hx Zo 6. When X 2m + 3jit §03 _ 8r _ 3^ as ~ ' r m + 6jLt (29). .(30). Since the expression for the efiect of a longitudiaal strain is "''V9i^"^3^j' So; X if we make VOL. I. „ 9mu ., Zx 1 E = ^, then — =o m+Sfi X E (31). 50 THE EQUILIBRIUM OF ELASTIC SOLIDS. The quantity E may be deduced from experiment on the extension of wires rods of the substance, and ju, is given in terms of m and E by the equation, .(32), or _ Em ^^%m-&E' -i J^=^ («^>' P being the extending force, h the length of the rod, s the sectional area, and hx the elongation, which may be determined by the deflection of a wire, as in the apparatus of S' Gravesande, or by direct measurement. Case IV. The only known direct method of finding the compressibility of liquids is that employed by Canton, GErsted, Perkins, Aim^, &c. The liquid is confined in a vessel with a narrow neck, then pressure is applied, and the descent of the liquid in the tube is observed, so that the difierence between the change of volume of liquid and the change of internal capacity of the vessel may be determined. Now, since the substance of which the vessel is formed is compressible, a change of the internal capacity is possible. If the pressure be applied only to the contained liquid, it is evident that the vessel will be distended, and the compressibility of the liquid will appear too great. The pressure, therefore, is commonly applied externally and internally at the same time, by means of a hydrostatic pressure produced by water compressed either in a strong vessel or in the depths of the sea. As it does not necessarily follow, from the equality of the external and internal pressures, that the capacity does not change, the equilibrium of the vessel must be determined theoretically. (Ersted, therefore, obtained from Poisson his solution of the problem, and applied it to the case of a vessel of lead. To find the cubical elasticity of lead, he apphed the theory of Poisson to the numerical results of Tredgold. As the compressibility of lead thus found was greater than that of water, (Ersted expected that the apparent compressibility of water in a lead vessel would be negative. On making the experiment the apparent compressibility was greater in lead than in glass. The quantity found THE EQUILIBRIUM OF ELASTIC SOLIDS. 51 by Tredgold from the extension of rods was that denoted by E, and the value of fjL deduced from E alone by the formulae of Poisson cannot be true, unless — = |-; and as — for lead is probably more than 3, the calculated compressi- bility is much too great. A similar experiment was made by Professor Forbes, who used a vessel of caoutchouc. As in this case the apparent compressibility vanishes, it appears that the cubical compressibility of caoutchouc is equal to that of water. Some who reject the mathematical theories as unsatisfactory, have conjec- tured that if the sides of the vessel be sufficiently thin, the pressure on both sides being equal, the compressibility of the vessel will not affect the result. The following calculations shew that the apparent compressibility of the liquid depends on the compressibUity of the vessel, and is independent of the thickness when the pressures are equal. A hollow sphere, whose external and internal radii are a, and a^, is acted on by external and internal normal pressures \ and h^, it is required to deter- mine the equilibrium of the elastic solid. The pressures at any point in the solid are : — 1. A pressure p in the direction of the radius. 2. A pressure q in the perpendicular plane. These pressures depend on the distance from the centre, which is denoted by T. d^T . ... OT The compressions at any point are -7— in the radial direction, and — in the tangent plane, the values of these compressions are : — Multiplying the last equation by r, differentiating with respect to r, and equating the result with that of the first equation, we find \9ju, 3m/ \dr dr/^m\ dr^ ^ ) 7—2 52 THE EQUILIBRIUM OF ELASTIC SOLIDS. Since the forces wHcK act on the particle in the direction of the radius must balance one another, or 2' dp , But ^ = 2^+P' and the equation becomes ^ + 3^-^ = 0. dr r r 1 c therefore P = c„ -^ + -^. ± ' ^ 3 Siace p = hi when *• = «!, and p = \ when r^a^, the value of p at any distance is found to be a% - a,% a^a^ h^ - h, ■^ a^-a^ 'f al-a,' ^"^^f- ttj /t'j "^ Ct-g /t'2 -. €fj-\ (a/q ivy ^~ /^a / ^^ <-a/ "^^^^^-a/ ^^^)- 8F_ 8r _ai%i — a^^j 1 ^a^a^ K — h^ 1 F r ~ a^—a^ fj- r^ a^ — a^ m When r = a„ 8^ = —h%('''-^') ("'■ The forces perpendicular to the axis are IdrX dp dr ^ d'r ,-. , . dx ^ . 'P [d^j +''ds ds-^''Pds^-^^^-^^^''Ts-^' = ''- Substituting for p its value, the equation gives _ {h^ — \) /dr dr dx\ (h^ — h^) /dr ds d'x ds c?V\ , . ^~ t \ds dx dsj It \dxdxdi dxds')""^ The equations of elasticity become dhs (l 1\/ K + h\ p Sr (1 l\/ K + h\ q Differentiating ^— = ^- i — r ) , and in this case ° dr dr\r I dBr _ dr dr dBs dr ds ds ds By a comparison of these values of -r— , dsj \9/A 3m p+q+~^l + ± + m ds m \9ju, 3m/ \dr dr) r dq dr m dr as THE EQiniJBRrUM OP ELASTIC SOLIDS. 59 a To obtain an expression for the curvature of the plate at the vertex, let be the radius of curvature, then, as an approximation to the equation of the plate, let x = - 2a By substituting the value of a; in the values of p and q, and in the equa- tion of elasticity, the approximate value of a is found to be <*■+*-> (li-sL)' \9/x, 3m/ m t -ISmfj. K + h, m-Six , , Since the focal distance of the mirror, or -, depends on the difference of pressures, a telescope on Mr Nasmyth's principle would act as an aneroid baro- meter, the focal distance varying inversely as the pressure of the atmosphere. Case VIL To find the conditions of torsion of a cyhnder composed of a great number of parallel wires bound together without adhering to one another. Let X be the length of the cylinder, a its radius, r the radius at any point, 8^ the angle of torsion, M the force producing torsion, Sa; the change of length, and P the longitudinal force. Each of the wires becomes a helix whose radius is r, its angular rotation hQ, and its length along the axis x — ^x. --i-^)' Its length is therefore j {rZ9) and the tension is =E\\— (l j+rM This force, resolved parallel to the axis, is SO X 1 d d p^^( 1 A rdOdr \ l/^_8xV^JSdV j 8—2 60 THE EQUILIBErUM OP ELASTIC SOLIDS. and since — and r — are small, we may assimie X X d0dr \x The force, when resolved in the tangential direction, is approximately r dddr \ x x 2 \x J } 3f=.£J^i«l5-^(SiY} (55). \2 X X &\x] ] By eliminating — between (54) and (55) we have M=t^ip.E.am (56). 2 X 2i\x) When P = Q, M depends on the sixth power of the radius and the cube of the angle of torsion, when the cylinder is composed of separate filaments. Since the force of torsion for a homogeneous cyliader depends on the fourth power of the radius and the first power of the angle of torsion, the torsion of a wire having a fibrous texture will depend on both these laws. The parts of the force of torsion which depend on these two laws may be found by experiment, and thus the difierence of the elasticities in the direction of the axis and in the perpendicular directions may be determined. A calculation of the force of torsion, on this supposition, may be found in Young's Mathematical Principles of Natural Philosophy; and it is introduced here to account for the variations from the law of Case II., which may be observed in a twisted rod. Case VIII. It is well known that grindstones and fly-wheels are often broken by the centrifugal force produced by their rapid rotation. I have therefore calculated the strains and pressure acting on an elastic cylinder revolving round its axis, and acted on by the centrifugal force alone. THE EQUILEBBIUM OP ELASTIC SOLIDS. 61 The equation of tlie equilibrium of a particle [see Equation (21)], becomes dp itr'h . -where q and p are the tangential and radial pressures, h is the weight in pounds of a cubic inch of the substance, g is twice the height in inches that a body falls in a second, t is the time of revolution of the cylinder in seconds. By substituting the value of q and -^ m. Equations (19), (20), and neglect- mg 0, 3to /J__ J. which gives ^"^^T^ + ^V^ + m)*^"^ dr gf di^j m\ dr gf dr' ) 1 ,'n'k f . 2E\„ q-p= ~Ci — + 7r-i (-4 + — Ir* q= -c. "■ r" 2gt 1 . .Tfk m (-2 + ^ r' + Co (57). r" 2gf If the radii of the surfaces of the hollow cylinder be a^ and a^, and the pressures acting on them ^j and \, then the values of Cj and c, are a^—a^ 2gt \ m When aj = 0, as in the case of a solid cylinder, c^ = 0, and ■'k I . E (58). , .TT'k f^^E\ When ^1 = 0, and r = ai, ? = I^f^-2 .(60). gt \m When q exceeds the tenacity of the substance in pounds per square inch, the cylinder will give way; and by making q equal to the number of pounds which a square inch of the substance will support, the velocity may be found at which the bursting of the cylinder will take place. 62 THE EQUILIBRIUM OF ELASTIC SOLIDS. Since 7=6a)(g-p) = — [— - 2] &r^ a transparent revolving cylinder, when polarised Hght is transmitted parallel to the axis, wiU exhibit rings whose diameters are as the square roots of an arithmetical progression, and brushes parallel and perpendicular to the plane of polarization. Case IX. A hoUow cylinder or tube is surrounded by a medium of a constant temperature while a liquid of a different temperature is made to flow through it. The exterior and interior surfaces are thus kept each at a constant tem- perature tni the transference of heat through the cylinder becomes uniform. Let V be the temperature at any point, then when this quantity has reached its limit, rdv _ v = c^logr + G2 (61). Let the temperatures at the surfaces be 6^ and 0^, and the radii of the surfaces a^ and a^, then _ ^1 — ^2 _ log <*i^2 — log (Xj^i ^ log tti — log 0.3 ' ^ log ttj — log a^ Let the coefficient of linear dilatation of the substance be Cg, then the proportional dilatation at any point will be expressed by c^v, and the equations of elasticity (18), (19), (20), become dSx ( 1 1 \ / , .X dhr /I 1 \ , X P Sr / I 1 \ , . q The equation of equilibrium is ^=P + ^£ •••••■•• (21), and siuce the tube is supposed to be of a considerable length dBx -J— = c^ 0, constant quantity. THE EQUILIBRIUM OF ELASTIC SOLIDS. 63 From these equations we find that 9ja 3m and hence v = Cilogr + c^, p may be found in terms of r. Hence Since I=hcoiq -p) = &C0 (A + A.) " c,03 - 2&..C, 1 , the rings seen in this case will differ from those described in Case III. only by the addition of a constant quantity. When no pressures act on the exterior and interior surfaces of the tube hi — h^ = 0, and /^ 1_\ -' A a,%' log g,- log g, a/ log a, - < log aA P-W^Sm) ''^^3^iogr+-^;, ___+ _____ 1^ .(62). There will, therefore, be no action on polarized light for the ring whose radius is r when r^ = 2 Ct^ 0^2 ttj 2 2 log - - Case X. Sir David Brewster has observed (Edinburgh Transactions, Vol. viii.), that when a solid cylinder of glass is suddenly heated at the cyhndric surface a polarizing force is developed, which is at any point proportional to the square of the distance from the axis of the cylinder; that is to say, that the dif- 64 THE EQUILIBBIUM OF ELASTIC SOLIDS, ference of retardation of the oppositely polarized rays of light is proportional to the square of the radius r, or 1= Ic^^r^ = hoy {q -p) = boyr -£ , dp Cj , , Since if a be the radius of the cylinder, p = when r = a, _p = 3 (r= -«==). Hence ? = J (3^' - a')- By substituting these values of p and q in equations (19) and (20), and T . d 8r dSr -r n ■, makmfif -^ r = -y- , 1 nnd, ° dr r dr -|(^+l.)'-+"" c^)- Ci being the temperature of the axis of the cyUnder, and Cg the coefficient of linear expansion for glass. Case XI. Heat is passing uniformly through the sides of a spherical vessel, such as the ball of a thermometer, it is required to determine the mechanical state of the sphere. As the methods are nearly the same as in Case IX., it will be sufficient to give the results, using the same notation. „ dv c, _ ^1 — ^2 ^ A — ^ A Cj — a-fl,^ — , Cj = , «! — ttj «! — ttj 1 /2 1 \-' 1 , When ^j = ^2 = the expression for p becomes P-W + Bm) '^^^^ ^^1 1^73^ ? - ^:3^ r + ^'^^ (a, - a,) («,» ^ a/)| ^^^)- From this value of p the other quantities may be found, as in Case IX., from the equations of Case lY. THE EQUILIBRIUM OF ELASTIC SOLIDS, 65 Case XII. When a long beam is bent into the form of a closed circular ring (as in Case v.), aU the pressures act either parallel or perpendicular to the direction of the length of the beam, so that if the beam were divided into planks, there would be no tendency of the planks to slide on one another. But when the beam does not form a closed circle, the planks into which it may be supposed to be divided will have a tendency to slide on one another, and the amount of sliding is determined by the linear elasticity of the sub- stance. The deflection of the beam thus arises partly from the bending of the whole beam, and partly from the sliding of the planks ; and since each of these deflections is small compared with the length of the beam, the total deflection wUl be the smn of the deflections due to bending and sliding. Let A=Mc = ELfdy (65). A is the stifiBiess of the beam as found in Case V., the equation of the transverse section being expressed in terms of x and y, y being measured from the neutral surface. Let a horizontal beam, whose length is 2l, and whose weight is 2w, be supported at the extremities and loaded at the middle with a weight W. Let the deflection at any point be expressed by 8^y, and let this quantity be small compared with the length of the beam. At the middle of the beam, S^y is found by the usual methods to be %=3 W^+¥'W) (66). Let B— — \xdy — -— (sectional area) i^*^)- B is the resistance of the beam to the sliding of the planks. The de- flection of the beam arising from this cause is Ky=^{w+W) (68). VOL. L 9 66 THE EQUILIBEIUM OP ELASTIC SOLIDS. This quantity is small compared with h,y, when the depth of the beam is small compared with its length. The whole deflection A.y = 8^y + S,y Ay=-^{iw+W) + ^{w+W) A.-(^.^ + i^)^-^(^+ii) ('')■ Case XIII. When the values of the compressions at any point have been found, when two different sets of forces act on a solid separately, the compressions, when the forces act at the same time, may be found by the composition of com- pressions, because the small compressions are independent of one another. It appears from Case I., that if a cylinder be twisted as there described, the compressions wiU be inversely proportional to the square of the distance from the centre. If two cylindric surfaces, whose axes are perpendicular to the plane of an indefinite elastic plate, be equally twisted in the same direction, the resultant compression in any direction may be found by adding the compression due to each resolved in that direction. The result of this operation may be thus stated geometrically. Let ^j and A^ (Fig. 1) be the centres of the twisted cylinders. Join A^A„, and bisect A^A„_ in O. Draw OBC at right angles, and cut off OB^ and OB^ each equal to OA^. Then the difference of the retardation of oppositely polarized rays of light passing perpendicularly through any point of the plane varies directly as the product of its distances from B^ and B.^, and inversely as the square of the product of its distances from A^ and A,^. The isochromatic lines are represented in the figure. The retardation is infinite at the points A^ and A^; it vanishes at B^ and B^ ; and if the retardation at be taken for unity, the isochromatic curves 2, 4, surround A^ and A^ ; that in which the retardation is unity has two loops, and passes through 0; the curves ^-, ^ are continuous, and have points of contrary flexure ; the curve ^ has multiple points at Cj and C,_, where THE EQUILIBRIUM OF ELASTIC SOLIDS. 67 Afi^ = A^A^, and two loops surrounding B^ and B^; the other curves, for which I—^, ^, &c., consist each of two ovals surrounding B^ and B^, and an exterior portion surrounding all the former curves. I have produced these curves in the jelly of isinglass described in Case I. They are best seen by using circxilarly polarized light, as the curves are then seen without interruption, and their resemblance to the calculated curves is more apparent. To avoid crowding the curves toward the centre of the figure, I have taken the values of I for the different curves, not in an arithmetical, but in a geometrical progression, ascending by powers of 2. 9—2 68 THE EQUILIBEIUM OF ELASTIC SOLIDS. Case XIV. On the determination of the pressures which act in the interior of trans- parent solids, from observations of the action of the solid on polarized light. Sir David Brewster has pointed out the method by which polarized light might be made to indicate the strains in elastic solids ; and his experiments on bent glass confirm the theories of the bending of beams. The phenomena of heated and unannealed glass are of a much more complex nature, and they cannot be predicted and explained without a knowledge of the laws of cooling and solidification, combined with those of elastic equilibrium. In Case X. I have given an example of the inverse problem, in the case of a cylinder in which the action on light followed a simple law ; and I now go on to describe the method of determining the pressures in a general case, applying it to the case of a triangle of unannealed plate-glass. D D D n The lines of equal intensity of the action on light are seen without interruption, by using circularly polarized light. They are represented in Fig. 2, where A, BBB, DDD are the neutral points, or points of no action on light, and CCC, EEE are the points where that action is greatest ; and the intensity THE EQUILIBRIUM OF ELASTIC SOLIDS. 69 of the action at any other point is determined by its position with respect to the isochromatic curves. The direction of the principal axes of pressure at any point is found by transmitting plane polarized light, and analysing it in the plane perpendicular to that of polarization. The light is then restored in every part of the triangle, except in those points at which one of the principal axes is parallel to the plane of polarization. A dark band formed of all these points is seen, which shifts its position as the triangle is turned round in its own plane. Fig. 3 represents these curves for every fifteenth degree of inclination. They correspond to the lines of equal variation of the needle in a magnetic chart. From these curves others may be found which shall indicate, by their own direction, the direction of the principal axes at any point. These curves of direction of compression and dilatation are represented in Fig. 4 ; the curves whose direction corresponds to that of compression are concave toward the centre of the triangle, and intersect at right angles the curves of dilatation. Let the isochromatic lines in Fig. 2 be determined by the equation <^,{x,y) = I- = o,{q-p)-, where / is the difference of retardation of the oppositely polarized rays, and q and p the pressures in the principal axes at any point, z being the thick- ness of the plate. Let the lines of equal inclination be determined by the equation ^2 {x, y) = tan 6, 6 being the angle of inclination of the principal axes ; then the differential equation of the curves of direction of compression and dUatation (Fig. 4) is By considering any particle of the plate as a portion of a cylinder whose axis passes through the centre of curvature of the curve of compression, we find ^-TP=i'£. (21)- 70 THE EQUILIBKIUM OF ELASTIC SOLIDS. Let R denote the radius of curv^ature of the curve of compression at any point, and let S denote the length of the curve of dilatation at the same point, dp and since {q -p), R and S are known, and since at the surface, where (j), {x, y) = 0, p = 0, all the data are given for determining the absolute value of p by inte- gration. Though this is the best method of finding p and q by graphic construc- tion, it is much better, when the equations of the curves have been found, that is, when (f>, and (f)^ are known, to resolve the pressures in the direction of the axes. The new quantities are p„ p,, and q,; and the equations are tan 9 = -^^ , {p - qf = q-! + [p^ -p.)', p. +p, =P + q- Pi -P2 It is therefore possible to find the pressures from the curves of equal tint and equal inclination, in any case in which it may be required. In the mean- time the curves of Figs. 2, 3, 4 shew the correctness of Sir John HerscheU's ingenious explanation of the phenomena of heated and unannealed glass. Note A. As the mathematical laws of compressions and pressures have been very thoroughly investigated, and as they are demonstrated with great elegance in the very complete and elaborate memoir of MM. Lam^ and Clapeyron, I shall state as briefly as possible their results. Let a solid be subjected to compressions or pressures of any kind, then, if through any point in the solid lines be drawn whose lengths, measured from the given point, are pro- portional to the compression or pressure at the point resolved in the directions in which the lines are drawn, the extremities of such lines will be in the surface of an ellipsoid, whose centre is the given point. The properties of the system of compressions or pressures may be deduced from those of the ellipsoid. THE BQUILIBEIUM OP ELASTIC SOLIDS. 71 There are three diameters having perpendicular ordinates, which are called the principal axes of the ellipsoid. Similarly, there are always three directions in the compressed particle in which there is no tangential action, or tendency of the parts to slide on one another. These directions are called the principal axes of compression or of pressure, and in homogeneous solids they always coincide with each other. The compression or pressure in any other direction is equal to the sum of the products of the compressions or pressures in the principal axes multiplied into the squares of the cosines of the angles which they respectively make with that direction. KoTE B. The fundamental equations of this paper differ from those of Navier, Poisson, &c., only in not assuming an invariable ratio between the linear and the cubical elasticity; but since I have not attempted to deduce them from the laws of molecular action, some other reasons must be given for adopting them. The experiments from which the laws are deduced are — 1st. Elastic solids put into motion vibrate isochronously, so that the sound does not vary with the amplitude of the vibrations. 2nd. Kegnault's experiments on hollow spheres shew that both linear and cubic com- pressions are proportional to the pressures. 3rd. Experiments on the elongation of rods and tubes immersed in water, prove that the elongation, the decrease of diameter, and the increase of volume, are proportional to the tension. 4th. In Coulomb's balance of torsion, the angles of torsion are proportional to the twisting forces. It would appear from these experiments, that compressions are always proportional to pressures. Professor Stokes has expressed this by making one of his coefficients depend on the cubical elasticity, while the other is deduced from the displacement of shifting produced by a given tangential force. M. Cauchy makes one coefficient depend on the linear compression produced by a force acting in one direction, and the other on the change of volume produced by the same force. Both of these methods lead to a correct result ; but the coefficients of Stokes seem to have more of a real signification than those of Cauchy ; I have therefore adopted those of Stokes, using the symbols m and /a, and the fundamental equations (4) and (5), which define them. 72 THE EQUILIBRIUM OE ELASTIC SOLIDS. Note C. As the coefficient co, which determines the optical effect of pressure on a substance, varies from one substance to another, and is probably a function of the linear elasticity, a determination of its value in different substances might lead to some explanation of the action of media on light. This paper commenced by pointing out the insufiSciency of all theories of elastic solids, in which the equations do not contain two independent constants deduced from experiments. One of these constants is common to liquids and solids, and is called the modulus of cubical elasticity. The other is peculiar to solids, and is here called the modulus of linear elasticity. The equations of Navier, Poisson, and Lamd and Clapeyron, contain only one coefficient ; and Professor G. G. Stokes of Cambridge, seems to have formed the first theory of elastic solids which recognised the independence of cubical and linear elasticity, although M. Cauchy seems to have suggested a modification of the old theories, which made the ratio of linear to cubical elasticity the same for all substances. Professor Stokes has deduced the theory of elastic solids from that of the motion of fluids, and his equations are identical with those of this paper, which are deduced from the two following assumptions. In an element of an elastic solid, acted on by three pressures at right angles to one another, as long as the compressions do not pass the limits of perfect elasticity — 1st. The sum of the pressures, in three rectangular axes, is proportional to the sum of the compressions in those axes. 2nd. The difference of the pressures in two axes at right angles to one another, is proportional to the difference of the compressions in those axes. Or, in symbols : fi being the modulus of cubical, and m that of linear elasticity. These equations are found to be very convenient for the solution of problems, some of which were given in the latter part of the paper. THE EQUILIBEIUM OP ELASTIC SOLIDS. 73 These particular cases were — That of an elastic hollow cylinder, the exterior surface of which was fixed, while the interior was turned through a small angle. The action of a transparent solid thus twisted on polarized light, was calculated, and the calculation confirmed by experiment. The second case related to the torsion of cylindric rods, and a method was given by which m may be found. The quantity E = — was found by elongating, or by bending the rod used to determine m, and fi is found by the equation, _ Em '^~9m-QE' The effect of pressure on the surfaces of a hollow sphere or cylinder was calculated, and the result applied to the determination of the cubical compressibility of liquids and solids. An expression was found for the curvature of an elastic plate exposed to pressure on one side ; and the state of cylinders acted on by centrifugal force and by heat was determined. The principle of the superposition of compressions and pressures was applied to the case of a bent beam, and a formula was given to determine E from the deflection of a beam supported at both ends and loaded at the middle. The paper concluded with a conjecture, that as the quantity co (which expresses the relation of the inequality of pressure in a solid to the doubly-refracting force produced) is probably a function of m, the determination of these quantities for different substances might lead to a more complete theory of double refraction, and extend our knowledge of the laws of optics. VOL. I, 10 [Extracted from the Cambridge and Dublin Mathematical Journal, Vol. viii. p. 188, February, 1854.] Solutions of Problems. 1. If from a point in the circumference of a vertical circle two heavy particles be suc- cessively projected along the curve, their initial velocities being equal and either in the same or in opposite directions, the subsequent, motion will be such that a straight line joining the particles at any instant will touch a circle. Fote. The particles are supposed not to interfere with each other's motion. The direct analytical proof would involve the properties of elliptic integrals, but it may be made to depend upon tbe following geometrical theorems. (l) If from a point in one of two circles a right line be drawn cutting the other, the rectangle contained by the segments so formed is double of the rectangle contained by a line drawn from the point perpendicular to the radical axis of the two circles, and the line joining their centres. The radical axis is the line joining the points of intersection of the two circles. It is always a real line, whether the points of intersection of the circles be real or imaginary, and it has the geometrical property — that if from any point on the radical axis, straight lines be drawn cutting the circles, the rectangle con- tained by the segments formed by one of the circles is equal to the rectangle contained by the segments formed by the other. The analytical proof of these propositions is very simple, and may be resorted to if a geometrical proof does not suggest itself as soon as the requisite figure is constructed. If ^, B be the centres of the circles, P the given point in the circle whose centre is A, a line drawn from P cuts the first circle in p, the second in Q SOLUTIONS OF PROBLEMS. 75 and q, and the radical axis in R. If PH be drawn perpendicular to the radical axis, then PQ.Pq = 2AB,HP. CoR. If the line be drawn from P to touch the circle in T, instead of cutting it in Q and q, then the square of the tangent PT is equal to the rectangle 2AB . HP. Similarly, if ph be drawn from p perpendicular to the radical axis pr' = 2AB.hp. Hence, if a line be drawn touching one circle in T, and cutting the other in P and p, then {PTf : {pTf :: HP : hp. (2) If two straight lines touching one circle and cutting another be made to approach each other indefinitely, the small arcs intercepted by their inter- sections with the second circle will be ultimately proportional to their distances from the point of contact. This result may easily be deduced from the properties of the similar triangles FTP and ppT. CoR. If particles P, p be constrained to move in the circle A, while the line Pp joining them continually touches the circle B, then the velocity of P at any instant is to that of p as PT to pT; and conversely, if the velocity of P at any instant be to that of P as PT to pT, then the line Pp win continue to be a tangent to the circle B. Now let the plane of the circles be vertical and the radical axis horizontal, and let gravity act on the particles P, p. The particles were projected from the same point with the same velocity. Let this velocity be that due to the depth of the point of projection below the radical axis. Then the square of the velocity at any other point will be proportional to the perpendicular from that point on the radical axis ; or, by the corollary to (l), if P and p be at any time at the extremities of the Hne PTp, the square of the velocity of P win be to the square of the velocity of p as PH to ph, that is, as {PT)'' to {pT)\ Hence, the velocities of P and p are in the proportion of PT to pT, and therefore, by the corollary to (2), the Hne joining them will continue a tangent to the circle B during each instant, and will therefore remam a tangent during the motion. 10—2 76 SOLUTIONS OF PBOBLEMS, The circle A, the radical axis, and one position of the line Pp, are given by the circumstances of projection of P and p. From these data it is easy to determine the circle 5 by a geometrical construction. It is evident that the character of the motion will determine the position of the circle B. If the motion is oscillatory, B wiU intersect A. If P and p make complete revolutions in the same direction, B will lie entirely within A, but if they move in opposite dhections, B wiU lie entirely above the radical axis. If any number of such particles be projected from the same point at equal intervals of time with the same direction and velocity, the lines joining successive particles at any instant will be tangents to the same circle; and if the time of a complete revolution, or oscillation, contain n of these intervals, then these lines will form a polygon of n sides, and as this is true at any instant, any number of such polygons may be formed. Hence, the following geometrical theorem is true : "If two cu-cles be such that n lines can be drawn touching one of them and having their successive intersections, including that of the last and first, on the circumference of the other, the construction of such a system of lines wiU be possible, at whatever point of the first circle we draw the first tangent." 2. A transparent medium is suda that the path of a ray of light within it is a given circle, the index of refraction being a function of the distance from a given point in the plane of the circle. Find the form of this function and shew that for light of the same refrangibility — (1) The path of every ray within the medium is a circle. (2) All the rays proceeding from any point in the medium will meet accurately in another point. (3) If rays diverge from a point without the medium and enter it through a spherical surface having that point for its centre, they will be made to converge accurately to a point within the medium. Lemma I. Let a transparent medium be so constituted, that the refractive index is the same at the same distance from a fixed point, then the path of any ray of light within the medium will be in one plane, and the perpen- SOLUTIONS OP PROBLEMS. 1*1 dicular from the fixed point on the tangent to the path of the ray at any point wUl vary inversely as the refractive index of the medium at that point. We may easily prove that when a ray of light passes through a spherical surface, separating a medium whose refractive index is \l^ from another where it is /1A2, the plane of incidence and refraction passes through the centre of the sphere, and the perpendiculars on the direction of the ray before and after refraction are in the ratio of jlij to [i^. Since this is true of any number of spherical shells of different refractive powers, it is also true when the index of refraction varies continuously from one shell to another, and therefore the proposition is true. Lemma II. If from any fixed point in the plane of a circle, a perpen- dicular be drawn to the tangent at any point of the circumference, the rectangle contained by this perpendicular and the diameter of the circle is equal to the square of the line joining the point of contact with the fixed point, together with the rectangle contained by the segments of any chord through the fixed point. Let APB be the circle, the fixed point ; then OY.FR^OF'^AO.OB. Produce PO to Q, and join QR, then the triangles OYP, PQR are similar; therefore OY.PR=OP.PQ ^OP' + OP,.OQ; .-. OY.PR = OR + AO.OB. If we put in this expression AO . OB = du ' All these quantities being functions of u, l, 6, cr and <^, are functions of u and of each other; and if the forms of these functions be known, the positions of all the generating lines may be successively determined, and the equation to the surface may be found by integrating the equations containing the values of t,, 6, cr and ^. When the surface is bent in any manner about the generating lines, t„ 0, and or remain unaltered, but (f> is changed at every point. The form of ^ as a function of u will depend on the nature of the bending ; but since this is perfectly arbitrary, may be any arbitrary function of u. In this way we may find the form of any surface produced by bending the given surface along its generating lines. By making ^ = 0, we make all the generating lines parallel to the same plane. Let this plane be that of xy, and let the first generating line coincide with the axis of x, then C will be the height of any other generating line above the plane of xy, and the angle which its projection on that plane makes with the axis of x. The ultimate intersections of the projections of the generating lines on the plane of xy will form a curve, whose length, measured from the axis of x, will be cr. 11—2 84 TRANSFORMATION OF SURFACES BY BENDING. Since in this case the quantities {, 9, and o- are represented by distinct geometrical quantities, we may simphfy the consideration of all surfaces generated by straight lines by reducing them by bending to the case in which those lines are parallel to a given plane. In the class of surfaces in which the generating lines ultimately intersect, ^ = 0, and I constant. If these surfaces be bent so that <^ = 0, the whole of the generating lines will lie in one plane, and their ultimate intersections will form a plane curve. The surface is thus reduced to one plane, and therefore belongs to the class usually described as "developable surfaces." The form of a developable surface may be defined by means of the three quantities 6, cr and (f). The generating lines form by their ultimate intersections a curve of double curvature to which they are all tangents. This curve has been called the cuspidal edge. The length of this curve is represented by cr, its absolute curvature at any point by t— , and its torsion at the same point by -r- . When the surface is developed, the cuspidal edge becomes a plane curve, and every part of the surface coincides with the plane. But it does not follow that every part of the plane is capable of being bent into the original form of the surface. This may be easily seen by considering the surface when the position of the cuspidal edge nearly coincides with the plane curve but is not confounded with it. It is evident that if from any point in space a tangent can be drawn to the cuspidal edge, a sheet of the surface passes through that point. Hence the number of sheets which pass through one point is the same as the number of tangents to the cuspidal edge which pass through tliat point ; and since the same is true in the limit, the number of sheets which coincide at any point of the plane is the same as the number of tangents which can be drawn from that point to the plane curve. In constructing a developable surface of paper, we must remove those parts of the sheet from which no real tangents can be drawn, and provide additional sheets where more than one tangent can be drawn. In the case of developable surfaces we see the importance of attending to the position of the hues of bending; for though all developable surfaces may be produced from the same plane surface, their distinguishing properties depend on the form of the plane curve which determines the lines of bending. TRANS b^ORMATION OF SURFACES BY BENDING. 85 II. On the Bending of Surfaces of Revolution. In the cases previously considered, the bending in one part of the surface may take place independently of that in any other part. In the case now before us the bending must be simultaneous over the whole surface, and its nature must be investigated by a different method. The position of any point P on a surface of revolution may be deter- mined by the distance PV from the vertex, measured along a generating Une, and the angle A VO which the plane of the generating line makes with a fixed plane through the axis. Let FV=s and AVO = d. Let r be the distance (Pp) of P from the axis; r will be a function of s depending on the form of the generating curve. Now consider the small rectangular element of the surface at P. PR = Ss, and its breadth PQ — rS9, where r is a function of s. If in another surface of revolution r is some other function of s, then the length and breadth of the new element will be Ss and r'BO', and if Its length r — fir, and ff = -6, r'W = rhe, and the dimensions of the two elements will be the same. Hence the one element may be applied, to the other, and the one surface may be applied to the other surface, element to element, by bending it. To effect this, the surface must be divided by cutting it along one of the generating lines, and the parts opened out, or made to overlap, according as /a is greater or less than unity. To find the effect of this transformation on the form of the surface we must find the equation to the original form of the generating line in terms of s and r, then putting r' = [jir, the equation between s and r will give the form of the generating line after bending. 86 TRANSFORMATION OF SURFACES BY BENDING. dv When ^ is greater than 1 it may happen that for some values of s, ^ is greater than -. In this case ^ = „^ is greater than 1 ; a result which indicates that the curve becomes impossible for such values of A- and /x. The transformation is therefore impossible for the corresponding part of the surface. If, however, that portion of the original surface be removed, the remainder may be subjected to the required transformation. The theory of bending when apphed to the case of surfaces of revolution presents no geometrical difBculty, and little variety; but when we pass to the consideration of surfaces of a more general kind, we discover the insufficiency of the methods hitherto employed, by the vagueness of our ideas with respect to the nature of bending in such cases. In the former case the bending is of one kind only, and depends on the variation of one variable ; but the surfaces we have now to consider may be bent in an infinite variety of ways, depending on the variation of three variables, of which we do not yet know the nature or interdependence. We have therefore to discover some method sufficiently general to be appli- cable to every possible case, and yet so definite as to limit each particular case to one kind of bending easily understood. The method adopted in the following investigations is deduced from the consideration of the surface as the limit of the inscribed polyhedron, when the size of the sides is indefinitely diminished, and their number indefinitely increased. A method is then described by which such a polyhedron may be inscribed in any surface so that all the sides shall be triangles, and all the solid angles composed of six plane angles. The problem of the bending of such a polyhedron is a question of trigo- nometry, and equations might be found connecting the angles of the different edges which meet in each solid angle of the polyhedron. It vdll be shewn that TRANSFOKMATION OF SURFACES BY BENDING. 87 the conditions thus obtained would be equivalent to three equations between the six angles of the edges belonging to each solid angle. Hence three addi- tional conditions would be necessary to determine the value of every such angle, and the problem would remain as indefinite as before. But if by any means we can reduce the number of edges meeting in a point to four, only one con- dition would be necessary to determine them all, and the problem would be reduced to the consideration of one kind of bending only. This may be done by drawing the polyhedron in such a manner that the planes of adjacent triangles coincide two and two, and form quadrilateral facets, four of which meet in every solid angle. The bending of such a polyhedron can take place only in one way, by the increase of the angles of two of the edges which meet in a pomt, and the diminution of the angles of the other two. The condition of such a polyhedron being inscribed in any surface is then found, and it is shewn that when two forms of the same surface are given, a perfectly definite rule may be given by which two corresponding polyhedrons of this kmd may be inscribed, one in each surface. Since the hind of bending completely defines the nature of the quadrilateral polyhedron which must be described, the lines formed by the edges of the quadrilateral may be taken as an indication of the kind of bending performed on the surface. These lines are therefore defined as " Lines of Bending." When the lines of bending are given, the forms of the quadrilateral facets are completely determined ; and if we know the angle which any two adjacent facets make with one another, we may determine the angles of the three edges which meet it at one of its extremities. From each of these we may find the angles of three other edges, and so on, so that the form of the polyhedron after bending wUl be completely determined when the angle of one edge is given. The bending is thus made to depend on the change of one variable only. In this way the angle of any edge may be calculated from that of any given edge ; but since this may be done in two different ways, by passing along two different sets of edges, we must have the condition that these results may be consistent with each other. This condition is satisfied by the method of inscribing the polyhedron. Another condition will be necessary that the change of the angle of any edge due to a small change of the given angle, produced by bending, may be the same by both calculations. This is the con- dition of " Instantaneous Lines of Bending." That this condition may continue 88 TRANSFORMATION OF SURFACES BY BENDING. to be satisfied during the whole process we must have another, which is the condition for "Permanent Lines of Bending." The use of these hues of bending in simplifying the theory of surfaces is the only part of the present method which is new, although the investigations connected with them naturally led to the employment of other methods which had been used by those who have already treated of this subject. A state- ment of the principal methods and results of these mathematicians will save repetition, and will indicate the different points of view under which the subject may present itself The first and most complete memoir on the subject is that of M. Gauss, already referred to. The method which he employs consists in referring every point of the surface to a corresponding point of a sphere whose radius is unity. Normals are drawn at the several points of the surface toward the same side of it, then lines drawn through the centre of the sphere in the direction of each of these normals intersect the surface of the sphere in points corresponding to those points of the original surface at which the normals were drawn. If any line be drawn on the surface, each of its points will have a corresponding point on the sphere, so that there will be a corresponding fine on the sphere. If the line on the surface return into itself, so as to enclose a finite area of the surface, the corresponding curve on the sphere will enclose an area on the sphere, the extent of which will depend on the form of the surface. This area on the sphere has been defined by M. Gauss as the measure of the " entire curvature " of the area on the surface. This mathematical quantity is of great use in the theory of surfaces, for it is the only quantity connected with curvature which is capable of being expressed as the sum of all its parts. The sum of the entire curvatures of any number of areas is the entire curvature of their sum, and the entire curvature of any area depends on the form of its boundary only, and is not altered by any change in the form of the surface within the boundary line. The curvature of the surface may even be discontinuous, so that we may speak of the entire curvature of a portion of a polyhedron, and calculate its amount. If the dimensions of the closed curve be duninished so that it may be treated as an element of the surface, the ultimate ratio of the entire curvature TRANSFORMATION OF SURFACES BY BENDING. 89 to the area of the element on the surface is taken as the measure of the " specific curvature " at that point of the surface. The terms "entire" and "specific" curvature when used in this paper are adopted from M. Gauss, although the use of the sphere and the areas on its surface formed an essential part of the original design. The use of these terms will save much explanation, and supersede several very cumbrous expressions. M. Gauss then proceeds to find several analytical expressions for the measure of specific curvature at any point of a surface, by the consideration of three points very near each other. The co-ordinates adopted are first rectangular, x and y, or x, y and z, being regarded as independent variables. Then the points on the surface are referred to two systems of curves drawn on the surface, and their position is defined by the values of two independent variables p and q, such that by varying p while q remains constant, we obtain the difierent points of a line of the first system, while p constant and q variable defines a line of the second system. By means of these variables, points on the surface may be referred to lines on the surface itself instead of arbitrary co-ordinates, and the measure of cur- vature may be found in terms of p and q when the surface is known. In this way it is shewn that the specific curvature at any point is the reciprocal of the product of the principal radii of curvature at that point, a result of great interest. From the condition of bending, that the length of any element of the curve must not be altered, it is shewn that the specific curvature at any point is not altered by bending. The rest of the memoir is occupied with the consideration of particular modes of describing the two systems of lines. One case is when the lines of the first system are geodesic, or " shortest " lines having their origin in a point, and the second system is drawn so as to cut ofi" equal lengths from the curves of the first system. The angle which the tangent at the origin of a line of the first system makes with a fixed line is taken as one of the co-ordinates, and the distance of the point measured along that Kne as the other. It is shewn that the two systems intersect at right angles, and a simple expression is found for the specific curvature at any point. M, Liouville [Journal, Tom, xii.) has adopted a different mode of simpli- VOL. I. 12 90 TRANSFORMATION OF SURFACES BY BENDING. lying the problem. He has shewn that on every surface it is possible to find two systems of curves intersecting at right angles, such that the length and breadth of every element into which the surface is thus divided shall be equal, and that an infinite number of such systems may be found. By means of these curves he has found a much simpler expression for the specific curvature than that given by M. Gauss. He has also given, in a note to his edition of Monge, a method of testing two given surfaces in order to determine whether they are applicable to one another. He first draws on both surfaces lines of equal specific curvature, and determines the distance between two corresponding consecutive lines of curvature in both surfaces. If by assuming the origin properly these distances can be made equal for every part of the surface, the two surfaces can be applied to each other. He has developed the theorem analytically, of which this is only the geometrical interpretation. When the lines of equal specific curvature are equidistant throughout their whole length, as in the case of surfaces of revolution, the surfaces may be applied to one another in an infinite variety of ways. When the specific curvature at every point of the surface is positive and equal to a% the surface may be applied to a sphere of radius a, and when the specific curvature is negative = —a" it may be applied to the surface of revo- lution which cuts at right angles all the spheres of radius a, and whose centres are in a straight line. M. Bertrand has given in the Xlllth Vol. of Liouville's Journal a very simple and elegant proof of the theorem of M. Gauss about the product of the radii of curvature. He supposes one extremity of an inextensible thread to be fixed at a point in a surface, and a closed curve to be described on the surface by the other extremity, the thread being stretched all the while. It is evident that the length of such a curve cannot be altered by bending the surface. He then calculates the length of this curve, considering the length of the thread small, and finds tha,t it depends on the product of the principal radii of curvature of the surface at the fixed point. His memoir is followed by a note of M. Diguet, who deduces the same resiilt from a consideration of the area of the same curve; and by an independent memoir of M. Puiseux, who seems to give the same proof at somewhat greater length. TRANSFORMATION OF SURFACES BY BENDING. 91 Note. Since this paper was written, I have seen the Rev. Professor Jellett's Memoir, On the Properties of Inextensihle Surfaces. It is to be found in the Transactions of the Royal Irish Academy, Vol. XXII. Science, &c., and was read May 23, 1853. Professor Jellett has obtained a system of three partial differential equations which express the conditions to which the displacements of a continuous inextensihle membrane are subject. From these he has deduced the two theorems of Gauss, relating to the invariability of the product of the radii of curvature at any point, and of the " entire curvature" of a finite portion of the surface. He has then applied his method to the consideration of cases in which the flexibility of the surface is limited by certain conditions, and he has obtained the following results : — If the displacements of an inextensihle surface he all parallel to the same plane, the surface moves as a rigid body. Or, more generally, If the movement of an inextensihle surface, parallel to any one line, be that of a rigid body, the entire movement is that of a rigid body. The following theorems relate to the case in which a curve traced on the surface is rendered rigid : — If any curve he traced upon an inextensihle surface whose principal radii of curvature are finite and of the same sign, and if this curve be rendered immoveable, the entire surface will become immoveable also. In a developable surface composed of an inextensihle membrane, any one of its rectilinear sections may be fi,xed without destroying the flexibility of the membrane. In convexo-concave surfaces, there are two directions passing through every point of the surface, such that the curvature of a normal section taken in these directions vanishes. We may therefore conceive the entire surface to be crossed by two series of curves, such that a tangent drawn to either of them at any point shall coincide with one of these direc- tions. These curves Professor Jellett has denominated Curves of Flexure, from the following properties : — Any curve of flexure may be fixed without destroying the flexibility of the surface. If an arc of a curve traced upon an inextensihle surface he rendered fixed or rigid, the entire of the quadrilateral, formed by drawing the two cu/rves of flexure through each extremity of the curve, becomes fixed or rigid also. Professor Jellett has also investigated the properties of partially inextensihle surfaces, and of thin material laminae whose extensibility is small, and in a note he has demonstrated the following theorem.: — If a closed oval surface he perfectly inextensihle, it is also perfectly rigid. A demonstration of one of Professor Jellett's theorems will be found at the end of this paper. J. C. M. Aug. 30, 1854. 12—2 / I — ^ — 7^ /il V IX /\/' y ,/\ / v^ 92 TEANSFORMATION OF SURFACES BY BENDING. On the properties of a Surface considered as the limit of the inscribed Polyhedron. 1. To inscribe a polyhedron in a given surface, all whose sides shall he triangles, and all whose solid angles shall be hexahedral. On the given surface describe a series of curves according to any assumed law. Describe a second series intersecting these in any manner, so as to divide the whole surface into quadrilaterals. Lastly, describe a third series (the dotted lines in the figure), so as to pass through aU the intersections of the first and second series, forming the diagonals of the quadrilaterals. The surface is now covered with a network of curvilinear triangles. The plane triangles which have the same angular points will form a polyhedron fulfilling the required conditions. By increasing the number of the curves in each series, and diminishing their distance, we may make the polyhedron approximate to the surface without limit. At the same time the polygons formed by the edges of the polyhedron will approximate to the three systems of intersecting curves. 2. To find the measure of the " entire curvature " of a solid angle of the polyhedron, and of a finite portion of its sicrface. From the centre of a sphere whose radius is unity draw perpendiculars to the planes of the six sides forming the solid angle. These lines will meet the surface in six points on the same side of the centre, which being joined by arcs of great circles will form a hexagon on the surface of the sphere. The area of this hexagon represents the entire curvature of the solid angle. It is plain by spherical geometry that the angles of this hexagon are the supplements of the six plane angles which form the solid angle, and that the arcs forming the sides are the supplements of those subtended by the angles of the six edges formed by adjacent sides. The area of the hexagon is equal to the excess of the sum of its angles above eight right angles, or to the defect of the sum of the six plane angles from four right angles, which is the same thing. Since these angles are TRANSFORMATION OF SURFACES BY BENDING. 93 invariable, the bending of the polyhedron cannot alter the measure of curvature of each of its solid angles. If perpendiculars be drawn to the sides of the polyhedron which contain other solid angles, additional points on the sphere will be found, and if these be joined by arcs of great circles, a network of hexagons will be formed on the sphere, each of which corresponds to a solid angle of the polyhedron and represents its " entire curvature." The entire curvature of any assigned portion of the polyhedron is the sum of the entire curvatures of the solid angles it contains. It is therefore repre- sented by a polygon on the sphere, which is composed of all the hexagons corresponding to its solid angles. If a polygon composed of the edges of the polyhedron be taken as the boundary of the assigned portion, the sum of its exterior angles will be the same as the sum of the exterior angles of the polygon on the sphere ; but the area of a spherical polygon is equal to the defect of the sum of its exterior angles from four right angles, and this is the measure of entire curva- ture. Therefore the entire curvature of the portion of the polyhedron enclosed by the polygon is equal to the defect of the sum of its exterior angles from four right angles. Since the entire curvature of each solid angle is unaltered by bending, that of a finite portion of the surface must be also invariable. 3. On the " Conic of Contact," and its use in determining the curvature of normal sections of a surface. Suppose the plane of one of the triangular facets of the polyhedron to be produced till it cuts the surface. The form of the curve of intersection ^dll depend on the nature of the surface, and when the size of the triangle is indefinitely diminished, it will approximate to the form of a conic section. For we may suppose a surface of the second order constructed so as to have a contact of the second order with the given surface at a point within the angular points of the triangle. The curve of intersection with this surface will be the conic section to which the other curve of intersection approaches. This curve will be henceforth called the " Conic of Contact," for want of a better name. 94 TRANSFORMATION OF SURFACES BY BENDING. To find the radius of curvature of a normal section of the surface. Let ARa be the conic of contact, C its centre, and CP perpendicular to its plane. rPR a normal section, and its centre of curvature, then PR PO = l ^ CP CR- — Ittd ii^ ^'^^ limit, when CR and PR coincide, -8 (jp' or calling CP the "sagitta," we have this theorem: " The radius of curvature of a normal section is equal to the square of the corresponding diameter of the conic of contact divided by eight times the sagitta." 4. To inscribe a j^olyhedron in a given surface, all ivhose sides shcdl he ■plane quadrilaterals, and all whose solid angles shall he tetrahedral. Suppose the three systems of curves drawn as described in sect. (1), then each of the quadrilaterals formed by the intersection of the first and second systems is divided into two triangles by the third system. If the planes of these two triangles coincide, they form a plane quadrilateral, and if every such pair of triangles coincide, the polyhedron will satisfy the required condition. Let ahc be one of these triangles, and acd the other, which is to be in the same plane with ahc. Then if the plane of ahc be produced to meet the surface in the conic of contact, the curve will pass _ through ahc and d. Hence ahcd must be a quad- rilateral inscribed in the conic of contact. But since ah and do belong to the same system of curves, they will be ultimately parallel when the size of the facets is diminished, and for a similar reason, ad and ho will be ultimately parallel. Hence ahcd will become a paral- lelogram, but the sides of a parallelogram inscribed in a conic are parallel to conjugate diameters. TRANSFORMATION OF SURFACES BY BENDING. 95 Therefore the directions of two curves of the first and second system at their point of intersection must be parallel to two conjugate diameters of the conic of contact at that point in order that such a polyhedron may be inscribed. Systems of curves intersecting in this manner wUl be referred to as "conju- gate systems." 5. On the elementary cotiditions of the appliGobility of two surfaces. It is evident, that if one surface is capable of being applied to another by bending, every point, line, or angle in the first has its corresponding point, line, or angle in the second. If the transformation of the surface be effected without the extension or contraction of any part, no line drawn on the surface can experience any change in its length, and if this condition be fulfilled, there can be no extension or contraction. TheiacLore the condition of bending is, that if any line whatever be drawn on the first surface, the corresponding curve on the second surface is equal to it in length. All other conditions of bending may be deduced from this. 6. If two curves on the first surface intersect, the corresponding curves on the second surface intersect at the same angle. On the first surface draw any curve, so as to form a triangle with the curves already drawn, and let the sides of this triangle be indefinitely dimin- ished, by making the new curve approach to the intersection of the former curves. Let the same thing be done on the second surface. We shall then have two corresponding triangles whose sides are equal each to each, by (5), and since their sides are indefinitely small, we may regard them as straight lines. Therefore by Euclid i. 8, the angle of the first triangle formed by the intersection of the two curves is equal to the corresponding angle of the second. 7. At any given point of the first surface, two directions can he found, which are conjugate to each other with respect to the conic of contact at that point, and continue to he conjugate to each other ivhen the first surface is transformed into the second. For let the first surface be transferred, without changing its form, to a position such that the given point coincides with the corresponding point of the second surface, and the normal to the first surface coincides with that of the 96 TRANSFOKMATION OF SURFACES BY BENDING. second at the same point. Then let the first surface be turned about the normal as an axis tiU the tangent of any line through the point coincides with the tangent of the corresponding line in the second surface. Then by (6) any pair of corresponding lines passing through the point wUl have a common tangent, and wiU therefore coincide in direction at that point. If we now draw the conies of contact belonging to each surface we shall have two conies with the same centre, and the problem is to determine a pair of conjugate diameters of the first which coincide with a pair of conjugate diameters of the second. The analytical solution gives two directions, real, coincident, or impossible, for the diameters required. In our investigations we can be concerned only with the case in which these directions are real. When the conies intersect in four points, P, Q, R, S, PQBS is a r^prallelo- gram inscribed in both conies, and the axes CA, CB, parallel to the sides, are conjugate in both conies. If the conies do not intersect, describe, through any point F of the second conic, a conic similar to and con- centric with the first. If the conies intersect in four points, we must proceed as before; if they touch in two points, the diameter through those points and its conju- gate must be taken. If they intersect in two points only, then the problem is impossible ; and if they coincide altogether, the conies are similar and similarly situated, and the problem is indeterminate. 8. Two surfaces being given as before, one pair of conjugate systems of curves may be drawn on the first surface, which shall correspond to a 'pair of conjugate systems on the second surface. By article (7) we may find at every point of the first surface two directions conjugate to one another, corresponding to two conjugate directions on the second surface. These directions indicate the directions of the two systems of curves which pass through that point. Knowing the direction which every curve of each system must have at every point of its course, the systems of curves may be either drawn by some direct geometrical method, or constructed from their equations, which may be found by solving their differential equations. TRANSFORMATION OF SURFACES BY BENDING. 97 Two systems of curves being drawn on the first surface, the corresponding systems may be drawn on the second surface. These systems being conjugate to each other, fuMl the condition of Art. (4), and may therefore be made the means of constructing a polyhedron with quadrilateral facets, by the bending of which the transformation may be effected. These systems of curves will be referred to as the "first and second systems of Lines of Bending." 9. General considerations applicable to Lines of Bending. It has been shewn that when two forms of a surface are given, one of which may be transformed into the other by bending, the nature of the lines of bending is completely determined. Supposing the problem reduced to its analytiosil expression, the equations of these curves would appear under the form of dou solutions of differential eqiiations of the first order and second degree, each of wnich would involve one arbitrary quantity, by the variation of which we should pass from one curve to another of the same system. Hence the position of any curve of either system depends on the value assumed for the arbitrary constant ; to distinguish the systems, let us call one the first system, and the other the second, and let all quantities relating to the second system be denoted by accented letters. Let the arbitrary constants introduced by integration be u for the first system, and u for the second. Then the value of u will determine the position of a curve of the first system, and that of u a curve of the second system, and therefore u and u will suffice to determine the point of intersection of these two curves. Hence we may conceive the position of any point on the surface to be determined by the values of u and u for the curves of the two systems which intersect at that point. By taking into account the equation to the surface, we may suppose x, y, and z the co-ordinates of any point, to be determined as functions of the two variables u and u. This being done, we shall have materials for calculating everything connected with the surface, and its lines of bending. But before entering on such calculations let us examine the principal properties of these lines which we must take into account. Suppose a series of values to be given to u and u, and the corresponding curves to be drawn on the surface. VOL, I. 13 98 TRANSFOEMATION OF SUBFACES BY BENDING. The surface will then be covered with a system of quadrilaterals, the size of which may be diminished indefinitely by interpolating values of u and u' between those already assumed; and in the limit each quadrilateral may be regarded as a parallelogram coinciding with a facet of the inscribed polyhedron. The length, the breadth, and the angle of these parallelograms will vary at different parts of the surface, and will therefore depend on the values of u and u. The curvature of a line drawn on a surface may be investigated by consider- ing the curvature of two other lines depending on it. The first is the projection of the line on a tangent plane to the surface at a given point in the line. The curvature of the projection at the point of contact may be called the tangential curvature of the line on the surface. It has also been called the geodesic curvature, because it is the measure of its deviation from a geodesic or shortest line on the surface. The other projection necessary to define the curvature of a line on the surface is on a plane passing through the tangent to the curve and the normal to the surface at the point of contact. The curvature of this projection at that point may be called the normal curvature of the line on the surface. It is easy to shew that this normal curvature is the same as the curvature of a normal section of the surface passing through a tangent to the curve at the same point. 10. General considerations applicable to the inscribed polyhedron. When two series of lines of bending belonging to the first and second systems have been described on the surface, we may proceed, as in Art. (l), to describe a third series of curves so as to pass through all their intersections and form the diagonals of the quadrilaterals formed by the first pair of systems. Plane triangles may then be constituted within the surface, having these points of intersection for angles, and the size of the facets of this polyhedron may be diminished indefinitely by increasing the number of curves in each series. But by Art. (8) the first and second systems of lines of bending are conju- gate to each other, and therefore by Art. (4) the polygon just constructed will liave every pair of triangular facets in the same plane, and may therefore be TRANSFORMATION OF SURFACES BY BENDING. 99 considered as a polyhedron with plane quadrilateral facets all whose solid angles are formed by four of these facets meeting in a point. When the number of curves in each system is increased and their distance diminished indefinitely, the plane facets of the polyhedron will ultimately coincide with the curved surface, and the polygons formed by the successive edges between the facets, will coincide with the lines of bending. These quadrilaterals may then be considered as parallelograms, the length of which is determined by the portion of a curve of the second system inter- cepted between two curves of the first, while the breadth is the distance of two curves of the second system measured along a curve of the first. The expressions for these quantities wUl be given when we come to the calculation of our results along with the other particulars which we only specify at present. The angle of the sides of these parallelograms wiU be ultimately the same as the angle of intersection of the first and second systems, which we may call (j) ; but if we suppose the dimensions of the facets to be small quantities of the first order, the angles of the four facets which meet in a point wUl differ from the angle of intersection of the curves at that point by small angles of the first order depending on the tangential curvature of the lines of bending. The sum of these four angles will differ from four right angles by a small angle of the second order, the circular measure of which expresses the entire curvature of the solid angle as in Art. (2). The angle of incHnation of two adjacent facets will depend on the normal curvature of the lines of bending, and wiU be that of the projection of two con- secutive sides of the polygon of one system on a plane perpendicular to a side of the other system. 11. Explanation of the Notation to he employed in calculation. Suppose each system of lines of bend- ing to be determined by an equation con- taining one arbitrary parameter. Let this parameter be u for the first system, and u for the second. Let two curves, one from each system, be selected as curves of reference, and let their parameters be «„ and u\. 13—2 100 TBANSrORMATION 03? SURFACES BY BENDING. Let ON and OM in the figure represent these two curves. Let PM be any curve of the first system whose parameter is u, and PN any curve of the second whose parameter is u, then their intersection P may be defined as the point (^^ u'), and all quantities referring to the point P may be expressed as functions of u and u. Let PN, the length of a curve of the second system {u), from N (w„) to P {u), be expressed by s, and PM the length of the curve {u) from {u\) to {u), by /; then s and s will be functions of u and u. Let {u + hu) be the parameter of the curve QF of the first system consecu- tive to PM. Then the length of PQ, the part of the curve of the second system intercepted between the curves [u) and [u + hu), will be ds 5, -1- ou. du Similarly PR may be expressed by CtS c* / du These values of PQ and PR will be the ultimate values of the length and breadth of a quadrilateral facet. The angle between these lines will be ultimately equal to . The sides of this parallelogram will be I and l', the supplements of the angles of the edges of the polyhedron, and we may therefore express its area as a plane parallelogram k = ir sin ou on. pp' sin = ah, TRANSFOEMATION OF SURFACES BY BENDING. 103 and the expression for p becomes _4Ji' or if the area of the circumscribing parallelogram be called A, The priacipal radii of curvature of the surface are parallel to the axes of the conic of contact. Let R and R' denote these radii, then a^ If ^ = i J and R = ^j; and therefore substituting in the expression for p, I P- RR" or the specific curvature is the reciprocal of the product of the principal radii of curvature. This remarkable expression was introduced by Gauss in the memoir referred to in a former part of this paper. His method of investigation, though not so elementary, is more direct than that here given, and wHl shew how this result can be obtained without reference to the geometrical methods necessary to a more extended inquiry into the modes of bending. 14. On the variation of normal curvature of the lines of bending as we pass from one point of the surface to another. We have determined the relation between the normal curvatures of the lines of bending of the two systems at their points of intersection ; we have now to find the variation of normal curvature when we pass from one line of the first system to another, along a line of the Second. In analytical language we have to find the value of du \p/ Referring to the figure in Art. (11), we shall see that this may be done if we can determine the difference between the angle of inclination of the facets a and 6, and that of c and d : for the angle I between a and b is Zl as o / = r—j -J-, OU , psiJKp au 104 TRANSFOEMATION OF SURFACES BY BENDING. and therefore the difference between the angle of a and h and that of c and d is d ds' -A SttStt'; 07 f^" r. U^-T-OU— -. , . , 7 , au du \p8in

/ 1 ds'\ ^ 5 , cot 4> ds' 1 ds ^ -, , 1 ds 1 ds' ^ ^ , \p sm (p du / p sin (p du r du p sin du r du Finally, substituting the values of I, V, and SZ from Art. (14) du which may be put under the more convenient form ^ n \_ *^ 1 ( ^ ^^'\ .IcZ-s fjLiP^^* ^ du^ ° "^ du ° \sin ' We may simplify these equations by putting p for the specific curvature of the surface, and q for the ratio -, , which is the only quantity altered by bending. We have then p= / ■ 2 , , and 3 = 4, ^ pp sur

P^ = ^T3> and the equations become d „ , d , I ds' ^aog2)=^iog(p^, d ,, . d , I ds 1 ds ^ , 2 ds 1 r' du r du sin <^ ^' 2 ^*' f ,A_ ^ cZ/ 1 1 r du' " r' cZ-w' sin ^ g' ' In this way we may reduce the problem of bending a surface to the consideration of one variable q, by means of the lines of bending. VOL. I. 1^ lOG TRANSFOKMATION OF SUEFACES BY BENDING. 16. To obtain the condition of Instantaneous lines of bending. We have now obtained the values of the differential coefficients of q with respect to each of the variables u, u. From the equation we might find an equation which would give certain conditions of lines of bending. These conditions however would be equivalent to those which we have already assumed when we drew the systems of lines so as to be conjugate to each other. To find the true conditions of bending we must suppose the form of the surface to vary continuously, so as to depend on some variable t which we may call the time. Of the different quantities which enter into our equations, none are changed by the operation of bending except q, so that in differentiating with respect to ^aU the rest may be considered constant, q being the only variable. Differentiating the equations of last article with respect to t, we obtain c?^ ,, . i ds 1 d ,, , Whence d^ „ -.2 ds' 1 1 c^ ,, > 5^7d^7^(^°S^) = d (2 ds I \ 2 ds I d ,, ^ d ,, . 2 ds 1 d „ dv:[rd^ sln^j + r Tu ^<^ du' (^"8" ^) [ ^ ^ (^^^ ^^ + r ^ diT? ^ ^d« (^°g ?)' and d^ : (log l) dudu'dt d (2ds' 1 \ 2ds 1 d \ld., . 2 ds I 1 d ,, , du\r'du'^m4>j ^'^^'sin.^c^z/^^^l^^i^^^S^'' + r'd^'Z^g^w^(l«g?)' two independent values of the same quantity, whence the required conditions may be obtained. TRANSFORMATION OF SURFACES BY BENDING. 107 Substituting in these equations the values of those quantities which occur in the original equations, we obtain I ds ( d , I ds . ,\ 2 ds , ,] 1 1 ds' ( d , I ,ds . ,\ 2 ds , , 1 which is the condition which must hold at every instant during the process of bending for the lines about which the bending takes place at that instant. When the bending is such that the position of the lines of bending on the surface alters at every instant, this is the only condition which is required. It is therefore called the condition of Instantaneous lines of bending. 1 7. To find the condition of Permanent lines of bending. Since q changes with the time, the equation of last article will not be satisfied for any finite time unless both sides are separately equal to zero, that case we have the two conditions In d ^ I ds . ,\ 2 ds' , , '> log [W TT.. sin <^ ) + ^ ^' cot <^ = 0, d%( du 1 ds or — ?- = 0. r du d , I ,ds' . ,\ 2 ds , log [pr ^, sm <^ j + ^ ^ cot <^ = 0, .(1). J du du 1 ds or -, -T-, = 0. r du .(2). If the lines of bending satisfy these conditions, a finite amount of bending may take place without changing the position of the system on the surface. Such lines are therefore called Permanent lines of bending. The only case in which the phenomena of bending may be exhibited by means of the polyhedron with quadrilateral facets is that in which permanent lines of bending are chosen as the boundaries of the facets. In all other cases the bending takes place about an instantaneous system of lines which is con- tinually in motion with respect to the surface, so that the nature of the poly- hedron would need to be altered at every instant. 14—2 108 TRANSFORMATION OF SURFACES BY BENDING. We are now able to determine whether any system of lines drawn on a given surface is a system of instantaneous or permanent lines of bending. We are also able, by the method of Article (8), to deduce from two con- secutive forms of a surface, the lines of bending about which the transformation must have taken place. If our analytical methods were sufficiently powerful, we might apply our results to the determination of such systems of lines on any known surface, but the necessary calculations even in the simplest cases are so complicated, that, even if useful results were obtained, they would be out of place in a paper of this kind, which is intended to afford the means of forming distinct canceptions rather than to exhibit the results of mathematical labour. 18. On the application of the ordinary methods of analytical geometry to the consideration of lines of bending. It may be interesting to those who may hesitate to accept results derived from the consideration of a polyhedron, when applied to a curved surface, to inquire whether the same results may not be obtained by some independent method. As the following method involves only those operations which are most famUiar to the analyst, it will be sufficient to give the rough outline, which may be filled up at pleasure. The proof of the invariability of the specific curvature may be taken from any of the memoirs above referred to, and its value in terms of the equation of the surface will be found in the memoir of Gauss. Let the equation to the surface be put under the form then the value of the specific curvature is P = dh dh ~dh~ dx^ dy^ dx dy dz dy The definition of conjugate systems of curves may be rendered independent of the reasoning formerly employed by the following modification. TRANSFORMATION OF SURFACES BY BENDING. 109 Let a tangent plane move along any line of the first system, then if the line of ultimate intersection of this plane with itself be always a tangent to some line of the second system, the second system is said to be conjugate to the first. It is easy to show that the first system is also conjugate to the second. Let the system of curves be projected on the plane of xy, and at the point (a;, y) let a be the angle which a projected curve of the first system makes vsdth the axis of x, and y8 the angle which the projected curve of the second system which intersects it at that point makes with the same axis. Then the condition of the systems being conjugate will be found to be a and )Q being known as functions of x and y, we may determine the nature of the curves projected on the plane of xy. Supposing the surface to touch that plane at the origin, the length and tangential curvature of the lines on the surface near the point of contact may be taken the same as those of their projections on the plane, and any change of form of the surface due to bending will not alter the form of the projected lines indefinitely near the point of contact. We may therefore consider z as the only variable altered by bending; but in order to apply our analysis with facility, we may assume ^ = PQ sin^ a + PQ-' sin^ A = — PQ sin a cos a — PQ'^ sin yS cos jS, dxdy d'z , = PQ cos' a + PQ-'co^'fi. It will be seen that these, values satisfy the condition last given. Near the origin we have d^z d\ dh " d3(? dy^ dxdy and q=Q~\ ■■P' sin' {a -13), 110 TRANSFOKMATION OF SUBPACES BY BENDING. Differentiating these values of -r-,, &c., we shall obtain two values of ^^, , and of 1 1 2 > which being equated will give two equations of condition. Now if / be measured along a curve of the first system, and R be any function of x and y, then dE dR dR . -r-7 — -^r- cos a + -^— sm a, as ax ay , dR dR ds' du ds' du ' We may also show that -^ = - , , ,, , da . da d , Ids' . ,\ and that cos a ^ sm a ^- = -r log -j—, sm 4> . ay dx ds ° \du / By substituting these values in the equations thus obtained, they are reduced to the two equations given at the end of (Art. 15). This method of investigation introduces no difficulty except that of somewhat long equations, and is therefore satisfactory as supplementary to the geometrical method given at length. As an example of the method given in page (2), we may apply it to the case of the surface whose equation is ^\ + (JL\ = (^ G — ZJ \C + Zl \g This surface may be generated by the motion of a straight line whose equation is of the form x = acoBtil — j, y = asiutil + -\ , t being the variable, by the change of which we pass from one position of the line to another. This line always passes through the circle z = 0, x' + y^ = a\ and the straight lines z = c, x = Q, and z= —c, y=0, which may therefore be taken as the directors of the surface. TRANSFORMATION OF SURFACES BY BENDING. Ill Taking two consecutive positions of this line, in which the values of t are t and t + ht, we may find by the ordinary methods the equation to the shortest line between them, its length, and the co-ordinates of the point in which it intersects the first line. Calling the length 8^, Jd' + c' and the co-ordinates of the point of intersection are a? = 2a. cos'* t, y — 'ia sin^ t, z= —c cos 2t. The angle 8^ between the consecutive lines is Ja' + c The distance Scr between consecutive shortest lines is Oct = ,^ sm 2tot, Ja' + d" and the angle 8<^ between these latter lines is \/a +c Hence if we suppose ^, 6, a, , and t to vanish together, we shall have by integration ^ J a' + c sla^ + e _{1 —cos 2t), ''^Ja? + c By bending the surface about its generating lines we alter the value of ^ in any manner without changing t„ 6, or cr. For instance, making <;& = 0, all the generating lines become parallel to the same plane. Let this plane be that of xy, then t, is the distance of a generating line from that plane. The projections 112 TRANSFORMATION OP SURFACES BY BENDING. of the generating lines on the plane of xy will, by their ultimate intersections, form a curve, the length of which is measured by cr, and the angle which its tangent makes with the axis of a; by ^, ^ and cr being connected by the equation a- = _ . 1 - cos , which shows the curve to be an epicycloid. The generating lines of the surface when bent into this form are therefore tangents to a cyhndrical surface on an epicycloidal base, touching that surface along a curve which is always equally inclined to the plane of the base, the tangents themselves being drawn parallel to the base. We may now consider the bending of the surface of revolution ^7s?+^ + Z^ = C^ Putting r = Ja? + 'if, then the equation of the generating line is 9 2 2 r^ + z^ = c^. This is the well-known hypocycloid of four cusps. Let s be the length of the curve measured from the cusp in the axis of 2, then, „ 1 2 s = f c^r^, wherefore, r = (f )^ c ~ ^ s^. Let 6 be the angle which the plane of any generating line makes with that of xz, then s and 9 determine the position of any point on the surface. The length and breadth of an element of the surface wUl be Ss and rZd. Now let the surface be bent in the manner formerly described, so that 6 becomes 6', and r, r', when 6' = [10 and r'=-r, then / = (|)^c~^ju,"'s^ provided c' = fi^c. The equation between / and s being of the same form as that between r and s shows that the surface when bent is similar to the original surface, its dimensions being multipHed by /x^ TEANSFORMATION OP SURFACES BY BENDING. 113 This, however, is true only for one half of the surface when bent. The other half is precisely symmetrical, biit belongs to a surface which is not con- tinuous with the first. The surface in its original form is divided by the plane of xy into two parts which meet in that plane, forming a kind of cuspidal edge of a circular form which limits the possible value of s and r. After being bent, the surface still consists of the same two parts, but the edge in which they meet is no longer of the cuspidal form, but has a finite angle = 2 cos"'-, and the two sheets of the surface become parts of two different surfaces which meet but are not continuous. NOTE. As an example of the application of the more general theory of " lines of bending," let us consider the problem which has been already solved by Professor Jellett. To determine the conditions under which one portion of a surface may he rendered rigid, while the remainder is flexible. Suppose the lines of bending to be traced on the surface, and the corresponding poly- hedron to be formed, as in (9) and (10), then if the angle of one of the four edges which meet at any solid angle of the polyhedron be altered by bending, those of the other three must be also altered. These edges terminate in other solid angles, the forms of which will also be changed, and therefore the effect of the alteration of one angle of the polyhedron will be communicated to every other angle within the system of lines of bending which defines the form of the polyhedron. If any portion of the surface remains unaltered it must lie beyond the limits of the system of lines of bending. We must therefore investigate the conditions of such a system being bounded. The boundary of any system of lines on a surface is the curve formed by the ultimate inter- section of those lines, and therefore at any given point coincides in direction with the curve of the system which passes through that point. In this case there are two systems of lines of bending, which are necessarily coincident in extent, and must therefore have the same boundary. At any point of this boundary therefore the directions of the lines of bending of the first and second systems are coincident. But, by (7), these two directions must be "conjugate" to each other, that is, must corre- spond to conjugate diameters of the " Conic of Contact." Now the only case in which con- VOL. I. 15 114 TRANSFORMATION OF SURFACES BY BENDING. jugate diameters of a conic can coincide, is when the conic is an hyperbola, and both diameters coincide with one of the asymptotes ; therefore the boundary of the system of lines of bending must be a curve at every point of which the conic of contact is an hyperbola, one of whose asymptotes lies in the direction of the curve. The radius of "normal curvature" must there- fore by (8) be infinite at every point of the curve. This is the geometrical property of what Professor Jellett calls a " Curve of Flexure," so that we may express the result as follows : If one 'portion of a surface he fixed, while the remainder is bent, the boundary of the fixed "portion is a curve of flexure. This theorem includes those given at p. (92), relative to a fixed curve on a surface, for in a surface whose curvatures are of the same sign, there can be no "curves of flexure," and in a developable surface, they are the rectilinear sections. Although the cuspidal edge, or arete de rehroussement, satisfies the analytical condition of a curve of flexure, yet, since its form determines that of the whole surface, it cannot remain fixed while the form of the surface is changed. In concavo-convex surfaces, the curves of flexure must either have tangential curvature or be straight lines. Now if we put (f)=0 in the equations of Art. (17), we find that the lines of bending of both systems have no tangential curvature at the point where they touch the curve of flexure. They must therefore lie entirely on the convex side of that . curve, and therefore If a curve of flexure be fixed, the surface on the concave side of the curve is not flexible. I have not yet been able to determine whether the surface is inflexible on the convex side of the curve. It certainly is so in some cases which I have been able to work out, but I have no general proof. When a surface has one or more rectilinear sections, the portions of the surface between them may revolve as rigid bodies round those lines as axes in any manner, but no other motion is possible. The case in which the rectihnear sections form an infinite series has been discussed in Sect. (I.). [From the Cambridge and Dublin Mathematical Journal, Vol. ix. V. On a particular case of the descent of a heavy body in a resisting medium. Every one must liave observed that when a slip of paper falls through the air, its motion, though undecided and wavering at first, sometimes becomes regular. Its general path is not in the vertical direction, but inclined to it at an angle which remains nearly constant, and its fluttering appearance will be found to be due to a rapid rotation round a horizontal axis. The direction of deviation from the vertical depends on the direction of rotation. If the positive directions of an axis be toward the right hand and upwards, and the positive angular direction opposite to the direction of motion of the hands of a watch, then, if the rotation is in the positive direction, the hori- zontal part of the mean motion will be positive. These effects are commonly attributed to some accidental peculiarity in the form of the paper, but a few experiments with a rectangular slip of paper (about two inches long and one broad), will shew that the direction of rotation is determined, not by the irregularities of the paper, but by the initial circum- stances of projection, and that the symmetry of the form of the paper greatly increases the distinctness of the phenomena. We may therefore assume that if the form of the body were accurately that of a plane rectangle, the same effects would be produced. The following investigation is intended as a general explanation of the true cause of the phenomenon. I suppose the resistance of the air caused by the motion of the plane to be in the direction of the normal and to vary as the square of the velocity estimated in that direction. Now though this may be taken as a sufficiently near approximation to the magnitude of the resisting force on the plane taken as a whole, the pressure 15—2 116 DESCENT OP A HEAVY BODY IN A EESISTING MEDIUM. on any given element of the surface will vary with its position so that the resultant force will not generally pass through the centre of gravity. It is found by experiment that the position of the centre of pressure depends on the tangential part of the motion, that it lies on that side of the centre of gravity towards which the tangential motion of the plane is directed, and that its distance from that point increases as the tangential velocity in- creases. I am not aware of any mathematical investigation of this effect. The explanation may be deduced from experiment. Place a body similar in shape to the slip of paper obliquely in a current of some visible fluid. Call the edge where the fluid first meets the plane the first edge, and the edge where it leaves the plane, the second edge, then we may observe that (1) On the anterior side of the plane the velocity of the fluid increases as it moves along the surface from the first to the second edge, and therefore by a known law in hydrodynamics, the pressure must diminish from the first to the second edge. (2) The motion of the fluid behind the plane is very unsteady, but may be observed to consist of a series of eddies diminishing in rapidity as they pass behind the plane from the first to the second edge, and therefore relieving the posterior pressure most at the first edge. Both these causes tend to make the total resistance greatest at the first edge, and therefore to bring the centre of pressure nearest to that edge. Hence the moment of the resistance about the centre of gravity will always tend to turn the plane towards a position perpendicular to the direction of the current, or, in the case of the slip of paper, to the path of the body itself. It will be shewn that it is this moment that maintains the rotatory motion of the falling paper. When the plane has a motion of rotation, the resistance will be modified on account of the unequal velocities of difierent parts of the surface. The magnitude of the whole resistance at any instant will not be sensibly altered if the velocity of any point, due to angular motion be small compared with that due to the motion of the centre of gravity. But there will be an additional moment of the resistance round the centre of gravity, which will always act in the direction opposite to that of rotation, and will vary directly as the normal and angular velocities together. DESCENT OF A HEAVY BODY IN A RESISTING MEDIUM. 117 The part of the moment due to the obliquity of the motion will remain nearly the same as before. We are now prepared to give a general explanation of the motion of the slip of paper after it has become regular. Let the angular position of the paper be determined by the angle between the normal to its surface and the axis of x, and let the angular motion be such that the normal, at first coinciding with the axis of x, passes towards that of y. The motion, speaking roughly, is one of descent, that is, in the negative direction along the axis of y. The resolved part of the resistance in the vertical direction will always act upwards, being greatest when the plane of the paper is horizontal, and vanishing when it is vertical. When the motion has become regular, the effect of this force during a whole revolution will be equal and opposite to that of gravity during the same time. Since the resisting force increases while the normal is in its first and third quadrants, and diminishes when it is in its second and fourth, the maxima of velocity will occur when the normal is in its first and third quadrants, and the minima when it is in the second and fourth. The resolved part of the resistance in the horizontal direction will act in the positive direction along the axis of x in the first and third quadrants, and in the negative direction during the second and fourth; but since the resistance increases with the velocity, the whole effect during the first and third quadrants will be greater than the whole effect during the second and fourth. Hence the horizontal part of the resistance wiU act on the whole in the positive direction, and wiU therefore cause the general path of the body to inchne in that direction, that is, toward the right. That part of the moment of the resistance about the centre of gravity which depends on the angular velocity will vary in magnitude, but will always act in the negative direction. The other part, which depends on the obliquity of the plane of the paper to the direction of motion, will be positive in the first and third quadrants and negative in the second and fourth ; but as its magnitude increases with the velocity, the positive effect will be greater than the negative. When the motion has become regular, the effect of this excess in the 118 DESCENT OF A HEAVY BODY IN A RESISTING MEDIUM. positive direction will be equal and opposite to the negative effect due to the angular velocity during a whole revolution. The motion will then consist of a succession of equal and similar parts performed in the same manner, each part corresponding to half a revolution of the paper. These considerations will serve to explain the lateral motion of the paper, and the maintenance of the rotatory motion. Similar reasoning will shew that whatever be the initial motion of the paper, it cannot remain uniform. Any accidental oscillations will increase till their amplitude exceeds half a revolution. The motion will then become one of rotation, and will continually approximate to that which we have just considered. It may be also shewn that this motion will be unstable unless it take place about the longer axis of the rectangle. If this axis is inclined to the horizon, or if one end of the slip of paper be different from the other, the path will not be straight, but in the form of a helix. There will be no other essential difference between this case and that of the symmetrical arrangement. Trinity College, April 5, 1853. [From the Transactiojis of the Royal Scottish Society of Arts, Vol. iv. Part iii, VI. O/i the Tlieory of Colours in relation to Colour- Blindness. A letter to Dr G. Wilson. Deae Sir, — As you seemed to think that the results which I have obtained in the theory of colours naight be of service to you, I have endeavoured to arrange them for you in a more convenient form than that in which I first obtained them. I must premise, that the first distinct statement of the theory of colour which I adopt, is to be found in Young's Lectures on Natural Philo- sophy (p. 345, Kelland's Edition) ; and the most philosophical enquiry into it which I have seen is that of Helmholtz, which may be found in the Annals of Philosophy for 1852. It is well known that a ray of light, from any source, may be divided by means of a prism into a number of rays of different refrangibility, forming a series called a spectrum. The intensity of the light is different at different points of this spectrum ; and the law of intensity for different refrangibilities differs according to the nature of the incident light. In Sir John F. W. Herschel's Treatise on Light, diagrams will be found, each of which represents completely, by means of a curve, the law of the intensity and refrangibility of a beam of solar light after passing through various coloured media. I have mentioned this mode of defining and registering a beam of light, because it is the perfect expression of what a beam of light is in itself, con- sidered with respect to all its properties as ascertained by the most refined instruments. When a beam of light falls on the human eye, certain sensations are produced, from which the possessor of that organ judges of the colour and intensity of the light. Now, though every one experiences these sensations, and though they are the foundation of all the phenomena of sight, yet, on account of their absolute simplicity, they are incapable of analysis, and can never become in themselves objects of thought. If we attempt to discover them, we must 120 THE THEOBY OF COLOURS IN RELATION TO COLOUR-BLINDNESS. do SO by artificial means; and our reasonings on them must be guided by some theory. The most general form in which the existing theory can be stated is this, — There are certain sensations, finite in number, but infinitely variable in degree, which may be excited by the different kinds of light. The compound sensation resulting from all these is the object of consciousness, is a simple act of vision. It is easy to see that the number of these sensations corresponds to what may be called in mathematical language the number of independent variables, of which sensible colour is a function. This will be readUy understood by attending to the following cases : — 1. When objects are illuminated by homogeneous yellow light, the only thing which can be distinguished by the eye is difierence of intensity or brightness. If we take a horizontal line, and colour it black at one end, with increasing degrees of intensity of yellow light towards the other, then every visible object will have a brightness corresponding to some point in this line. In this case there is nothing to prove the existence of more than one sensation in vision. In those photographic pictures in which there is only one tint of which the different intensities correspond to the different degrees of illumination of the object, we have another illustration of an optical effect depending on one variable only. 2. Now, suppose that different kinds of light are emanating from dififerent sources, but that each of these sources gives out perfectly homogeneous light, then there will be two things on which the nature of each ray wUl depend : — (l) its intensity or brightness ; (2) its hue, which may be estimated by its position in the spectrum, and measured by its wave length. If we take a rectangular plane, and illuminate it with the different kinds of homogeneous light, the intensity at any point being proportional to its hori- zontal distance along the plane, and its wave length being proportional to its height above the foot of the plane, then the plane will display every possible variety of homogeneous light, and will furnish an instance of an optical effect depending on two variables. THE THEORY OF COLOURS IN RELATION TO COLOUR-BLINDNESS. 121 3. Now, let US take the case of nature. We find that colours diflfer not only in intensity and hue, but also in tint ; that is, they are more or less pure. We might arrange the varieties of each colour along a line, which should begin with the homogeneous colour as seen in the spectrum, and pass through all gradations of tint, so as to become continually purer, and terminate in white. We have, therefore, three elements in our sensation of colour, each of which may vary independently. For distinctness sake I have spoken of intensity, hue, and tint ; but if any other three independent qualities had been chosen, the one set might have been expressed in terms of the other, and the results identified. The theory which I adopt assumes the existence of three elementary sen- sations, by the combination of which all the actual sensations of colour are produced. It will be shewn that it is not necessary to specify any given colours as typical of these sensations. Young has called them red, green, and violet ; but any other three colours might have been chosen, provided that white resulted from their combination in proper proportions. Before going farther I would observe, that the important part of the theory is not that three elements enter into our sensation of colour, but that there are only three. Optically, there are as many elements in the composition of a ray of Hght as there are different kinds of light in its spectrum; and, therefore, strictly speaking, its nature depends on an infinite number of independent variables. I now go on to the geometrical form into which the theory may be thrown. Let it be granted that the three pure sensations corre- spond to the colours red, green, and violet, and that we can estimate the intensity of each of these sensations numerically. Let V, r, g be the angular points of a triangle, and conceive the three sensations as having their positions at these points. If we find the numerical measure of the red, green, and violet parts of the sensation of a given colour, and then place weights proportional to these parts at r, g, and v, and find the centre of gravity of the three weights by the ordinary process, that point will be the position of the given colour, and the numerical measure of its intensity will be the sum of the three primitive sensations. In this way, every possible colour may have its position and intensity VOL. L 1^ 122 THE THEORY OP COLOURS IN RELATION TO COLOUR-BLINDNESS. ascertained; and it is easy to see that when two compound colours are com- bined, their centre of gravity is the position of the new colour. The idea of this geometrical method of investigating colours is to be found in Newton's Optichs (Book I., Part 2, Prop. 6), but I am not aware that it has been ever employed in practice, except in the reduction of the experiments which I have just made. The accuracy of the method depends entirely on the truth of the theory of three sensations, and therefore its success is a testimony in favour of that theory. Every possible colour must be included within the triangle rgv. White will be found at some point, ui, within the triangle. If lines be drawn through IV to any point, the colour at that point will vary in hue according to the angular position of the line drawn to w, and the purity of the tint will depend on the length of that line. Though the homogeneous rays of the prismatic spectrum are absolutely pure in themselves, yet they do not give rise to the "pure sensations" of which we are speaking. Every ray of the spectrum gives rise to all three sensations, though in different proportions ; hence the position of the colours of the spectrum is not at the boundary of the triangle, but in some curve C R Y G B V considerably within the triangle. The nature of this curve is not yet determined, but may form the subject of a future investigation *- All natural colours must be withia this curve, and all ordinary pigments do in fact lie very much within it. The experiments on the colours of the spectrum which I have made are not brought to the same degree of accuracy as those on coloured papers. I therefore proceed at once to describe the mode of making those experiments which I have found most simple and convenient. The coloured paper is cut into the form of discs, each with a small hole in the centre, and divided along a radius, so as to admit of several of them being placed on the same axis, so that ( \=r ) part of each is exposed. By slipping one disc over another, ^ we can expose any given portion of each colour. These ^ — ^^ ' It is by the use of analogies of this kind that I have attempted to bring before the mind, in a convenient and manageable form, those mathematical ideas which are necessary to the study of the phenomena of electricity./' The methods are generally those suggested by the processes of reasoning which are found in the researches of Faraday*, and which, though they have been interpreted mathematically by Prof. Thomson and others, are very generally supposed to be of an indefinite and unmathematical character, when compared with those em- ployed by the professed mathematicians. By the method which I adopt, I hope to render it evident that I am not attempting to establish any physical theory of a science in which I have hardly made a single experiment, and that the limit of my design is to shew how, by a strict application of the ideas and * See especially Series xxxviii. of the Experimental Researches, and Phil. Mag. 1852. 158 ON Faraday's lines of foece. methods of Faraday, the connexion of the very different orders of phenomena which he has discovered may be clearly placed before the mathematical mind. I shall therefore avoid as much as I can the introduction of anything which does not serve as a direct illustration of Faraday's methods, or of the mathe- matical deductions which may be made from them. In treating the simpler parts of the subject I shall use Faraday's mathematical methods as well as his ideas. When the complexity of the subject requires it, I shall use analytical notation, still confining myself to the development of ideas originated by the same philosopher. I have in the first place to explain and illustrate the idea of "lines of force." When a body is electrified in any manner, a small body charged with posi- tive electricity, and placed in any given position, will experience a force urging it in a certain direction. If the small body be now negatively electrified, it will be urged by an equal force in a direction exactly opposite. The same relations hold between a magnetic body and the north or south poles of a small magnet. If the north pole is urged in one direction, the south pole is urged in the opposite direction. In this way we might find a line passing through any point of space, such that it represents the direction of the force acting on a positively electrified particle, or on an elementary north pole, and the reverse direction of the force on a negatively electrified particle or an elementary south pole. Since at every point of space such a direction may be found, if we commence at any point and draw a line so that, as we go along it, its direction at any point shall always coincide with that of the resultant force at that point, this curve will indicate the direction of that force for every point through which it passes, and might be called on that account a line of force. We might in the same way draw other lines of force, till we had filled all space with curves indicating by their direction that of the force at any assigned point. We should thus obtain a geometrical model of the physical phenomena, which would tell us the direction of the force, but we should still require some method of indicating the intensity of the force at any point. /If we consider these curves not , as mere lines, but as fine tubes of variable section carrying an incompressible fluid, then, since the velocity of the fluid is inversely as the section of the tube, we may make the velocity vary according to any given law, by regulating the section of the tube, and in this way we might represent the ON paraday's lines of force. 159 intensity of the force as well as its direction by the motion of the fluid in these tubes. } This method of representing the intensity of a force by the velocity of an imaginary fluid in a tube is applicable to any conceivable system of forces, but it is capable of great simplification in the case in which the forces are such as can be explained by the hypothesis of attractions varying inversely as the square of the distance, such as those observed in electrical and magnetic pheno- mena. In the case of a perfectly arbitrary system of forces, there wdl generally be interstices between the tubes ; but in the case of electric and magnetic forces it is possible to arrange the tubes so as to leave no interstices. The tubes will then be mere surfaces, directing the motion of a fluid fiJhng up the whole space. It has been usual to commence the investigation of the laws of these forces by at once assuming that the phenomena are due to attractive or repulsive forces acting between certain points. We may however obtain a different view of the subject, and one more suited to our more difficult inquiries, by adopting for the definition of the forces of which we treat, that they may be represented in magnitude and direction by the uniform motion of an incompressible fluid. I propose, then, first to describe a method by which the motion of such a fluid can be clearly conceived ; secondly to trace the consequences of assuming certain conditions of motion, and to point out the application of the method to some of the less complicated phenomena of electricity, magnetism, and galvanism ; and lastly (to shew how by an extension of these methods, and the introduction of another idea due to Faraday, the laws of the attractions and inductive actions of magnets and currents may be clearly conceived, without making any assump- tions as to the physical nature of electricity, or adding anything to that which has been already proved by experiment. By referring everything to the purely geometrical idea of the motion of an imaginary fluid, I hope to attain generality and precision, and to avoid the dangers arising from a premature theory professing to explain the cause of the phenomena. If the results of mere speculation which I have collected are found to be of any use to experimental philosophers, in arranging and interpreting their results, they wiU have served their purpose, and a mature theory, in which physical facts wdl be physically explained, will be formed by those who by interrogating Nature herself can obtain the only true solution of the questions which the mathematical theory suggests. 160 ON Faraday's lines of force. I. Theory of the Motion of an incompressible Fluid. (1) The substance here treated of must not be assumed to possess any of the properties of ordinary fluids except those of freedom of motion and resistance to compression. It is not even a hypothetical fluid which is introduced to explain actual phenomena. It is merely a collection of imaginary properties which may be employed for establishing certain theorems in pure mathematics in a way more intelligible to many minds and more applicable to physical problems than that in which algebraic symbols alone are used. The use of the word "Fluid" wUl not lead us into error, if we remember that it denotes a purely imaginary substance with the following property : The portion of fluid which at any instant occupied a given volume, will at any succeeding instant occupy an equal volume. This law expresses the incompressibility of the fluid, and furnishes us with a convenient measure of its quantity, namely its volume. The unit of quantity of the fluid wUl therefore be the unit of volume. (2) The direction of motion of the fluid will in general be difierent at difierent points of the space which it occupies, but since the direction is deter- minate for every such point, we may conceive a line to begin at any point and to be continued so that every element of the line indicates by its direction the direction of motion at that point of space. Lines drawn in such a manner that their direction always indicates the direction of fluid motion are called lines of fluid motion. If the motion of the fluid be what is called steady motion, that is, if the direction and velocity of the motion at any fixed point be independent of the time, these curves will represent the paths of individual particles of the fluid, but if the motion be variable this will not generally be the case. The cases of motion which will come under our notice will be those of steady motion. (3) If upon any surface which cuts the lines of flmd motion we draw a closed curve, and if from every point of this curve we draw a line of motion, these lines of motion will generate a tubular surface which we may call a tube of fluid motion. Since this surface is generated by lines in the direction of fluid ON Faraday's lines of force. 161 motion no part of the fluid can flow across it, so that this imaginary surface is as impermeable to the fluid as a real tube. (4) The quaritity of fluid which in unit of time crosses any fixed section of the tube is the same at whatever part of the tube the section be taken. For the fluid is incompressible, and no part runs through the sides of the tube, therefore the quantity which escapes from the second section is equal to that which enters through the first. If the tube be such that unit of volume passes through any section in unit of time it is called a unit tube of fluid motion. (5) In what follows, various units will be referred to, and a finite number of lines or surfaces will be drawn, representing in terms of those units the motion of the fluid. Now in order to define the motion in every part of the fluid, an infinite number of lines would have to be drawn at indefinitely small intervals ; but since the description of such a system of lines would involve continual reference to the theory of limits, it has been thought better to suppose the lines drawn at intervals depending on the assumed unit, and afterwards to assume the unit as small as we please by taking a small submultiple of the standard unit. (6) To define the motion of the whole fluid by means of a system of unit tubes. Take any fixed surface which cuts all the lines of fluid motion, and draw upon it any system of curves not intersecting one another. On the same surface draw a second system of curves intersecting the first system, and so arranged that the quantity of fluid which crosses the surface within each of the quadri- laterals formed by the intersection of the two systems of curves shall be unity in unit of time. From every point in a curve of the first system let a line of fluid motion be drawn. These lines will form a surface through which no fluid passes. Similar impermeable surfaces may be drawn for all the curves of the first system. The curves of the second system wUl give . rise to a second system of impermeable surfaces, which, by their intersection with the first system, will form quadrilateral tubes, which wUl be tubes of fluid motion. Since each quadrilateral of the cutting surface transmits unity of fluid in unity of time, every tube in the system will transmit unity of fluid through any of its sections in unit of time. The motion of the fluid at every part of the space it occupies VOL. L 21 162 ON FARADAY S LINES OF FORCE. is determined by this system of unit tubes; for the direction of motion is that of the tube through the pomt in question, and the velocity is the reciprocal of the area of the section of the unit tube at that point. (7) We have now obtained a geometrical construction which completely defines the motion of the fluid by dividing the space it occupies into a system of unit tubes. We have next to shew how by means of these tubes we may ascertain various points relating to the motion of the fluid. A unit tube may either return into itself, or may begin and end at differ- ent points, and these may be either in the boundary of the space in which we investigate the motion, or within that space. In the first case there is a con- tinual circulation of fluid in the tube, in the second the fluid enters at one end and flows out at the other. If the extremities of the tube are in the bound- ing surface, the fluid may be supposed to be continually supplied from without from an unknown source, and to flow out at the other into an unknown reser- voir ; but if the origin of the tube or its termination be within the space under consideration, then we must conceive the fluid to be supplied by a source within that space, capable of creating and emitting unity of fluid in unity of time, and to be afterwards swallowed up by a sink capable of receiving and destroying the same amount continually. There is nothing self-contradictory in the conception of these sources where the fluid is created, and sinks where it is annihilated. The properties of the fluid are at our disposal, we have made it incompressible, and now we suppose it produced from nothing at certain points and reduced to nothing at others. The places of production will be called sources, and their numerical value will be the number of units of fluid which they produce in unit of time. The places of reduction will, for want of a better name, be called sinlcs, and will be esti- mated by the number of units of fluid absorbed in unit of time. Both places will sometimes be called sources, a source being understood to be a sink when its sign is negative. (8) It is evident that the amount of fluid which passes any fixed surface is measured by the number of unit tubes which cut it, and the direction in which the fluid passes is determined by that of its motion in the tubes. If the surface be a closed one, then any tube whose terminations lie on the same side of the surface must cross the surface as many times in the one direction as in the other, and therefore must carry as much fluid out of the surface as ON FARADAY S LINES OF FORCE. 163 it carries in. A tube which begins within the surface and ends without it will carry out unity of fluid ; and one which enters the surface and terminates within it will carry in the same quantity. In order therefore to estimate the amount of fluid which flows out of the closed surface, we must subtract the number of tubes which end within the surface from the number of tubes which begin there. If the result is negative the fluid will on the whole flow inwards. If we call the beginning of a unit tube a unit source, and its termination a unit sink, then the quantity of fluid produced within the surface is estimated by the number of unit sources minus the number of unit sinks, and this must flow out of the surface on account of the incompressibility of the fluid. In speaking of these unit tubes, sources and sinks, we must remember what was stated in (5) as to the magnitude of the unit, and how by diminishing their size and increasing their number we may distribute them according to any law however complicated. (9) If we know the direction and velocity of the fluid at any point in two different cases, and if we conceive a third case in which the direction and velocity of the fluid at any point is the resultant of the velocities in the two former cases at corresponding points, then the amount of fluid which passes a given fixed surface in the third case wiU be the algebraic sum of the quantities which pass the same surface in the two former cases. For the rate at which the fluid crosses any surface is the resolved part of the velocity normal to the surface, and the resolved part of the resultant is equal to the sum of the resolved parts of the components. Hence the number of unit tubes which cross the surface outwards in the third case must be the algebraical sum of the numbers which cross it in the two former cases, and the number of sources within any closed surface will be the sum of the numbers in the two former cases. Since the closed surface may be taken as small as we please, it is evident that the distribution of sources and sinks in the third case arises from the simple superposition of the distri- butions in the two former cases. II. Theory of the uniform motion of an imponderable incompressible fluid through a resisting medium. (10) The fluid is here supposed to have no inertia, and its motion is opposed by the action of a force which we may conceive to be due to the resistance of a 21—2 164 ON Faraday's lines of force. medium througli which the fluid is supposed to flow. This resistance depends on the nature of the medium, and will in general depend on the direction in which the fluid moves, as well as on its velocity. For th6 present we may restrict ourselves to the case of a uniform medium, whose resistance is the same in aU directions. The law which we assume is as follows. Any portion of the fluid moving through the resisting medium is directly opposed by a retarding force proportional to its velocity. If the velocity be represented by v, then the resistance will be a force equal to hv acting on unit of volume of the fluid in a direction contrary to that of motion. In order, therefore, that the velocity may be kept up, there must be a greater pressure behind any portion of the fluid than there is in front of it, so that the difference of pressures may neutraUse the effect of the resistance. Con- ceive a cubical unit of fluid (which we may make as small as we please, by (5)), and let it move in a direction perpendicular to two of its faces. Then the resist- ance will be hv, and therefore the difference of pressures on the first and second faces is hv, so that the pressure diminishes in the direction of motion at the rate of hv for every unit of length measured along the line of motion ; so that if we measure a length equal to h units, the difference of pressure at its extremities will be hvh. (11) Since the pressure is supposed to vary continuously in the fluid, all the points at which the pressure is equal to a given pressure p will lie on a certain surface which we may call the surface (p) of equal pressure. If a series of these surfaces be constructed in the fluid corresponding to the pressures 0, 1, 2, 3 &c., then the number of the surface will indicate the pressure belonging to it, and the surface may be referred to as the surface 0, 1, 2 or 3. The unit of pressure is that pressure which is produced by unit of force acting on unit of surface. In order therefore to diminish the unit of pressure as in (5) we must diminish the unit of force in the same proportion. (12) It is easy to see that these surfaces of equal pressure must be perpen- dicular to the lines of fluid motion; for if the fluid were to move in any other direction, there would be a resistance to its motion which could not be balanced by any difference of pressures. (We must remember that the fluid here con- sidered has no inertia or mass, and that its properties are those only which are formally assigned to it, so that the resistances and pressures are the only things ON Faraday's lines of force. 165 to be considered.) There are therefore two sets of surfaces which by their inter- section form the system of unit tubes, and the system of surfaces of equal pres- sure cuts both the others at right angles. Let h be the distance between two consecutive surfaces of equal pressure measured along a line of motion, then since the difference of pressures = 1, Tcvh = 1, which determines the relation of v to h, so that one can be found when the other is known. Let s be the sectional area of a unit tube measured on a surface of equal pressure, then since by the definition of a unit tube vs = l, we find by the last equation (13) The surfaces of equal pressure cut the unit tubes into portions whose length is h and section s. These elementary portions of unit tubes will be called unit cells. In each of them unity of volume of fluid passes from a pressure p to a pressure (jp — 1) in unit of time, and therefore overcomes unity of resistance in that time. The work spent in overcoming resistance is therefore unity in every ceU in every unit of time. (14) If the surfaces of equal pressure are known, the direction and magni- tude of the velocity of the fluid at any point may be found, after which the complete system of unit tubes may be constructed, and the beginnings and end- ings of these tubes ascertained and marked out as the sources whence the fluid is derived, and the sinks where it disappears. In order to prove the converse of this, that if the distribution of sources be given, the pressure at every point may be found, we must lay down certain preliminary propositions. (15) If we know the pressures at every point in the fluid in two difierent cases, and if we take a third case in which the pressure at any point is the sum of the pressures at corresponding points in the two former cases, then the velocity at any point in the third case is the resultant of the velocities in the other two, and the distribution of sources is that due to the simple superposition of the sources in the two former cases. For the velocity in any direction is proportional to the rate of decrease of the pressure in that direction; so that if two systems of pressures be added 166 ON i*aeaday's lines of foece. together, since the rate of decrease of pressure along any line will be the sum of the combined rates, the velocity in the new system resolved in the same direction will be the sum of the resolved parts in the two original systems. The velocity in the new system will therefore be the resultant of the velocities at corresponding points in the two former systems. It follows from this, by (9), that the quantity of fluid which crosses any fixed surface is, in the new system, the sum of the corresponding quantities in the old ones, and that the sources of the two original systems are simply combined to form the third. It is evident that in the system in which the pressure is the diflerence of pressure in the two given systems the distribution of sources will be got by changing the sign of all the sources in the second system and adding them to those in the first. (16) If the pressure at every point of a closed surface be the same and equal to p, and if there be no sources or sinks within the surface, then there will be no motion of the fluid within the surface, and the pressure within it will be uniform and equal to p. For if there be motion of the fluid within the surface there will be tubes of fluid motion, and these tubes must either return into themselves or be terminated either within the surface or at its boundary. Now since the fluid always flows from places of greater pressure to places of less pressure, it cannot flow in a re-entering curve ; since there are no sources or sinks within the surface, the tubes cannot begin or end except on the surface ; and since the pressure at all points of the surface is the same, there can be no motion in tubes having both extremities on the surface. Hence there is no motion within the surface, and therefore no difference of pressure which would cause motion, and since the pressure at the bounding surface is p, the pressure at any point within it is also p. (17) If the pressure at every point of a given closed surface be known, and the distribution of sources within the surface be also known, then only one distribution of pressures can exist within the surface. For if two different distributions of pressures satisfying these conditions could be found, a third distribution could be formed in which the pressure at any point should be the difference of the pressures in the two former distri- butions. In this case, since the pressures at the surface and the sources within ON FABAPAy's lines of force, 1(J7 it are the same in both distributions, the pressure at the surface in the third, distribution would be zero, and all the sources within the surface would vanish, by (15). Then by (16) the pressure at every point in the third distribution must be zero; but this is the difference of the pressures in the two former cases, and therefore these cases are the same, and there is only one distribution of pressure possible. (18) Let us next determine the pressure at any point of an infinite body of fluid in the centre of which a unit source is placed, the pressure at an infinite distance from the source being supposed to be zero. The fluid will flow out from the centre symmetrically, and since unity of volume flows out of every spherical surface surrounding the point in unit of time, the velocity at a distance r from the source will be 1 •v = 4777^' The rate of decrease of pressure is therefore ho or - — -^, and since the pressure = when r is infinite, the actual pressure at any point will be l ^ Awr' The pressure is therefore inversely proportional to the distance from the sou3:ce. It is evident that the pressure due to a unit sink will be negative and k equal to — -- — . ^ iirr If we have a source formed by the coalition of S unit sources, then the hS resulting pressure will be ^ = - — ; , so that the pressure at a given distance varies as the jesistance and number of sources conjointly. (19) If a number of sources and sinks coexist in the fluid, then in order to determine the resultant pressure we have only to add the pressures which each source or sink produces. For by (15) this will be a solution of the problem, and by (17) it will be the only one. By this method we can determine the pressures due to any distribution of sources, as by the method 168 ON taraday's lines or foece. of (14) we can determine the distribution of sources to wHcli a given distri- bution of pressures is due. (20) We have next to shew that if we conceive any imaginary surface as fixed in space and intersecting the lines of motion of the fluid, we may substitute for the fluid on one side of this surface a distribution of sources upon the surface itself without altering in any way the motion of the fluid on the other side of the surface. For if we describe the system of unit tubes which defines the motion of the fluid, and wherever a tube enters through the surface place a unit source, and wherever a tube goes out through the surface place a unit sink, and at the same time render the surface impermeable to the fluid, the motion of the fluid in the tubes will go on as before. (21) If the system of pressures and the distribution of sources which pro- duce them be known in a medium whose resistance is measured by h, then in order to produce the same system of pressures in a medium whose resistance is unity, the rate of production at each source must be multiplied by k. For the pressure at any point due to a given source varies as the rate of produc- tion and the resistance conjointly ; therefore if the pressure be constant, the rate of production must vary inversely as the resistance. (22) On the conditions to he fulfilled at a surface which separates two media whose coefficients of resistance are k and ¥. These are found from the consideration, that the quantity of fluid which flows out of the one medium at any point flows into the other, and that the pressure varies continuously from one medium to the other. The velocity normal to the surface is the same in both media, and therefore the rate of diminution of pressure is proportional to the resistance. The direction of the tubes of motion and the surfaces of equal pressure will be altered after passing through the surface, and the law of this refraction will be, that it takes place in the plane passing through the direction of incidence and the normal to the sur&ce, and that the tangent of the angle of incidence is to the tangent of the angle of refraction as k' is to k. (23) Let the space within a given closed surface be filled with a medium different from that exterior to it, and let the pressures at any point of this compound system due to a given distribution of sources within and without ON FAEADAYS LINES OF FORCE. 169 the surface be given; it is required to determine a distribution of sources which would produce the same system of pressures in a medium whose coefficient of resistance is unity. Construct the tubes of fluid motion, and wherever a unit tube enters either medium place a unit source, and wherever it leaves it place a unit sink. Then if we make the surface impermeable all will go on as before. Let the resistance of the exterior medium be measured by k, and that of the interior by h'. Then if we multiply the rate of production of all the sources in the exterior medium (including those in the surface), by h, and make the coefficient of resistance unity, the pressures will remain as before, and the same will be true of the interior medium if we multiply all the sources in it by h', including those in the surface, and make its resistance unity. Since the pressures on both sides of the surface are now equal, we may suppose it permeable if we please. We have now the original system of pressures produced in a uniform medium by a combination of three systems of sources. The first of these is the given external system multiplied by h, the second is the given internal system multi- plied by Tc, and the third is the system of sources and sinks on the surface itself. In the original case every source in the external medium had an equal sink in the internal medium on the other side of the surface, but now the source is multiplied by h and the sink by Ic, so that the result is for every external unit source on the surface, a source —{k — lc). By means of these three systems of sources the original system of pressures may be produced in a medium for which ^=1. (24) Let there be no resistance in the medium within the closed surface, that is, let F = 0, then the pressure within the closed surface is uniform and equal to p, and the pressure at the surface itself is also p. If by assuming any distribution of pairs of sources and sinks within the surface in addition to the given external and internal sources, and by supposing the medium the same within and without the surface, we can render the pressure at the surface uni- form, the pressures so found for the external medium, together with the uniform pressure p in the internal medium, will be the true and only distribution of pressures which is possible. For if two such distributions could be found by taking different imaginary distributions of pairs of sources and sinks within the medium, then by taking VOL. L 22 170 ON paraday's lines of rOECE. the difference of the two for a third distribution, we should have the pressure of the bounding surface constant in the new system and as many sources as sinks within it, and therefore whatever fluid flows in at any point of the surface, an equal quantity must flow out at some other point. In the external medium all the sources destroy one another, and we have an infinite medium without sources surrounding the internal medium. The pres- sure at infinity is zero, that at the surface is constant. If the pressure at the surface is positive, the motion of the fluid must be outwards from every point of the surface ; if it be negative, it must flow inwards towards the surface. But it has been shewn that neither of these cases is possible, because if any fluid enters the surface an equal quantity must escape, and therefore the pressure at the surface is zero in the third system. The pressure at all points in the boundary of the internal medium in the third case is therefore zero, and there are no sources, and therefore the pressure is everywhere zero, by (16). The pressure in the bounding surface of the internal medium is also zero, and there is no resistance, therefore it is zero throughout ; but the pressure in the third case is the difference of pressures in the two given cases, therefore these are equal, and there is only one distribution of pressure which is possible, namely, that due to the imaginary distribution of sources and sinks. (25) When the resistance is infinite in the internal medium, there can be no passage of fluid through it or into it. The bounding surface may therefore be considered as impermeable to the fluid, and the tubes of fluid motion will run along it without cutting it. If by assuming any arbitrary distribution of sources within the surface in addition to the given sources in the outer medium, and by calculating the resulting pressures and velocities as in the case of a uniform medium, we can fulfil the condition of there being no velocity across the surface, the system of pressures in the outer medium will be the true one. For since no fluid passes through the surface, the tubes in the interior are independent of those outside, and may be taken away without altering the external motion. (26) If the extent of the internal medium be small, and if the difference of resistance in the two media be also small, then the position of the unit tubes will not be much altered from what it would be if the external medium filled the whole space. ON FARADAY S LINES OE FORCE. 171 On this supposition we can easily calculate the kind of alteration which the introduction of the internal medium will produce ; for wherever a unit tube h' — Ic enters the surface we must conceive a source producing fluid at a rate -^ — , and wherever a tube leaves it we must place a sink annihilating fluid at the h' — h rate — r — , then calculating pressures on the supposition that the resistance in both media is h, the same as in the external medium, we shall obtain the true distribution of pressures very approximately, and we may get a better result by repeating the process on the system of pressures thus obtained. (27) If instead of an abrupt change from one coefficient of resistance to another we take a case in which the resistance varies continuously from point to point, we may treat the medium as if it were composed of thin shells each of which has uniform resistance. By properly assuming a distribution of sources over the surfaces of separation of the shells, we may treat the case as if the resistance were equal to unity throughout, as in (23). The sources wiU then be distributed continuously throughout the whole medium, and will be positive whenever the motion is from places of less to places of greater resistance, and negative when in the contrary direction. (28) Hitherto we have supposed the resistance at a given point of the medium to be the same in whatever direction the motion of the fluid takes place ; but we may conceive a case in which the resistance is different in different directions. In such cases the lines of motion will not in general be perpendicular to the surfaces of equal pressure. If a, h, c be the components of the velocity at any point, and a, /S, y the components of the resistance at the same point, these quantities will be connected by the following system of linear equations, which may be called "equations of conduction," and will be referred to by that name. a = P,a + QS + R,y, h = P„^+Q,y + B,a, c = P,y+Q,a + RS- In these equations there are nine independent coefficients of conductivity. In order to simplify the equations, let us put Q, + R,^2S„ Q,-R, = 1lT, &c &c. 22—2 172 ON FARAD ay's LINES OF FORCE. where 4.r = {Q,-R^y + {Q,-B,Y + {Q,-R:)\ and I, m, n are direction-cosines of a certain fixed line in space. The equations then become a = P,a + SS + S,y + {nfi - my) T, h=P,fi + S,y + S,a + { ly - no) T, c = P,y + S,a + Sj3 + {ma- IP) T. By the ordinary transformation of co-ordinates we may get rid of the coeflScients marked S. The equations then become a = P;a + {n'^-m'y)T, b = P:i3 + {l'y-n'a)T, c = P:y + {m'a- l'^)T, where I', m', n are the direction-cosines of the fixed line with reference to the new axes. If we make _dp o_dp J __dp ~ dx' dy' '^~ dz' the equation of continuity becomes da db <^c _ dx dy dz ' ^' dx''^ ' dy- ' dz'~^' and if we make x — JP^^, y = JP^'q, z = JP^t„ d^p d^p d'p „ the ordinary equation of conduction. It appears therefore that the distribution of pressures is not altered by the existence of the coefficient T. Professor Thomson has shewn how to conceive a substance in which this coefficient determines a property having reference to an axis, which unhke the axes of P^, P^, Pj is dipolar. For further information on the equations of conduction, see Professor Stokes On the Conduction of Heat in Crystals (Cambridge and Dublin Math. Journ.), and Professor Thomson On the Dynamical Theory of Heat, Part v. [Transactions of Royal Society of Edinburgh, Vol. xxi. Part i.). ON FAKADAYS LINES OF FORCE. 173 It is evident that all ttat has been proved in (14), (15), (16), (17), with respect to the superposition of different distributions of pressure, and there being only one distribution of pressures corresponding to a given distribution of sources, will be true also in the case in which the resistance varies from point to point, and the resistance at the same point is different in different directions. For if we examine the proof we shall find it applicable to such cases as weU as to that of a uniform medium. (29) We now are prepared to prove certain general propositions which are true in the most general case of a medium whose resistance is different in different directions and varies from point to point. We may by the method of (28), when the distribution of pressures is known, construct the surfaces of equal pressure, the tubes of fluid motion, and the sources and sinks. It is evident that since in each cell into which a unit tube is divided by the surfaces of equal pressure unity of fluid passes from pressure p to pressure (p — 1) in unit of time, unity of work is done by the fluid in each cell in overcoming resistance. The number of cells in each unit tube is determined by the number of surfaces of equal pressure through which it passes. If the pressure at the beginning of the tube be p and at the end p, then the number of cells in it will be p—p'. Now if the tube had extended from the source to a place where the pressure is zero, the number of cells would have been p, and if the tube had come from the sink to zero, the number would have been p', and the true number is the difference of these. Therefore if we find the pressure at a source S from which S tubes proceed to be p, Sp is the number of cells due to the source S ; but if S' of the tubes terminate in a sink at a pressure p, then we must cut off ^p cells from the number previously obtained. Now if we denote the source of S tubes by S, the sink of S' tubes may be written -S', sinks always being reckoned negative, and the general expression for the number of cells in the system wUl be % (Sp). (30) The same conclusion may be arrived at by observing that unity of work is done on each cell. Now in each source S, S units of fluid are expelled against a pressure p, so that the work done by the fluid in over- coming resistance is Sp. At each sink in which S' tubes terminate, S' units of fluid sink into nothing under pressure p ; the work done upon the fluid by 174 ON Faraday's lines of force. the pressure is therefore S' p. The whole work done by the fluid may there- fore be expressed by W = tSp-tS'p', or more concisely, considering sinks as negative sources, W = t{Sp). (31) Let S represent the rate of production of a source in any medium, and let p be the pressure at any given point due to that source. Then if we superpose on this another equal source, every pressure will be doubled, and thus by successive superposition we find that a source nS would produce a pressure np, or more generally the pressure at any point due to a given source varies as the rate of production of the source. This may be expressed by the equation p^RS, where ^ is a coefficient depending on the nature of the medium and on the positions of the source and the given point. In a uniform medium whose resistance is measured by h, ■' 477r 4Trr R may be called the coefficient of resistance of the medium between the source and the given point. By combining any number of sources we have generally p = %{RS). (32) In a uniform medium the pressure due to a source S k S 477 r At another source S' at a distance r we shall have S'p^]^^ = Sp\ if p be the pressure at S due to S'. If therefore there be two systems of sources t{S) and t{S'), and if the pressures due to the first be p and to the second p', then t[S'p) = t{Sp'). For every term S'p has a term Sp' equal to it. ON Faraday's lines op force. 175 (33) Suppose that in a uniform medium the motion of the fluid is every- where parallel to one plane, then the surfaces of equal pressure will be perpendicular to this plane. If we take two parallel planes at a distance equal to k from each other, we can divide the space between these planes into unit tubes by means of cylindric surfaces perpendicular to the planes, and these together with the surfaces of equal 'pressure will divide the space into cells of which the length is equal to the breadth. For if h be the distance between consecutive surfaces of equal pressure and s the section of the unit tube, we have by (13) s = kh. But s is the product of the breadth and depth ; but the depth is k, therefore the breadth is h and equal to the length. If two systems of plane curves cut each other at right angles so as to divide the plane into little areas of which the length and breadth are equal, then by taking another plane at distance k from the first and erecting cyhndric surfaces on the plane curves as bases, a system of cells will be formed which will satisfy the conditions whether we suppose the fluid to run along the first set of cutting lines or the second*. Application of the Idea of Lines of Force. I have now to shew how the idea of lines of fluid motion as described above may be modified so as to be applicable to the sciences of statical elec- tricity, permanent magnetism, magnetism of induction, and uniform galvanic currents, reserving the laws of electro-magnetism for special consideration. I shall assume that the phenomena of statical electricity have been already explained by the mutual action of two opposite kinds of matter. If we consider one of these as positive electricity and the other as negative, then any two particles of electricity repel one another with a force which is measured by the product of the masses of the particles divided by the square of their distance. Now we found in (18) that the velocity of our imaginary fluid due to a source S at a distance r varies inversely as r^ Let us see what will be the effect of substituting such a source for every particle of positive electricity. The velocity due to each source would be proportional to the attraction due to the corresponding particle, and the resultant velocity due to all the sources would * See Cambridge wnd Dublin Mathematical Journal, Vol. iii. p. 286. 176 ON FAKADAYS LINES OF FOECE. be proportional to the resultant attraction of all the particles. Now we may find the resultant pressure at any point by adding the pressures due to the given sources, and therefore we may find the resultant velocity in a given direction from the rate of decrease of pressure in that direction, and this will be proportional to the resultant attraction of the particles resolved in that direction. Since the resultant attraction in the electrical problem is proportional to the decrease of pressure in the imaginary problem, and since we may select any values for the constants in the imaginary problem, we may assume that the resultant attraction in any direction is numerically equal to the decrease of pressure in that direction, or dx' By this assumption we find that if V be the potential, d V= Xdx + Ydy + Zdz = - dp, or since at an infinite distance F= and p = 0, V= —p. In the electrical problem we have In the fluid p^l, (- F= -2 (— r S= -7^ dm. If k be supposed very great, the amount of fluid produced by each source in order to keep up the pressures will be very small. The potential of any system of electricity on itself wUl be %{pdm) = ^, %{pS) = ^W. If t (dm), % (dm) be two systems of electrical particles and p, p the potentials due to them respectively, then by (32) Mpdm) = ~t{pS') = ^^%{p'S)=.X{p'dm), or the potential of the first system on the second is equal to that of the second system on the first. ON Faraday's lines op force. 177 So ttat in the ordinary electrical problems the analogy in fluid motion is of this kind : V=-p, dm = -- S, k whole potential of a system = — 2 Vdm = ~-W, where W is the work done by the fluid in overcoming resistance. The lines of forces are the unit tubes of fluid motion, and they may be estimated numerically by those tubes. Theory of Dielectrics. The electrical induction exercised on a body at a distance depends not only on the distribution of electricity in the inductric, and the form and posi- tion of the inducteous body, but on the nature of the interposed medium, or dielectric. Faraday'^ expresses this by the conception of one substance having a greater inductive capacity, or conducting the lines of inductive action more freely than another. If we suppose that in our analogy of a fluid in a resisting medium the resistance is different in different media, then by making the resistance less we obtain the analogue to a dielectric which more easily conducts Faraday's lines. It is evident from (23) that in this case there will always be an apparent distribution of electricity on the surface of the dielectric, there being negative electricity where the lines enter and positive electricity where they emerge. In the case of the fluid there are no real sources on the surface, but we use them merely for purposes of calculation. In the dielectric there may be no real charge of electricity, but only an apparent electric action due to the surface. If the dielectric had been of less conductivity than the surrounding medium, we should have had precisely opposite effects, namely, positive electricity where lines enter, and negative where they emerge. * Series xi. VOL. I. 23 178 ON Faraday's lines of foece. If the conduction of the dielectric is perfect or nearly so for the small quantities of electricity with which we have to do, then we have the case of (24). The dielectric is then considered as a conductor, its surface is a surface of equal potential, and the resultant attraction near the surface itself is per- pendicular to it. Theory of Permanent Magnets. A magnet is conceived to be made up of elementary magnetized particles, each of which has its own north and south poles, the action of which upon other north and south poles is governed by laws mathematically identical with those of electricity. Hence the same application of the idea of lines of force can be made to this subject, and the same analogy of fluid motion can be employed to illustrate it. But it may be useful to examine the way in which the polarity of the elements of a magnet may be represented by the unit cells in fluid motion. In each unit cell unity of fluid enters by one face and flows out by the opposite face, so that the first face becomes a unit sink and the second a unit source with respect to the rest of the fluid. It may therefore be compared to an elementary magnet, having an equal quantity of north and south magnetic matter distributed over two of its faces. If we now consider the cell as forming part of a system, the fluid flowing out of one cell will flow into the next, and so on, so that the source will be transferred from the end of the cell to the end of the unit tube. If all the unit tubes begin and end on the bounding surface, the sources and sinks will be distributed entirely on that surface, and in the case of a magnet which has what has been called a solenoidal or tubular distribution of magnetism, all the imaginary magnetic matter will be on the surface'*". Theory of Paramagnetic and Diamagnetic Induction. Faraday t has shewn that the effects of paramagnetic and diamagnetic bodies in the magnetic field may be explained by supposing paramagnetic bodies to * See Professor Thomson On the Mathematical Theory of Magnetism, Chapters in. and v. Phil. Trans. 1851. t Experimental Researches (3292). ON FARADAY S LINES OF FORCE. 179 conduct the lines of force better, and diamagnetic bodies worse, than the surrounding medium. By referring to (23) and (26), and supposing sources to represent north magnetic matter, and sinks south magnetic matter, then if a paramagnetic body be in the neighbourhood of a north pole, the lines of force on entering it will produce south magnetic matter, and on leaving it they will produce an equal amount of north magnetic matter. Since the quantities of magnetic matter on the whole are equal, but the southern matter is nearest to the north pole, the result will be attraction. If on the other hand the body be diamagnetic, or a worse conductor of lines of force than the surrounding medium, there will be an imaginary distribution of northern magnetic matter where the lines pass into the worse conductor, and of southern where they pass out, so that on the whole there will be repulsion. We may obtain a more general law from the consideration that the poten- tial of the whole system is proportional to the amount of work done by the fluid in overcoming resistance. The introduction of a second medium increases or diminishes the work done according as the resistance is greater or less than that of the first medium. The amount of this increase or diminution will vary as the square of the velocity of the fluid. Now, by the theory of potentials, the moving force in any direction is measured by the rate of decrease of the potential of the system in passing along that direction, therefore when ¥, the resistance within the second medium, is greater than h, the resistance in the surrounding medium, there is a force tend- ing from places where the resultant force v is greater to where it is less, so that a diamagnetic body moves from greater to less values of the resultant force *. In paramagnetic bodies h' is less than k, so that the force is now from points of less to points of greater resultant magnetic force. Since these results depend only on the relative values of h and ¥, it is evident that by changing the surrounding medium, the behaviour of a body may be changed from para- magnetic to diamagnetic at pleasure. It is evident that we should obtain the same mathematical results if we had supposed that the magnetic force had a power of exciting a polarity in bodies which is in the same direction as the lines in paramagnetic bodies, and * Experimental Researches (2797), (2798). See Thomson, Cambridge and Dublin Mathematiccd Journal, May, 1847. 23—2 180 ON faeaday's lines of force. in tlie reverse direction in diamagnetic bodies*. In fact we have not as yet come to any facts which would lead us to choose any one out of these three theories, that of Imes of force, that of imaginary magnetic matter, and that of induced polarity. As the theory of lines of force admits of the most precise, and at the same time least theoretic statement, we shall allow it to stand for the present. Tlieory of Magnecrystallic Induction. The theory of Faradayt with respect to the behaviour of crystals in the magnetic field may be thus stated. In certain crystals and other substances the lines of magnetic force are conducted with different facility in different directions. The body when suspended in a uniform magnetic field will turn or tend to turn into such a position that the fines of force shall pass through it with least resist- ance. It is not diflScult by means of the principles in (28) to express the laws of this kind of action, and even to reduce them in certain cases to numerical formulae. The principles of induced polarity and of imaginary magnetic matter are here of little use ; but the theory of lines of force is capable of the most perfect adaptation to this class of phenomena. Theory of the Conduction of Current Electricity. It is in the calculation of the laws of constant electric currents that the theory of fluid motion which we have laid down admits of the most direct apph- cation. In addition to the researches of Ohm on this subject, we have those of M. Kirchhoff, An7i. de Chim. XLi. 496, and of M. Quincke, XLVii. 203, on the Conduction of Electric Currents in Plates. According to the received opinions we have here a current of fluid moving uniformly in conducting circuits, which oppose a resistance to the current which has to be overcome by the application of an electro-motive force at some part of the circuit. On account of this resistance to the motion of the fluid the pressure must be different at different points in the circuit. This pressure, which is commonly called electrical tension, * Exp. Res. (2429), (3320). See Weber, Poggendorff, Lxxxvii. p. 145. Prof. TyndaU, Phil. Trans. 1856, p. 237. t Exp. Res. (2836), &c. ON PAB.ADAYS LINES OF FOBCB. 181 is found to be physically identical with the potential in statical electricity, and thus we have the means of connecting the two sets of phenomena. If we knew what amount of electricity, measured statically, passes along that current which we assume as our unit of current, then the connexion of electricity of tension with current electricity would be completed'*. This has as yet been done only approximately, but we know enough to be certain that the conducting powers of different substances differ only in degree, and that the difference between glass and metal is, that the resistance is a great but finite quantity in glass, and a small but finite quantity in metal. Thus the analogy between statical electricity and fluid motion turns out more perfect than we might have supposed, for there the induction goes on by conduction just as in current electricity, but the quan- tity conducted is insensible owing to the great resistance of the dielectricst. On Electro-motive Forces. When a uniform current exists in a closed circuit it is evident that some other forces must act on the fluid besides the pressures. For if the current were due to difference of pressures, then it would flow from the point of greatest pressure in both directions to the point of least pressure, whereas in reality it circulates in one direction constantly. We must therefore admit the existence of certain forces capable of keeping up a constant current in a closed circuit. Of these the most remarkable is that which is produced by chemical action. A cell of a voltaic battery, or rather the surface of separation of the fluid of the cell and the zinc, is the seat of an electro-motive force which can maintain a current in opposition to the resistance of the circuit. If we adopt the usual convention in speaking of electric currents, the positive current is from the fluid through the platinum, the conducting circuit, and the zinc, back to the fluid again. If the electro-motive force act only in the surface of separation of the fluid and zinc, then the tension of electricity in the fluid must exceed that in the zinc by a quantity depending on the nature and length of the circuit and on the strength of the current in the conductor. In order to keep up this difference of pressure there must be an electro-motive force whose intensity is measured by that difference of pressure. If F be the electro-motive force, / the quantity of the current or the number of electrical * See Exp. Res. (371). t Exp. Res. Vol iii. p. 513. 182 ON FARADAY S LINES OF FORCE. units delivered in unit of time, and K a quantity depending on the length and resistance of the conducting circuit, then F=IK=p-p', where p is the electric tension in the fluid and p' in the zinc. If the circuit be broken at any point, then since there is no current the tension of the part which remains attached to the platinum will be p, and that of the other will be p', p—p' or F affords a measure of the intensity of the current. This distinction of quantity and intensity is very useful *, but must be distinctly understood to mean nothing more than this : — The quantity of a current is the amount of electricity which it transmits in unit of time, and is measured by / the number of unit currents which it contains. The intensity of a current is its power of overcoming resistance, and is measured by F or IK, where K is the resistance of the whole circuit. The same idea of quantity and intensity may be applied to the case of magnetism f. The quantity of magnetization in any section of a magnetic body is measured by the number of lines of magnetic force which pass through it. The intensity of magnetization in the section depends on the resisting power of the section, as weU as on the number of lines which pass through it. If k be the resisting power of the material, and S the area of the section, and / the number of lines of force which pass through it, then the whole intensity throughout the section =F=I~ s- When magnetization is produced by the influence of other magnets only, we may put p for the magnetic tension at any point, then for the whole magnetic solenoid F^illi^^iK.,-,: When a solenoidal magnetized circuit returns into itself, the magnetization does not depend on difference of tensions only, but on some magnetizing force of which the intensity is F. If i be the quantity of the magnetization at any point, or the number of lines of force passing through unit of area in the section of the solenoid, then * Exp. Mes. Vol. III. p. 519. f Exp. Ees. (2870), (3293). ON FARADAY S LINES OF FORCE. 183 the total quantity of magnetisation in the circuit is the number of hnes which pass through any section, I='Zidydz, where dydz is the element of the section, and the summation is performed over the whole section. The intensity of magnetization at any point, or the force required to keep up the magnetization, is measured by hi=f, and the total intensity of magnetization in the circuit is measured by the sum of the local intensities all round the circuit, F=t{fdx), where dx is the element of length in the circuit, and the summation is extended round the entire circuit. In the same circuit we have always F—IK, where K is the total resistance of the circuit, and depends on its form and the matter of which it is composed. On the Action of closed Currents at a Distance. The mathematical laws of the attractions and repulsions of conductors have been most ably investigated by Ampere, and his results have stood the test of subsequent experiments. From the single assumption, that the action of an element of one current upon an element of another current is an attractive or repulsive force acting in the direction of the line joining the two elements, he has determined by the simplest experiments the mathematical form of the law of attraction, and has put this law into several most elegant and usefxil forms. We must recoUect however that no experiments have been made on these elements of currents except under the form of closed currents either in rigid conductors or in fluids, and that the laws of closed currents can only be deduced from such experiments. Hence if Ampere's formulae applied to closed currents give true results, their truth is not proved for elements of currents unless we assume that the action between two such elements must be along the line which joins them. Although this assumption is most warrantable and philosophical in the present state of science, it will be more conducive to freedom of investi- gation if we endeavour to do without it, and to assume the laws of closed currents as the Tiltimate datum of experiment. ]84 ON fahaday's lines of force. Ampere has shewn that when currents are combined according to the law of the parallelogram of forces, the force due to the resultant current is the resultant of the forces due to the component currents, and that equal and opposite currents generate equal and opposite forces, and when combined neutralize each other. He has also shewn that a closed circuit of any form has no tendency to turn a moveable circular conductor about a fixed axis through the centre of the circle perpendicular to its plane, and that therefore the forces in the case of a closed circuit render Xdx + Ydy + Zdz a complete difietential. Finally, he has shewn that if there be two systems of circuits similar and similarly situated, the quantity of electrical current in corresponding conductors being the same, the resultant forces are equal, whatever be the absolute dimensions of the systems, which proves that the forces are, cceteris paribus, inversely as the square of the distance. From these results it follows that the mutual action of two closed currents whose areas are very small is the same as that of two elementary magnetic bars magnetized perpendicularly to the plane of the currents. The direction of magnetization of the equivalent magnet may be pre- dicted by remembering that a current travelling round the earth from east to west as the sun appears to do, would be equivalent to that magnetization which the earth actually possesses, and therefore in the reverse direction to that of a magnetic needle when pointing freely. If a number of closed unit currents in contact exist on a surface, then at all points in which two currents are in contact there will be two equal and opposite currents which will produce no eifect, but all round the boundary of the surface occupied by the currents there will be a residual current not neutrahzed by any other ; and therefore the result will be the same as that of a single unit current round the boundary of all the currents. From this it appears that the external attractions of a shell uniformly magnetized perpendicular to its surface are the same as those due to a current round its edge, for each of the elementary currents in the former case has the same effect as an element of the magnetic shell. If we examine the lines of magnetic force produced by a closed current, we shall find that they form closed curves passing round the current and embracing it, and that the total intensity of the magnetizing force all along the closed line of force depends on the quantity of the electric current only. ON FAHADAYS LINES OF FOECE. 185 The number of unit lines* of magnetic force due to a closed current depends on the form as well as the quantity of the current, but tbe number of unit ceUsf in each complete line of force is measured simply by the number of unit currents which embrace it. The unit cells in this case are portions of space in which unit of magnetic quantity is produced by unity of magnetizing force. The length of a cell is therefore inversely as the intensity of the magnetizing force, and its section inversely as the quantity of magnetic induction at that point. The whole number of cells due to a given current is therefore proportional to the strength of the current multiphed by the number of lines of force which pass through it. If by any change of the form of the conductors the number of cells can be increased, there will be a force tending to produce that change, so that there is always a force urging a conductor transverse to the lines of magnetic force, so as to cause more lines of force to pass through the closed circuit of which the conductor forms a part. The number of cells due to two given currents is got by multiplying the number of lines of inductive magnetic action which pass through each by the quantity of the currents respectively. Now by (9) the number of lines which pass through the first current is the sum of its own lines and those of the second current which would pass through the first if the second current alone were in action. Hence the whole number of cells will be increased by any motion which causes more lines of force to pass through either circuit, and therefore the resultant force will tend to produce such a motion, and the work done by this force during the motion will be measured by the number of new cells produced. All the actions of closed conductors on each other may be deduced from this principle. On Electric Currents produced by Induction. Faraday has shewn 4 that when a conductor moves transversely to the lines of magnetic force, an electro-motive force arises in the conductor, tending to produce a current in it. If the conductor is closed, there is a continuous current, if open, tension is the result. If a closed conductor move transversely to the lines of magnetic induction, then, if the number of lines which pass * Uxp. Res. (3122). See Art. (6) of this paper. t Art. (13). X Exp. Res. (3077), &c. VOL. L 24 186 ON pabaday's lines op force. througli it does not change during the motion, the electro-motive forces in the circuit will be in equilibrium, and there will be no current. Hence the electro- motive forces depend on the number of lines which are cut bj the conductor during the motion. If the motion be such that a greater number of lines pass through the circuit formed by the conductor after than before the motion, then the electro-motive force will be measured by the increase of the number of lines, and will generate a current the reverse of that which would have produced the additional lines. When the number of lines of inductive magnetic action through the circuit is increased, the induced current will tend to diminish the number of lines, and when the number is diminished the induced current will tend to increase them. That this is the true expression for the law of induced currents is shewn from the fact that, in whatever way the number of lines of magnetic induction passing through the circuit be increased, the electro-motive effect is the same, whether the increase take place by the motion of the conductor itself, or of other conductors, or of magnets, or by the change of intensity of other currents, or by the magnetization or demagnetization of neighbouring magnetic bodies, or lastly by the change of intensity of the current itself. In all these cases the electro-motive force depends on the cJmnge in the number of lines of inductive magnetic action which pass through the circuit*. * The electro-magnetic forces, whicli tend to produce motion of the matefiial conductor, must he carefully distinguished from the electro-motive forces, which tend to produce electric currents. Let an electric current be passed through a mags of metal of any form. The distribution of the currents within the metal will be determined by the laws of conduction. Now let a constant electric current be passed through another conductor near the first. If the two currents are in the same direction the two conductors will be attracted towards each other, and would come nearer if not held in their positions. But though the material conductors are attracted, the currents (which are free to choose any course within the metal) will not alter their original distribution, or incline towards each other. For, since no change takes place in the system, there will be no electro-motive forces to modify the original distribution of currents. In this case we have electro-magnetic forces acting on the material conductor, without any electro-motive forces tending to modify the current which it carries. Let us take as another example the case of a linear conductor, not forming a closed circuit, and let it be made to traverse the lines of magnetic force, either by its own motion, or by chan"es in the magnetic field. An electro -motive force will act in the direction of the conductor, and, as it cannot produce a current, because there is no circuit, it will produce electric tension at the extremi- ties. There will be no electro-magnetic attraction on the material conductor, for this attraction depends on the existence of the cun-ent within it, and this is prevented by the circuit not being closed. Here then we have the opposite case of an electro-motive force acting on the electricity in the conductor, but no attraction on its material particles. ON FARADAY S LINES OP FORCE. 187 It is natural to suppose that a force of this kind, which depends on a change in the number of lines, is due to a change of state which is measured by the number of these lines. A closed conductor in a magnetic field may- be supposed to be in a certain state arising from the magnetic action. As long as this state remains unchanged no effect takes place, but, when the state changes, electro-motive forces arise, depending as to their intensity and direction on this change of state. I cannot do better here than quote a passage from the first series of Faraday's Experimental Researches, Art. (60). "While the wire is subject to either volta-electric or magno-electric induction it appears to be in a peculiar state, for it resists the formation of an electrical current in it ; whereas, if in its common condition, such a current would be produced; and when left uninfluenced it has the power of originating a current, a power which the wire does not possess under ordinary circumstances. This electrical condition of matter has not hitherto been recognised, but it probably exerts a very important influence in many if not most of the phe- nomena produced by currents of electricity. For reasons which will immediately appear (7) I have, after advising with several learned friends, ventured to designate it as the electro-tonic state." Finding that all the phenomena could be otherwise explained without reference to the electro-tonic state, Faraday in his second series rejected it as not necessary ; but in his recent researches ''^ he seems still to think that there may be some physical truth in his conjecture about this new state of bodies. The conjecture of a philosopher so familiar with nature may sometimes be more pregnant with truth than the best established experimental law disco- vered by empirical inquirers, and though not bound to admit it as a physical truth, we may accept it as a new idea by which our mathematical conceptions maji be rendered clearer. r In this outline of Faraday's electrical theories, as they appear from a mathematical point of view, I can do no more than simply state the mathe- matical methods by which I believe that electrical phenomena can be best comprehended and reduced to calculation, and my aim has been to present the mathematical ideas to the mind in an embodied form, as systems of lines or surfaces, and not as mere symbols, which neither convey the same ideas, nor readily adapt themselves to the phenomena to be explained. The idea of the electro-tonic state, however, has not yet presented itself to my mind in such a * (3172) (3269). 24—2 188 ON Faraday's lines of force. form that its nature and properties may be clearly explained without reference to mere symbols, and therefore I propose- in the following investigation to use symbols freely, and to take for granted the ordinary mathematical operations. By a careful study of the laws of elastic solids and of the motions of viscous fluids, I hope to discover a method of forming a mechanical conception of this electro-tonic state adapted to general reasoning*. Part II. On Faraday's " ElectrO'tonic State." When a conductor moves in the neighbourhood of a current of electricity, or of a magnet, or when a current or magnet near the conductor is moved, or altered in intensity, then a force acts on the conductor and produces electric tension, or a continuous current, according as the circuit is open or closed. This current is produced only by changes of the electric or magnetic phenomena sur- rounding the conductor, and as long as these are constant there is no observed effect on the conductor. Still the conductor is in different states when near a current or magnet, and when away from its influence, since the removal or destruction of the current or magnet occasions a current, which would not have existed if the magnet or current had not been previously in action. Considerations of this kind led Professor Faraday to connect with his discovery of the induction of electric currents the conception of a state into which all bodies are thrown by the presence of magnets and currents. This state does not manifest itself by any known phenomena as long as it is undis- turbed, but any change in this state is indicated by a current or tendency towards a current. To this state he gave the name of the "Electro-tonic State," and although he afterwards succeeded in explaining the phenomena which suggested it by means of less hypothetical conceptions, he has on several occasions hinted at the probability that some phenomena might be discovered which would render the electro-tonic state an object of legitimate induction. These speculations, into which Faraday had been led by the study of laws which he has well established, and which he abandoned only for want of experi- * See Prof. W. Thomson On a Mechanical Representation of Electric, Magiietic and Galvanic Forces. Gamib. and Dub. Math. Jour. Jan. 1847. ON Faraday's lines of force. 189 mental data for the direct proof of the unknown state, have not, I think, been made the subject of mathematical investigation. Perhaps it may be thought that the quantitative determinations of the various phenomena are not suffi- ciently rigorous to be made the basis of a mathematical theory ; Faraday, however, has not contented himself with simply stating the numerical results of his experiments and leaving the law to be discovered by calculation. Where he has perceived a law he has at once stated it, in terms as unambiguous as those of pure mathematics ; and if the mathematician, receiving this as a physical truth, deduces from it other laws capable of being tested by experiment, he has merely assisted the physicist in arranging his own ideas, which is con- fessedly a necessary step in scientific induction. In the following investigation, therefore, the laws established by Faraday wiU be assumed as true, and it will be shewn that by following out his speculations other and more general laws can be deduced from them. If it should then appear that these laws, originally devised to include one set of phenomena, may be generalized so as to extend to phenomena of a different class, these mathematical connexions may suggest to physicists the means of establishing physical connexions ; and thus mere speculation may be turned to account in experimental science. On Quantity and Intensity as Properties of Electric Currents. It is found that certain effects of an electric current are equal at what- ever part of the circuit they are estimated. The quantities of water or of any other electrolyte decomposed at two different sections of the same circuit, are always found to be equal or equivalent, however different the material and form of the circuit may be at the two sections. The magnetic effect of a conducting wire is also found to be independent of the form or material of the wire in the same circuit. There is therefore an electrical effect which is equal at every section of the circuit. If we conceive of the conductor as the channel along which a fluid is constrained to move, then the quantity of fluid transmitted by each section will be the same, and we may define the quantity of an electric current to be the quantity of electricity which passes across a complete section of the current in unit of time. We may for the present measure quantity of electricity by the quantity of water which it would decom- pose in unit of time. 190 ON FARAD ay's LINES OF EOECB. In order to express mathematically the electrical currents in any conductor, we must have a definition, not only of the entire flow across a complete section, but also of the flow at a given point in a given direction. Def. The quantity of a current at a given point and in a given direction is measured, when uniform, by the quantity of electricity which flows across unit of area taten at that point perpendicular to the given direction, and when variable by the quantity which would flow across this area, supposing the flow uniformly the same as at the given point. In the following investigation, the quantity of electric current at the point {xyz) estimated in the directions of the axes x, y, z respectively will be denoted by a„ h„ c,. The quantity of electricity which flows in unit of time through the ele- mentary area dS = dS {la^ + mb^ + nc^, where I, m, n are the direction-cosines of the normal to dS. This flow of electricity at any point of a conductor is due to the electro- motive forces which act at that point. These may be either external or internal. External electro-motive forces arise either from the relative motion of currents and magnets, or from changes in their intensity, or from other causes acting at a distance. Internal electro-motive forces arise principally from diSerence of electric tension at points of the conductor in the immediate neighbourhood of the point in question. The other causes are variations of chemical composition or of tem- perature in contiguous parts of the conductor. Let 2^2 represent the electric tension at any point, and X^, Y^, Z^ the sums of the parts of all the electro-motive forces arising from other causes resolved parallel to the co-ordinate axes, then if a^, ySj, y^ be the effective electro-motive forces dp^ ""'^^^^-dx o _y _ dpi '^'~ ' dy „ dp^ (A). ON FARADAY S LINES OF FORCE. 191 Now the quantity of the current depends on the electro-motive force and on the resistance of the medium. If the resistance of the medium be uniform in all directions and equal to h^, (h = ka„ fi., = kfi^, 72 = ^262 (B), but if the resistance be different in different directions, the law will be more compHcated. These quantities a^, p.^, y^ may be considered as representing the intensity of the electric action in the directions of x, y, z. The intensity measured along an element da- of a curve is given by e = Za + m/3 + ny, where Z, m, n are the direction-cosines of the tangent. The integral \&1(t taken with respect to a given portion of a curve hne, represents the total intensity along that line. If the curve is a closed one, it represents the total intensity of the electro-motive force in the closed curve. Substituting the values of a, ^, y from equations (A) \&da- = l{Xdx + Ydy + Zdz) -p + C. If therefore [Xdx+Ydy + Zdz) is a complete differential, the value of \e.da- for a closed curve wUl vanish, and in all closed curves \edcr = \{Xdx + Ydy + Zdz), the integration being effected along the curve, so that in a closed curve the total intensity of the effective electro-motive force is equal to the total intensity of the impressed electro-motive force. The total quantity of conduction through any surface is expressed by \edS, where e = la-\- mh + no, I, m, n being the direction-cosines of the normal, . •. ledS = lladydz + \\hdzdx + \\cdxdy, the integrations being effected over the given surface. When the surface is a closed one, then we may find by integration by parts ^^^'^ = 11/(^ + 1 +1)^^^^^'- 192 ON Faraday's lines of foece. If we make da dh dc . /p\ -f- + T- +^ = 477/3 {^), dx dy dz \edS= itt\\\pdxdydz, where the integration on the right side of the equation is effected over every part of space within the surface. In a large class of phenomena, including aU cases of uniform currents, the quantity p disappears. Magnetic Quantity and Intensity. From his study of the lines of magnetic force, Faraday has been led to the conclusion that in the tubular surface'* formed by a system of such lines, the quantity of magnetic induction across any section of the tube is constant, aad that the alteration of the character of these lines in passing from one substance to another, is to be explained by a difference of inductive capacity in the two substances, which is analogous to conductive power in the theory of electric currents. In the following investigation we shall have occasion to treat of magnetic quantity and intensity in connection with electric. In such cases the magnetic symbols will be distinguished by the suifix 1, and the electric by the suffix 2. The equations connecting a, h, c, k, a, y8, y, p, and p, are the same in form as those which we have just given, a, h, c are the symbols of magnetic induction with respect to quantity ; k denotes the resistance to magnetic induction, and may be different in different directions ; a, /S, y, are the effective magnetizing forces, connected with a, b, c, by equations (B) ; p is the magnetic tension or potential which will be afterwards explained ; p denotes the density of real magnetic matter and is connected with a, b, c by equations (C). As aU the details of magnetic calculations wUl be more intelligible after the exposition of the connexion of magnetism with electricity, it will be sufficient here to say that all the definitions of total quantity, with respect to a surface, the total intensity to a curve, apply to the case of magnetism as weU as to that of electricity. * F. Exp. Res. 3271, definition of "Spliondyloid." ON Faraday's lines of force. 193 Electro-magnetism. Ampere has proved the following laws of the attractions and repulsions of electric currents : « I. Equal and opposite currents generate equal and opposite forces. II. A crooked current is equivalent to a straight one, provided the two currents nearly coincide throughout their whole length. III. Equal currents traversing similar and similarly situated closed curves act with equal forces, whatever be the linear dimensions of the circuits. IV. A closed current exerts no force tending to turn a circular conductor about its centre. It is to be observed, that the currents with which Ampere worked were constant and therefore re-entering. All his results are therefore deduced from experiments on closed currents, and his expressions for the mutual action of the elements of a current involve the assumption that this action is exerted in the direction of the line joining those elements. This assumption is no doubt warranted by the universal consent of men of science in treating of attractive forces considered as due to the mutual action of particles ; but at present we are proceeding on a different principle, and searching for the explanation of the phenomena, not in the currents alone, but also in the surrounding medium. The first and second laws shew that currents are to be combined like velocities or forces. The third law is the expression of a property of all attractions which may be conceived of as depending on the inverse square of the distance from a fixed system of points ; and the fourth shews that the electro-magnetic forces may always be reduced to the attractions and repulsions of imaginary matter properly distributed. In fact, the action of a very small electric circuit on a point in its neigh- bourhood is identical with that of a small magnetic element on a point outside it. If we divide any given portion of a surface into elementary areas, and cause equal currents to flow in the same direction round all these little areas, the effect on a point not in the surface will be the same as that of a shell coinciding with the surface, and uniformly magnetised normal to its surface. But by the first law all the currents forming the little circuits will destroy VOL. 1. 25 194 ON FARADAY S LINES OF FORCE, one another, and leave a single current running round the bounding line. So that the magnetic effect of a uniformly magnetized shell is equivalent to that of an electric current round the edge of the shell. If the direction of the current coincide with that of the apparent motion of the sun, then the direction of magnetization of the imaginary shell will be the same as that of the real mag- netization of the earth*. The total intensity of magnetizing force in a closed curve passing through and embracing the closed current is constant, and may therefore be made a measure of the quantity of the current. As this intensity is independent of the form of the closed curve and depends only on the quantity of the current which passes through it, we may consider the elementary case of the current which flows through the elementary area dydz. Let the axis of x point towards the west, z towards the south, and y upwards. Let x, y, z be the coordinates of a point in the middle of the area dydz, then the total intensity measured round the four sides of the element is -.-If)''- Total intensity = i-j^ ~~^] ^V ^^• The quantity of electricity conducted through the elementary area dydz is adydz, and therefore if we define the measure of an electric current to be the total intensity of magnetizing force in a closed curve embracing it, we shall have ^' dz dy ' " dx dz ' da^ c^/3i '' dy dx ' * See Experimental Researches (3265) for the relations between the electrical and magnetic cii-cuit, considered as roMtually embracing curves. ON FARADAY S LINES OF FORCE. 195 These equations enable us to deduce tlie distribution of the currents of electricity whenever we know the values of a, /3, y, the magnetic intensities. If a, /8, y be exact differentials of a function of x, y, % with respect to x, y and z respectively, then the values of a^, b^, c^ disappear; and we know that the magnetism is not produced by electric currents in that part of the field which we are investigating. It is due either to the presence of permanent magnetism within the field, or to magnetizing forces due to external causes. We may observe that the above equations give by differentiation da^ db^ dc^ dx dy dz ' which is the equation of continuity for closed currents. Our investigations are therefore for the present limited to closed currents ; and we know little of the magnetic effects of any currents which are not closed. Before entering on the calculation of these electric and magnetic states it may be advantageous to state certain general theorems, the truth of which may be established analytically. Theorem I. The equation d'V d'V d'V ^^ ^Q dx^ dy^ dz'' ^ ' (where V and p are functions of x, y, z never infinite, and vanishing for all points at an infinite distance), can be satisfied by one, and only one, value of V. See Art. (17) above. Theorem II. The value of V which will satisfy the above conditions is found by inte- grating the expression pdxdydz ///; {x — x'~f + y — y'l + z — z'\) where the limits of x, y, z are such as to include every point of space where p is finite. 25—2 196 ON Faraday's lines of force. The proofs of these theorems may be found in any work on attractions or electricity, and in particular in Green's Essay on the Application of Mathematics to Electricity. See Arts. 18, 19 of this paper. See also Gauss, on Attractions, translated in Taylor's Scientific Memoirs. Theorem III. Let U and V be two functions of x, y, z, then fd^v d^v div\ _ _ nudjid^ djjd^ djidv\ ^ W df ^ dz' ) "^"^"^y^^ - ]]]\dx dx^ dy dy^ dz dz j '^^'^^''^ where the integrations are supposed to extend over all the space in which U and V have values differing from 0. — (Green, p. 10.) This theorem shews that if there be two attracting systems the actions between them are equal and opposite. And by making U=V we find that the potential of a system on itself is proportional- to the integral of the square of the resultant attraction through all space ; a result deducible from Art. (30), since the volume of each cell is inversely as the square of the velocity (Arts. 12, 13), and therefore the number of cells in a given space is directly as the square of the velocity. Theorem IV. Let a, /3, y, p be quantities finite through a certain space and vanishing in the space beyond, and let h be given for all parts of space as a continuous or discontinuous function of x, y, z, then the equation in p dx k \ dx] dy k \ dyj dz h \' dz j ^ ' has one, and only one solution, in which p is always finite and vanishes at an infinite distance. The proof of this theorem, by Prof. W. Thomson, may be found in the Cambridge and Dublin Mathematical Journal, Jan. 1848. ON PAEADAYS LINES OP FORCE. 197 If a, P, y be the electro-motive forces, p tlie electric tension, and Tc the coefficient of resistance, then the above equation is identical with the equation of continuity dx dy dz P > and the theorem shews that when the electro-motive forces and the rate of production of electricity at every part of space are given, the value of the electric tension is determinate. Since the mathematical laws of magnetism are identical with those of elec- tricity, as far as we now consider them, we may regard a, ^, y as magnetizing forces, p as magnetic tetision, and p as real magnetic density, k being the coefficient of resistance to magnetic induction. The proof of this theorem rests on the determination of the minimum value where V is got from the equation d'V d'V d'V dyf dy" dz^ P~ > and p has to be determined. The meaning of this integral in electrical language may be thus brought out. If the presence of the media in which h has various values did not affect the distribution of forces, then the "quantity" resolved in x would be simply -7— and the intensity k -i— . But the actual quantity and intensity are - (a — -^) and o^--fi, and the parts due to the distribution of media alone are therefore dp\ dV , dp J dV -T I a — T^ — 5- and a — -r- — k — k \ dx/ dx dx dx Now the product of these . represents the work done on account of this distribution of media, the distribution of sources being determined, and taking in the terms in y and z we get the expression Q for the total work done 198 ON fabaday's lines or force. by that part of tlie whole effect at any point which is due to the distribution of conducting media, and not directly to the presence of the sources. This quantity Q is rendered a minimum by one and only one value of p, namely, that which satisfies the original equation. Theorem V. If a, h, c be three functions of x, y, z satisfying the equation da dh do _ dx dy dz~ ' it is always possible to find three functions a, ^, y which shall satisfy the equa- tions dfi dry _ dz dy ' dy da _ , dx dz ' da cZ/3 _ dy dx Let A = \cdy, where the integration is to be performed upon c considered as a function of y, treating x and z as constants. Let B = \adz, C~\hdx, A' = \hdz, B' = \cdx, C = \ady, integrated in the same way. Then a = ^-^' + ^, dx ^='B-B' + ^, dy' will satisfy the given equations ; for f-'^=lfdz-(^dx-i^dx+li'^dy, dz dy jdy ] dz jdy ^ J dy^' and ^ = /^^^+/|^^+/ dc , . dfi dy (da , , {da , (da , ■■^z-Ty=\drx^^^]dry^y^\i7,dz, = a. ON FARADAY S LINES OF FORCE. 199 In the same way it may be shewn that the values of a, fi, y satisfy the other given equations. The function t/> may be considered at present as perfectly indeterminate. The method here given is taken from Prof. W. Thomson's memoir on Magnetism {Phil. Trans. 1851, p. 283). As we cannot perform the required integrations when a, h, c are discon- tinuous functions of x, y, z, the following method, which is perfectly general though more comphcated, may indicate more clearly the truth of the proposition. Let A, B, C be determined from the equations (PA d^ dM dx'^ df^ dz' +*-0' dW d'B^ d^ dx^ dy" dz' ~ ' d'C d'C d'C dx'"^ df ^ dz''^^~^' by the methods of Theorems I. and II., so that A, B, C are never infinite, and vanish when x, y, or z is infinite. Also let _dB_dC dxj^ dz dy dx ' n_dC dA d\p dx dz dy ' then _ dA dB dxjj ' dy dx dz ' d^_dy_d^ldA dB dC\ _ /(PA d^ dM dz dy dx\dx dy dz/ \dx^ dy^ dz^ ^d^fdA dB dC\ dx\dx dy dz j ' If we find similar equations in y and z, and difierentiate the first by x, the second by y, and the third by z, remembering the equation between a, h, c, we shall have d^ d^ dr^fdA dB dC\_Q. daf dy^ dz^j \dx dy dz ) ' 200 ON FABADAYS LINES OP FORCE. and since A, B, C are always finite and vanish at an infinite distance, the only solution of this equation is dA dB dC^^ dx dy dz ' and we have finally dz dy ' with two similar equations, shewing that a, /8, y have been rightly determined. The function i/> is to be determined from the condition da d^ dY_/d' d^ ^\ i ■ dx dy dz \S^ dy^ dzy ^ ' if the left-hand side of this equation be always zero, rjj must be zero also. Theorem VI, Let rt, 6, c be any three functions of x, y, t, it is possible to find three functions a, /3, y and a fourth V, so that dd d^ dy dx dy dz'^ ' and a = ^_^ + ^ dz dy dx ' -L_dy do- dV dx dz dy ' _da d/3 dV ""-d^'dx^l^- Let da dh dc di^diy^dz^^^'^P' and let V be found from the equation d^V d'V d'V dx^'^^^'^d^ — ^^'f' then ON FARADAY S LINES OP FORCE. 201 dV satisfy the condition dx dy dz ' and therefore we can find three functions A, B, C, and from these a, fi, y, so as to satisfy the given equations. Theorem VII. The integral throughout infinity Q = \l\ (^lOi + Ml + Ciyi) dxdydz, where ap-fi^^, a^^y^ are any functions whatsoever, is capable of transformation into Q=+ \ll{^Trppi - (aof*2 + A&2 + y aT (Jj Pa'^'-P'a"'\' If at the centre of this sphere we place another source of the fluid, then the pressure due to this source must be added to that due to the other two ; and since this additional pressure depends only on the distance from the centre, it will be constant at the surface of the sphere, where the pressure due to the two other sources is zei'o. We have now the means of arranging a system of sources within a given sphere, so that when combined with a given system of sources outside the sphere, they shall produce a given constant pressure at the surface of the sphere. Let a be the radius of the sphere, and p the given pressure, and let the given sources be at distances h^, \, &c. from the centre, and let their rates of prodiiction be iirPi, iirP-^, &c. Then if at distances 5- , j- , &c. (measured in the same direction as 6„ &„, &c. from the centre) we place negative sources whose rates are -47rP, 1^, -47rP, p &c., ON Faraday's lines of force. 211 the pressure at the surface r = a will be reduced to zero. Now placing a source 4ir^ at the centre, the pressure at the surface will be uniform and equal to p. The whole amount of fluid emitted by the surface r = a may be found by adding the rates of production of the sources within it. The result is ^""{f-f-f-*-}' To apply this result to the case of a conducting sphere, let us suppose the external sources iirP^, iirP^ to be small electrified bodies, containing e^, e., of positive electricity. Let us also suppose that the whole charge of the con- ducting sphere \s =E previous to the action of the external points. Then all that is required for the complete solution of the problem is, that the surface of the sphere shall be a surface of equal potential, and that the total charge of the surface shall be E. If by any distribution of imaginary sources within the spherical surface we can effect this, the value of the. corresponding potential outside the sphere is the true and only one. The potential inside the sphere must really be constant and equal to that at the surface. We must therefore find the images of the external electrified points, that is, for every point at distance h from the centre we must find a point on the a^ same radius at a distance j- , and at that point we must place a quantity = — e y- of imaginary electricity. At the centre we must put a quantity E' such that E' = E + e, ^ + e^^-\-kG.; then if R be the distance from the centre, r^, r^, &c. the distances from the electrified points, and r\, t\, &c. the distances from their images at any point outside the sphere, the potential at that point will be E' (I a 1\ , (I a 1\ , _E e, (a K (t\ e, (a K a\,. 27—2 212 ON Faraday's lines op force. This is the value of the potential outside the sphere. At the surface we have R^a and — = -r , — = — r j &c. , so that at the surface ■^ a b, 0. and this must also be the value of j? for any point within the sphere. For the application of the principle of electrical images the reader is referred to Prof Thomson's papers in the Camhridge and Dublin Mathematical Journal. The only case which we shall consider is that in which j^ = I, and bj, is infi- nitely distant along the axis of x, and E = 0. The value p outside the sphere becomes then and inside p = 0. II. On the effect of a paramagnetic or diamagnetic sphere in a uniform field of magnetic force^'. The expression for the potential of a small magnet placed at the origin of co-ordinates in the direction of the axis of x is , d /m\ _ 7 « dx\r) 7^ ' The effect of the sphere in disturbing the lines of force may be supposed as a first hypothesis to be similar to that of a small magnet at the origin, whose strength is to be determined. (We shall find this to be accurately true.) * See Prof. Thomson, on the Theory of Magnetic Induction, Fhil Mag. March, 1851. The induc- tive capaciiy of the sphere, according to that paper, is the ratio of the quantity of magnetic induction (not the intensity) within the sphere to that without It is therefore equal to -^B^ = -^ , accord- I k 2k + k ing to our notation. ON PARADAY'S lines or FOECE. 213 Let the value of the potential undisturbed by the presence of the sphere be Let the sphere produce an additional potential, which for external points is and let the potential within the sphere be Pi — Bx. Let k' be the coefficient of resistance outside, and h inside the sphere, then the conditions to be fulfilled are, that the interior and exterior potentials should coincide at the surface, and that the induction through the surface should be the same whether deduced from the external or the internal potential. Putting x = r cos 6, we have for the external potential P = (/r+^j')cos^, and for the internal Pi = Br cos 6, and these must be identical when r = a, or I+A = B. The induction through the surface in the external medium is and that through the ruterior surface is 1 cfoi 1 „ ^ and .'. 1{I-2A)=^\b. These equations give A — T T) T 2k + k' ' 2k + F ■ The effect outside the sphere is equal to that of a little magnet whose length is I and moment ml, provided 214 ON faeaday's lines of force. Suppose this uniform field to be that due to terrestrial magnetism, then, if h is less than h' as in paramagnetic bodies, the marked end of the equi- valent magnet will be turned to the north. If h is greater than k' as in diamagnetic bodies, the unmarked end of the equivalent magnet would be turned to the north. III. Magnetic field of variable Intensity. Now suppose the intensity in the undisturbed magnetic field to vary in magnitude and direction from one point to another, and that its components in X, y, z are represented by a, ^, y, then, if as a first approximation we re- gard the intensity within the sphere as sensibly equal to that at the centre, the change of potential outside the sphere arising from the presence of the sphere, disturbing the lines of force, will be the same as that due to three smaU magnets at the centre, with their axes parallel to x, y, and z, and their moments equal to h-¥ 2k + ¥ aa, k-k' 2k + k :,a^/3, k-k' 2k + ¥ ay. The actual distribution of potential within and without the sphere may be conceived as the result of a distribution of imaginary magnetic matter on the surface of the sphere ; but since the external effect of this superficial magnetism is exactly the same as that of the three small magnets at the centre, the mechanical effect of external attractions will be the same as if the three magnets really existed. Now let three small magnets whose lengths are Zj, l^, l„ and strengths m^, mj, ms, exist at the point x, y, z with their axes parallel to the axes of X, y, z ; then resolving the forces on the three magnets in the direction of X, we have — X = mi da \ dx 2 da ?j dx 2 ■ + ^2 ' a + -a + da Zj dy 2 da I2 dy 2. + m. a + -a + da Zj dz 2 da Zj dz 2. J da T da -.da ON Faraday's lines of force. 215 Substituting the values of the moments of the imaginary magnets h-V ^J da d^ dy\ h-h' a' d , , ^ ,, The force impelling the sphere in the direction of x is therefore dependent on the variation of the square of the intensity or {a^' + ^ + 'f), as we move along the direction of x, and the same is true for y and z, so that the law is, that the force acting on diamagnetic spheres is from places of greater to places of less intensity of magnetic force, and that in similar distributions of magnetic force it varies as the mass of the sphere and the square of the intensity. It is easy by means of Laplace's CoeflBcients to extend the approximation to the value of the potential as far as we please, and to calculate the attrac- tion. For instance, if a north or south magnetic pole whose strength is M, be placed at a distance h from a diamagnetic sphere, radius a, the repulsion wiU be z? urn 7'.N a'/ 2. 1 , 3.2 a\ 4.3 a' , a When -r is small, the first term gives a sufficient approximation. The repul- sion is then as the square of the strength of the pole, and the mass of the sphere directly and the fifth power of the distance inversely, considering the pole as a point. IV. Two Spheres in uniform field. Let two spheres of radius a be connected together so that their centres are kept at a distance h, and let them be suspended in a uniform magnetic field, then, although each sphere by itself would have been in equilibrium at any part of the field, the disturbance of the field will produce forces tending to make the balls set in a particular direction. Let the centre of one of the spheres be taken as origin, then the undis- turbed potential is p = Ir cos 6, and the potential due to the sphere is 216 ON PABADAYS LINES OF FOKCE, The whole potential is therefore equal to -f ( ^ + ?ry-v-r> -J cos ^ =_p, cos^, 2A; + Fr rdd~ ' h — h'a?\.^ dp - I = dp dr + 1 dp ?de + 1 dp r" sin^ B d(f) k-k' a' k-k' '^2M'¥('-'''''^^'-2M' a" (1 + 3 cos' ,} This is the value of the square of the intensity at any point. The moment of the couple tending to turn the combination of balls in the direction of the original force L- d .„ / k — k' dd \2k + k' 7a.M when r = b, ^-^^ 2k + ¥ cf/^ k-k' a\ . „„ This expression, which must be positive, since h is greater than a, gives the moment of a force tending to turn the line joining the centres of the spheres towards the original lines of force. Whether the spheres are magnetic or diamagnetic they tend to set in the axial direction, and that without distinction of north and south. If, however, one sphere be mag-netic and the other diamagnetic, the line of centres will set equatoreally. The magnitude of the force depends on the square of (k — k'), and is therefore quite insensible except in iron"". V. Two Spheres between the poles of a Magnet. Let us next take the case of the same balls placed not in a uniform field but between a north and a south pole, ±M, distant 2c from each other in the direction of x. See Prof. Thomson in FMl Mag. March, 1851. ON FARADAY S LINES OP FORCE, 217 The expression for the potential, the middle of the line joining the poles being the origin, is Wc' + r' - 2cr cos d J& + f' + 2cr cos 6/ From this we find as the value of /", & \ c- c' I and the moment to turn a pair of spheres (radius a, distance 2&) in the direction in which Q is increased is This force, which tends to turn the line of centres equatoreally for diamagnetic and axiaUy for magnetic spheres, varies directly as the square of the strength of the magnet, the cube of the radius of the spheres and the square of the dis- tance of their centres, and inversely as the sixth power of the distance of the poles of the magnet, considered as points. As long as these poles are near each other this action of the poles will be much stronger than the mutual action of the spheres, so that as a general rule we may say that elongated bodies set axiaUy or equatoreally between the poles of a magnet according as they are mag- netic or diamagnetic. If, instead of being placed between two poles very near to each other, they had been placed in a uniform field such as that of terrestrial magnetism or that produced by a spherical electro-magnet (see Ex. viii.), an elongated body would set axiaUy whether magnetic or diamagnetic. In aU these cases the phenomena depend on Ic — h', so that the sphere con- ducts itself magnetically or diamagnetically according as it is more or less magnetic, or less or more diamagnetic than the medium in which it is placed. VI. On the Magnetic Phenomena of a Sphere cut from a substance whose coefficient of resistance is different in different directions. Let the axes of magnetic resistance be parallel throughout the sphere, and let them be taken for the axes of x, y, z. Let \, k, k, be the coefficients of resistance in these three directions, and let k" be that of the external medium, VOL. L 2^ 218 ON taeaday's lines of force. and a the radius of the sphere. Let I be the undisturbed magnetic intensity of the field into which the sphere is introduced, and let its direction-cosines be I, m, n. Let us now take the case of a homogeneous sphere whose coefficient is \ placed in a uniform magnetic field whose intensity is II in the direction of x. The resultant potential outside the sphere would be and for internal points So that in the interior of the sphere the magnetization is entirely in the direc- tion of X. It is therefore quite independent of the coefficients of resistance in the directions of x and y, which may be changed from \ into \ and h^ with- out disturbing this distribution of magnetism. We may therefore treat the sphere as homogeneous for each of the three components of /, but we must use a different coefficient for each. We find for external points and for internal points The external effect is the same as that which would have been produced if the small magnet whose moments are had been placed at the origin with their directions coinciding with the axes of x, y, z. The effect of the original force I in turning the sphere about the axis of x may be found by taking the moments of the components of that force on these equivalent magnets. The moment of the force in the direction of y acting on the third magnet is fCj K 7-21 and that of the force in z on the second magnet is "'2 "' 7-2 1 ON FARADAY S LINES OF FORCE. 219 The whole couple about the axis of x is therefore tending to turn the sphere round from the axis of y towards that of z. Sup- pose the sphere to be suspended so that the axis of x is vertical, and let / be horizontal, then if 6 be the angle which the axis of y makes with the direction of /, m = cos 6, n— — sin d, and the expression for the moment becomes tending to increase 6. The axis of least resistance therefore sets axiaUy, but with either end indifferently towards the north. Since in all bodies, except iron, the values of h are nearly the same as in a vacuum, the coefficient of this quantity can be but little altered by changing the value of M to h, the value in space. The expression then becomes i^^'iVsin2^, independent of the external medium ''^ VII. Permanent magnetism in a spherical shell. The case of a homogeneous shell of a diamagnetic or paramagnetic substance presents no difficulty. The intensity within the shell is less than what it would have been if the shell were away, whether the substance of the shell be dia- magnetic or paramagnetic. When the resistance of the shell is infinite, and when it vanishes, the intensity within the shell is zero. In the case of no resistance the entire effect of the shell on any point, internal or external, may be represented by supposing a superficial stratum of * Taking the more general case of magnetic induction referred to in Art. (28), we find, in the expression for the moment of the magnetic forces, a constant term depending on T, besides those terms which depend on sines and cosines of 6. The result is, that in every complete revolution in the negative direction round .the axis of T, a certain positive amount of work is gained ; but, since no inexhaustible source of work can exist in nature, we must admit that ^=0 in all substances, with respect to magnetic induction. This argument does not hold in the case of electric conduction, or in the case of a body through which heat or electricity is passing, for such states are main- tained by the continual expenditure of work. See Prof. Thomson, Phil. Mag. March, 1851, p. 186. 28—2 220 ON Faraday's lines op torce. magnetic matter spread over the outer surface, the density being given by the equation p = 3l cos 6. Suppose the shell now to be converted into a permanent magnet, so that the distribution of imaginary magnetic matter is invariable, then the external poten- tial due to the shell will be /=--f^.C03^, and the internal potential p^= —Ir gob 6. Now let us investigate the effect of filling up the shell with some substance of which the resistance is l, the resistance in the external medium being h\ The thickness of the magneti^^ed shell may be neglected. Let the magnetic moment of the permanent magnetism be la?, and that of the imaginary super- ficial distribution due to the medium Ic^^Aa?. Then the potentials are external p ={I+A) -, cos 9, internal p^ = ( J+ A) r cos d. The distribution of real magnetism is the same before and after the introduc- tion of the medium h, so that The external effect of the magnetized shell is increased or diminished according as k is greater or less than k\ It is therefore increased by filling up the shell with diamagnetic matter, and diminished by filling it with paramagnetic matter, such as iron. VIII. Electro-magnetic spherical shell. Let us take as an example of the magnetic effects of electric currents, an electro-magnet in the form of a thin spherical shell. Let its radius be a, and its thickness t, and let its external effect be that of a magnet whose moment is /a'. Both within and without the shell the magnetic effect may be represented by a potential, but within the substance of the shell, where there ON FARADAY S LINES OF FORCE. 221 are electric currents, tlie magnetic effects cannot be represented by a potential. Let 'p\ Pi be the external and internal potentials, a" p' = 1-^008$, p^ = Ar cos 6, and since there is no permanent magnetism, -^ = -p , when r = a, A=-2L If we draw any closed curve cutting the shell at the equator, and at some other point for which is known, then the total magnetic intensity roimd this curve will be Sla cos 9, and as this is a measure of the total electric current which flows through it, the quantity of the current at any point may be found by differentiation. The quantity which flows through the element td0 is — 3la sin 6dd, so that the quantity of the current referred to unit of area of section is -3/%in^. If the shell be composed of a wire coiled roimd the sphere so that the number of coUs to the inch varies as the sine of 6, then the external effect will be nearly the same as if the shell had been made of a uniform conducting sub- stance, and the currents had been distributed according to the law we have just given. If a wire conducting a current of strength I^ be wound round a sphere of radius a so that the distance between successive coils measured along the axis of ic is — , then there will be n coils altogether, and the value of /, for n the resulting electro-magnet will be I = -I The potentials, external and internal, wiU be The interior of the shell is therefore a uniform magnetic field. 222 ON Faraday's lines of force. IX. Effect of the core of the electro-magnet. Now let us suppose a sphere of diamagnetic or paramagnetic matter intro- duced into the electro-magnetic coil. The result may be obtained as in the last case, and the potentials become The external effect is greater or less than before, according as k' is greater or less than k, that is, according as the interior of the sphere is magnetic or diamagnetic with respect to the external medium, and the internal effect is altered in the opposite direction, being greatest for a diamagnetic medium. This investigation explains the effect of introducing an iron core into an electro-magnet. If the value of k for the core were to vanish altogether, the effect of the electro-magnet would be three times that which it has without the core. As k has always a finite value, the effect of the core is less than this. In the interior of the electro-magnet we have a uniform field of magnetic force, the intensity of which may be increased by surrounding the coil with a shell of iron. If k' — 0, and the shell infinitely thick, the effect on internal points would be tripled. The effect of the core is greater in the case of a cylindric magnet, and greatest of aU when the core is a ring of soft iron. X. Electro-tonic functions in spherical electro-magnet. Let us now find the electro-tonic functions due to this electro-magnet. They will be of the form ao = 0, I3, = 0)Z, y„= -coy, where w is some function of r. Where there are no electric currents, we must have ttj, 62, Cj each = 0, and this implies the solution of which is d /„ da)\ -=o.+^\ ON Faraday's lines of force, 223 Within the shell &> cannot become infinite ; therefore A> 7i symbols of magnetic quantity and intensity; a^, h^, c^, a,, /Sj, y^ of electric quantity and intensity. Let p^ be the electric tension at any point, _dp, y (1), y,- ^^ -t-A;c, dx dy dz ^ >' da^^d^^d^, dx dy dz ^ ON Faraday's lines of fobce. 227 The expressions for a^, ^„, y^ due to the magnetism of the field are a, = A, + - ycoaO, P^ = Bf, + -{zB\n.9 — x cos 6), A^, B^, Co being constants; and the velocities of the particles of the revolving sphere are dx dy dz -dr-'^y' dT''''' di=^- We have therefore for the electro-motive forces 1 da, 1 I „ a~,= —-, rr = ~ r~ t: cos Uox, ^ 4:TT dt 477 2 1 dy, 1 I . ^ Returning to equations (1), we get , /db^ _ ^\ _ ^A _ ^_ Q \dz dy) dz dy ' , /dc^ da\ _ dy^ da^ _ 1 I . ^ \dx dzj dx dz 4:Tr2 7ida^_ dhA _da^ _ d§2 _ q ^dy dx ) dy dx From which with equation (2) we find 117.. a, = — 7" V- T sm Ocaz, 11/.. ' ^ 47r 4 Pj = - — - Iq) {{x' + y^) cos 6-xz sin 6}. 29—2 228 ON Faraday's lines of force. These expressions would determine completely the motion of electricity in a revolving sphere if we neglect the action of these currents on themselves. They express a system of circular currents about the axis of y, the quantity of current at any point being proportional to the distance from that axis. The external magnetic effect will be that of a small magnet whose moment . TIP IS 7^7 co/ sin 0, with its direction along the axis of y, so that the magnetism of the field woidd tend to turn it back to the axis of x*. The existence of these currents will of course alter the distribution of the electro-tonic functions, and so they will react on themselves. Let the final result of this action be a system of currents about an axis in the plane of xy inclined to the axis of x at an angle ^ and producing an external effect equal to that of a magnet whose moment is FliJ'. The magnetic inductive components within the shell are /i sin 6 — 2T cos ^ in a;, — 2/' sin ) in y produces 2 , a (2 J' sin <^) in x ; /i COS ^ in z produces no currents. We must therefore have the following equations, since the state of the shell is the same at every instant, T Ji sin d — 2/' cos ^ = Zi sin ^ -I- , 0)2/' sin ^ The moment of the couple due to terrestrial magnetism tending to stop the rotation is Icr cos^ B, 2 2477^1^+ rv and the loss of work due to this in unit of time is I^Ttk Toy" j^j.^ _ i?P cos'' 6. 2 24i7^?+J'V This loss of work is made up by an evolution of heat in the substance of the shell, as is proved by a recent experiment of M. Foucault (see Comptes Rendus, xli. p. 450). [From the Transactions of the Royal Scottish Society of Arts, Vol iv. Part iv.] IX. Description of a New Form of the Platometer, an Instrument for measuring the Areas of Plane Figures drawn on Paper*. 1. The measurement of the area of a plane figure on a map or plan is an operation so frequently occurring in practice, that any method by which it may be easily and quickly performed is deserving of attention. A very able expo- sition of the principle of such instruments will be found in the article on Planimeters in the Reports of the Juries of the Great Exhibition, 1851. 2. In considering the principle of instruments of this kind, it will be most convenient to suppose the area of the figure measured by an imaginary straight line, which, by moving parallel to itself, and at the same time altering in length to suit the form of the area, accurately sweeps it out. Let AZ be a fixed vertical line, APQZ the boundary of the area, and let a variable horizontal line move parallel to itself from A to Z, so as to have its extremi- ties, P, M, in the curve and in the fixed straight line. Now, suppose the horizontal line (which we shall call the generating line) to move from the position PM to QN, MN being some small quantity, say one inch for distinct- ness. During this movement, the generating line will have swept out the narrow strip of the surface, PMNQ, which exceeds the portion PMNp by the small triangle PQp, But since MN, the breadth of the strip, is one inch, the strip will contain as many square inches as PM is iQches long; so that, when the generating Read to the Society, 22nd Jan. 1855. ON A NEW FORM OP THE PLATOMETEB. 231 line descends one inch, it sweeps out a number of square inches equal to the number of linear inches in its length. Therefore, if we have a machine with an index of any kind, which, while the generating line moves one inch downwards, moves forward as many degrees as the generating line is inches long, and if the generating line be alternately moved an inch and altered in length, the index will mark the number of square inches swept over during the whole operation, By the ordinary method of limits, it may be shown that, if these changes be made continuous instead of sudden, the index will still measure the area of the cm-ve traced by the extremity of the generating line. 3. When the area is bounded by a closed curve, as ABDC, then to determine the area we must carry the tra- cing point from some point A of the curve, completely round the circumference to A again. Then, while the tracing point moves from A to C, the index will go forward and mea- sure the number of square inches in ACRP, and, while it moves from C to D, the index will measure backwards the square inches in CRPD, so that it will now indicate the square inches in ACD. Similarly, during the other part of the motion from D to B, and from B to D, the part DBA will be measured; so that when the tracing point returns to D, the instrument will have measured the area ACDB. It is evident that the whole area wUl appear positive or negative according as the tracing point is carried round in the direction ACDB or ABDC. 4. We have next to consider the various methods of communicating the required motion to the index. The first is by means of two discs, the first having a flat horizontal rough surface, turning on a vertical axis, OQ, and the second vertical, with its circumference rest- ing on the flat surface of the first at P, so as to be driven round by the motion of the first disc. The velocity of the second disc will depend on OP, the distance of the point of contact from the centre of the first disc ; so that if OP be made always equal to the generating hne, the conditions of the instrument will be fulfiUed. This is accomplished by causing the index-disc to slip along the radius of 232 ON A NEW rOEM OF THE PLATOMETER. the horizontal disc; so that in working the instrument, the motion of the index- disc is compounded of a rolling motion due to the rotation^ of the first disc, and a slipping motion due to the variation of the generating line. 5. In the instrument presented by Mr Sang to the Society, the first disc is replaced by a cone, and the action of the instrument corresponds to a mathe- matical valuation of the area by the use of oblique co-ordinates. As he has himself explained it very completely, it will be enough here to say, that the index-wheel has still a motion of slipping as well as of rolling. 6. Now, suppose a wheel rolling on a surface, and pressing on it with a weight of a pound; then suppose the coefficient of friction to be |, it will require a force of 2 oz. at least to produce slipping at all, so that even if the resistance of the axis, &c., amounted to 1 oz., the rolling would be perfect. But if the wheel were forcibly pulled sideways, so as to slide along in the direction of the axis, then, if the friction of the axis, &c., opposed no resistance to the turning of the wheel, the rotation would still be that due to the forward motion ; but if there were any resistance, however small, it would produce its effect in diminishing the amount of rotation. The case is that of a mass resting on a rough surface, which requires a great force to produce the slightest motion ; but when some other force acts on it and keeps it in motion, the very smallest force is sufficient to alter that motion in direction. 7. This effect of the combination of slipping and rolling has not escaped the observation of Mr Sang, who has both measured its amount, and shown how to eliminate its effect. In the improved instrument as constructed by him, I beHeve that the greatest error introduced in this way does not equal the ordi- nary errors of measurement by the old process of triangulation. This accuracy, however, is a proof of the excellence of the workmanship, and the smoothness of the action of the instrument ; for if any considerable resistance had to be overcome, it would display itself in the results. 8. Having seen and admired these instruments at the Great Exhibition in 1851, and being convinced that the combination of slipping and rolling was a drawback on the perfection of the instrument, I began to search for some arrangement by which the motion should be that of perfect rolling in every ON A NEW FOEM OP THE PLATOMETER. 233 motion of whicli the instrument is capable. The forms of the rolling parts which I considered were — 1. Two equal spheres. 2. Two spheres, the diameters being as 1 to 2. 3. A cone and cylinder, axes at right angles. Of these, the first combination only suited my purpose. I devised several modes of mounting the spheres so as to make the principle available. That which I adopted is borrowed, as to many details, from the instruments already con- structed, so that the originality of the device may be reduced to this principle — The abolition of slipping by the use of two equal spheres. 9. The instrument (Fig. 1) is mounted on a frame, which rolls on the two connected wheels, MM, and is thus constrained to travel up and down the paper, moving parallel to itself CH is a horizontal axis, passing through two supports attached to the frame, and carrying the wheel K and the hemisphere LAP. The wheel K rolls on the plane on which the instrument travels, and communicates its motion to the hemisphere, which therefore revolves about the axis AH with a velocity proportional to that with which the instrument moves backwards or forwards. FCO is a framework (better seen in the other figures) capable of revolving about a vertical axis, Cc, being joined at C and c to the frame of the instru- ment. The parts CF and CO are at right angles to each other and horizontal. The part CO carries with it a ring, SOS, which turns about a vertical axis Oo. This rmg supports the index-sphere Bh by the extremities of its axis Ss, just as the meridian circle carries a terrestrial globe. By this arrangement, it will be seen that the axis of the sphere is kept always horizontal, while its centre moves so as to be always at a constant distance fi:om that of the hemisphere. This distance must be adjusted so that the spheres may always remain in con- tact, and the pressure at the point of contact may be regulated by means of springs or compresses at and o acting in the direction OC, oc. In this way the rotation of the hemisphere is made to drive the index-sphere. 10. Now, let us consider the working of the instrument. Suppose the arm CE placed so as to coincide with CD, then 0, the centre of the index-sphere will be in the prolongation of the axis HA. Suppose also that, when in this position, the equator bB of the index-sphere is in contact with the pole A of the hemisphere. Now, let the arch be turned into the position. CE as in the VOL. I. 30 334 ON A NEW FOEM OF THE PLATOMETER. figure, then the rest of the framework will be turned through an equal angle, a,nd the index-sphere will roll on the hemisphere till it come into the position represented in the figure. Then, if there be no slipping, the arc AP = BP, and the angle ACP = BOP. Next, let the instrument be moved backwards or forwards, so as to turn the wheel Kk and the hemisphere LI, then the index-sphere will be turned about its axis Ss by the action of the hemisphere, but the ratio of their veloci- ties will depend on their relative positions. If we draw PQ, PR, perpendiculars from the point of contact on the two axes, then the angular motion of the index-sphere will be to that of the hemisphere, as PQ is to PR ; that is, as PQ is to QC, by the equal triangles POQ, PQC ; that is, as ED is to DC, by the similar triangles CQP, CDE. Therefore the ratio of the angular velocities is as ED to DC, but since DC is constant, this ratio varies as ED. We have now only to contrive some way of making ED act as the generating line, and the machine is complete (see art. 2). 11. The arm CF is moved in the following manner: — Tt is a rectangular metal beam, fixed to the frame of the instrument, and parallel to the axis AH. eEe is a little carriage which rolls along it, having two rollers on one side and one on the other, which is pressed against the beam by a spring. This carriage carries a vertical pin, E, turning in its socket, and having a collar above, through which the arm CF works smoothly. The tracing point G is attached to the carriage by a jointed frame eGe, which is so arranged that the point may not bear too heavily on the paper. 12. When the machine is in action, the tracing point is placed on a point in the boundary of the figure, and made to move round it always in one direction till it arrives at the same point again. The up-and-down motion of the tracing point moves the whole instrument over the paper, turns the wheel K, the hemisphere LI, and the index-sphere Bb ; while the lateral motion of the tracing point moves the carriage E on the beam Tt, and so works the arm CF and the framework CO ; and so changes the relative velocities of the two spheres, as has been explained, 13. In this way the instrument works by a perfect rolling motion, in what- ever direction the tracing point is moved; but since the accuracy of the' result depends on the equality of the arcs AP and BP, and since the smallest error ON A NEW POEM OF THE PLATOMETEK. 235 of adjustment would, in the course of time, produce a considerable deviation from this equality, some contrivance is necessary to secure it. For this purpose a wheel is fixed on the same axis with the ring SOs, and another of the same size is fixed to the frame of the instrument, with its centre coinciding with the vertical axis through C. These wheels are connected by two pieces of watch- spring, which are arranged so as to apply closely to the edges of the wheels. The first is firmly attached to the nearer side of the fixed wheel, and to the farther side of the moveable wheel, and the second to the farther side of the fixed wheel, and the nearer side of the moveable wheel, crossing beneath the first steel band. In this way the spheres are maintained in their proper relative position ; but since no instrument can be perfect, the wheels, by preventing derangement, must cause some slight slipping, depending on the errors of work- manship. This, however, does not ruin the pretensions of the instiTiment, for it may be shown that the error introduced by slipping depends on the distance through which the lateral slipping takes place ; and since in this case it must be very small compared with its necessarily large amount in the other instru- ments, the error introduced by it must be diminished in the same proportion. 14. I have shewn how the rotation of the index-sphere is proportional to the area of the figure traced by the tracing point. This rotation must be measured by means of a graduated circle attached to the sphere, and read olf by means of a vernier. The result, as measured in degrees, may be interpreted in the following manner: — Suppose the instrument to be placed with the arm CF coinciding with CD, the equator Bb of the index-sphere touching the pole A of the hemisphere, and the index of the vernier at zero : then let these four operations be performed : — (1) Let the tracing point be moved to the right till DE = DC, and there- fore DCE, AGP, and FOB = 4.5". (2) Let the instrument be rolled upwards till the wheel K has made a complete revolution, carrying the hemisphere with it; then, on account of the equality of the angles SOF, FCA, the index-sphere will also make a complete revolution. (3) Let the arm CF be brought back again till F coincides with D. (4) Let the instrument be rolled back again through a complete revolution of the wheel K. The index-sphere will not rotate, because the point of contact is at the pole of the hemisphere. 30—2 236 ON A NEW FORM OF THE PLATOMETER. The tracing point has now traversed the boundary of a rectangle, whose length is the circumference of the wheel K, and its breadth is equal to CD; and during this operation, the index-sphere has made a complete revolution. 360° on the sphere, therefore, correspond to an area equal to the rectangle con- tained by the circumference of the wheel and the distance CD. The size of the wheel K being known, different values may be given to CD, so as to make the instrument measure according to any required scale. This may be' done, either by shifting the position of the beam Tt, or by having several sockets in the carriage E for the pin which directs the arm to work in. 15. If I have been too prolix in describing the action of an instrument which has never been constructed, it is because I have myself derived great satisfaction from following out the mechanical consequences of the mathematical theorem on which the truth of this method depends. Among the other forms of apparatus by which the action of the two spheres may be rendered available, is one which might be found practicable in cases to which that here given would not apply. In this instrument (Fig. 4) the areas are swept out by a radius-vector of variable length, turning round a fixed point in the plane. The area is thus swept out with a velocity varying as the angular velocity of the radius-vector and the square of its length conjointly, and the construction of the machine is adapted to the case as follows : — The hemisphere is fixed on the top of a vertical pillar, about which the rest of the instrument turns. The index-sphere is supported as before by a ring and framework. This framework turns about the vertical pillar along with the tra- cing point, but has also a motion in a vertical plane, which is communicated to it by a curved slide connected with the tracing point, and which, by means of a prolonged arm, moves the framework as the tracing point is moved to and from the pillar. The form of the curved slide is such, that the tangent of the angle of inclination of the line joining the centres of the spheres with the vertical is proportional to the square of the distance of the tracing point from the vertical axis of the instrument. The curve which fulfils this condition is an hyperbola, one of whose asymptotes is vertical, and passes through the tracing point, and the other horizontal through the centre of the hemisphere. The other parts of this instrument are identical with those belonging to that already described. VOL. I. PLATE IL '^"fm page 236 Cambridge University Press ON A NEW FORM OF THE PLATOMETEE. 237 When the tracing point is made to traverse the boundary of a plane figure, there is a continued rotation of the radius-vector combined with a change of length. The rotation causes the index-sphere to roll on the fixed hemisphere, while the length of the radius-vector determines the rate of its motion about its axis, so that its whole motion measures the area swept out by the radius-vector during the motion of the tracing point. The areas measured by this instrument may either lie on one side of the pillar, or they may extend all round it. In either case the action of the instrument is the same as in the ordinary case. In this form of the instrument we have the advantages of a fixed stand, and a simple motion of the tracing point ; but there seem to be difficulties in the way of supporting the spheres and arranging the slide ; and even then the instrument would require a tall pillar, in order to take in a large area. 16. It will be observed that I have said little or nothing about the prac- tical details of these instruments. Many useful hints will be found in the large work on Platometers, by Professor T. Gonnellu, who has given us an account of the difficulties, as well as the results, of the construction of his most elaborate instrument. He has also given some very interesting investigations into the errors produced by various irregularities of construction, although, as far as I am aware, he has not even suspected the error which the sliding of the index-wheel over the disc must necessarily introduce. With respect to this, and other points relating to the working of the instrument, the memou- of Mr Sang, in the Transactions of this Society, is the most complete that I have met with. It may, however, be as well to state, that at the time when I devised the improvements here suggested, I had not seen that paper, though I had seen the instrument standing at rest in the Crystal Palace. Edinbuegh, 30th January, 1855. Note. — Since the design of the ahove instrument was submitted to the Society of Arts, I have met with a description of an instrument combining simplicity of construction with the power of adaptation to designs of any size, and at the same time more portable than any other instrument of the kind. Although it does not act by perfect rolling, and there- fore belongs to a different class of instruments from that described in this paper, I think that its simplicity, and the beauty of the principle on which it acts, render it worth the attention of engineers and mechanists, whether practical or theoretical. A full account of this instrument is to be found in Moigno's " Cosmos," 5th year, Vol. viii.. Part Vlii., p. 213, published 20th February 1856. Description et TMorie du planimUre polaire, invents par J. Amsler, de Schaffouse en Suisse. Cambridge, 30th April, 1856. [From the Cambridge Philosophical Society Proceedings, Vol. i. pp. 173 — 175.] X. On the Elementary Tlieory of Optical Instruments. The object of this communication was to shew how the magnitude and position of the image of any object seen through an optical instrument could be ascertained without knowing the construction of the instrument, by means of data derived from two experiments on the instrument. Optical questions are generally treated of with respect to the pencils of rays which pass through the instrument. A pencil is a collection of rays which have passed through one point, and may again do so, by some optical contrivance. Now if we suppose all the points of a plane luminous, each will give out a pencil of rays, and that collection of pencils which passes through the instrument may be treated as a heam of light. In a pencil only one ray passes through any point of space, unless that point be the focus. In a beam an infinite number of rays, corresponding each to some point in the luminous plane, passes through any point ; and we may, if we choose, treat this collection of rays as a pencil proceeding from that point. Hence the same beam of light may be decomposed into pencils in an infinite variety of ways; and yet, since we regard it as the same collection of rays, we may study its properties as a beam independently of the particular way in which we conceive it analysed into pencils. Now in any instrument the incident and emergent beams are composed of the same light, and therefore every ray in the incident beam has a corresponding ray in the emergent beam. We do not know their path within the instrument, but before incidence and after emergence they are straight lines, and therefore any two points serve to determine the direction of each. Let us suppose the instrument such that it forms an accurate image of a plane object in a given position. Then every ray which passes through a given ON THE ELEMENTARY THEORY OF OPTICAL INSTRUMENTS. 239 point of the object before incidence passes through, the corresponding point of the image after emergence, and this determines one point of the emergent ray. If at any other distance from the instrument a plane object has an accurate image, then there will be two other corresponding points given in the incident and emergent rays. Hence if we know the points in which an incident ray meets the planes of the two objects, we may find the incident ray by joining the points of the two images corresponding to them. It was then shewn, that if the image of a plane object be distinct, flat, and similar to the object for two different distances of the object, the image of any other plane object perpendicular to the axis wiU be distinct, flat and similar to the object. When the object is at an infinite distance, the plane of its image is the principal focal plane, and the point where it cuts the axis is the principal focus. The line joining any point in the object to the corresponding point of the image cuts the axis at a fixed point called the focal centre. The distance of the principal focus from the focal centre is called the principal focal length, or simply the /ocaZ length. There are two principal foci, etc., formed by incident parallel rays passing in opposite directions through the instrument. If we suppose light always to pass in the same direction through the instrument, then the focus of incident rays when the emergent rays are parallel is the first principal focus, and the focus of emergent rays when the incident rays are parallel is the second principal focus. Corresponding to these we have first and second focal centres and focal lengths. Now let Q^ be the focus of incident rays, P^ the foot of the perpendicular from Q^ on the axis, Q^ the focus of emergent rays, P^ the foot of the corre- sponding perpendicular, F^F^ the first and second principal foci, A^A„ the first and second focal centres, then P,F\_P^,_F^ A,Fr p.Q-r FA.' lines being positive when measured in the direction of the light. Therefore the position and magnitude of the image of any object is found by a simple proportion. 240 ON THE ELEMENTARY THEORY OF OPTICAL INSTRUMENTS. In one important class of instruments there are no principal foci or focal centres. A telescope in which parallel rays emerge parallel is an instance. In such instruments, if m be the angular magnifying power, the linear dimensions of the image are — of the object, and the distance of the image of the object from the image of the object-glass is —^ of the distance of the object from Til/ the object-glass. Rules were then laid down for the composition of instruments, and suggestions for the adaptation of this method to second approximations, and the method itself was considered with reference to the labours of Cotes, Smith, Euler, Lagrange, and Gauss on the same subject. [From the Report of the British Association, 1856.] XI. On a Method of Drawing the Theoretical Forms of Faraday's Lines of Force without Calculation. The method applies more particularly to those cases in which, the lines are entirely parallel to one plane, such as the Hues of electric currents in a thin plate, or those round a system of parallel electric currents. In such cases, if we know the forms of the lines of forte in any two cases, we may combine them by simple addition of the functions on which the equations of the lines depend. Thus the system of lines in a uniform magnetic field is a series of parallel straight lines at equal intervals, and that for an infinite straight electric current perpendicular to the paper is a series of concentric circles whose radii are in geometric progression. Having drawn these two sets of fines on two separate sheets of paper, and laid a third piece above, draw a third set of lines through the intersections of the first and second sets. This will be the system of lines in a uniform field disturbed by an electric current. The most interesting cases are those of uniform fields disturbed by a small magnet. If we draw a circle of any diameter with the magnet for centre, and join those points in which the circle cuts the lines of force, the straight lines so drawn wUl be parallel and equidistant; and it is easily shown that they represent the actual lines of force in a paramagnetic, diamagnetic, or crystallized body, according to the nature of the original lines, the size of the circle, &c. No one can study Faraday's researches without wishing to see the forms of the lines of force. This method, therefore, by which they may be easily drawn, is recommended to the notice of electrical students. VOL. I. 31 [From the Report of the British Association, 1856.] XII. On the Unequal Sensibility of the Foramen Centrale to Light of different Colours. When observing the spectrum formed by looking at a long vertical slit through a simple prism, I noticed an elongated dark spot running up and down in the blue, and following the motion of the eye as it moved up and down the spectrum, but refusing to pass out of the blue into the other colours. It was plain that the spot belonged both to the eye and to the blue part of the spectrum. The result to which I have come is, that the appearance is due to the yeUow spot on the retina, commonly called the Foramen Centrale of Soem- mering. The most convenient method of observing the spot is by presenting to the eye in not too rapid succession, blue and yellow glasses, or, still better, allowing blue and yellow papers to revolve slowly before the eye. In this way the spot is seen in the blue. It fades rapidly, but is renewed every time the yellow comes in to relieve the effect of the blue. By using a Nicol's prism along with this apparatus, the brushes of Haidinger are well seen in connexion with the spot, and the fact of the brushes being the spot analysed by polarized light becomes evident. If we look steadily at an object behind a series of bright bars which move in front of it, we shall see a curious bending of the bars as they come up to the place of the yellow spot. The part which comes over the spot seems to start in advance of the rest of the bar, and this would seem to indicate a greater rapidity of sensation at the yeUow spot than in the surround- ing retina. But I find the experiment difficult, and I hope for better results from more accurate observers. [From the Report of the British Association, 1856.] XIII. On the Theory of Compound Colours with reference to Mixtures of Blue and Yellow Light. When we mix together blue and yellow paint, we obtain green paint. This fact is well known to all who have handled colours ; and it is universally admitted that blue and yellow make green. Eed, yellow, and blue, being the primary colours among painters, green is regarded as a secondary colour, arising from the mixture of blue and yellow. Newton, however, found that the green of the spectrum was not the same thing as the mixture of two colours of the spectrum, for such a mixture could be separated by the prism, while the green of the spectrum resisted further decomposition. But still it was believed that yellow and blue would make a green, though not that of the spectrum. As far as I am aware, the first experiment on the subject is that of M. Plateau, who, before 1819, made a disc with alternate sectors of prussian blue and gam- boge, and observed that, when spinning, the resultant tint was not green, but a neutral gray, inclining sometimes to yellow or blue, but never to green. Prof J. D. Forbes of Edinburgh made similar experiments in 1849, with the same result. Prof Helmholtz of Konigsberg, to' whom we owe the most complete investigation on visible colour, has given the true explanation of this phsenomenon. The result of mixing two coloured powders is not by any means the same as mixing the beams of light which flow from each separately. In the latter case we receive all the light which comes either from the one powder or the other. In the former, much of the light coming from one powder falls on particles of the other, and we receive only that portion which has escaped absorption by one or other. Thus the light coming from a mixture of blue and yellow powder, consists partly of light coming directly from blue particles or yellow particles, and partly of light acted on by both blue and yellow particles. This latter light is green, since the blue stops the red, yellow, and orange, and the yellow stops 31—2 244 ON THE THEORY OF COMPOUND COLOURS. the blue and violet. I have made experiments on the mixture of blue and yellow light — by rapid rotation, by combined reflexion and transmission, by view- ing them out of focus, in stripes, at a great distance, by throwing the colours of the spectrum on a screen, and by receiving them into the eye, directly ; and I have arranged a portable apparatus by which any one may see the result of this or any other mixture of the colours of the spectrum. In all these cases blue and yellow do not make green. I have also made experiments on the mixture of coloured powders. Those which I used principally were "mineral blue" (from copper) and "chrome-yellow." Other blue and yellow pigments gave curious results, but it was more difficult to make the mixtures, and the greens were less uniform in tint. The mixtures of these colours were made by weight, and were painted on discs of paper, which were afterwards treated in the manner described in my paper " On Colour as perceived by the Eye," in the Transactions of the Royal Society of Edinburgh, Vol. xxi. Part 2. The visible effect of the colour is estimatad in terms of the standard-coloured papers : — ^vermilion (Y), ultramarine (U), and emerald-green (E). The accuracy of the results, and their significance, can be best understood by referring to the paper before mentioned. I shall denote mineral blue by B, and chrome-yellow by Y ; and B3 Yj means a mixture of three parts blue and five parts yellow. Given 0( jlour. Y, Y3 Y, Y. Y, Y, Y, 100 100 100 100 ' 100 100 100 100 100 Standard Cc V. u. = 2 36 = 1 18 = 4 11 = 95 = 15 1 = 22-2 = 35-10 = 64-19 = 180 -27 )lours. E. 7 .. 17 .. Coefficient of brightness 45 B, 37 Be 34 .. 40 .. 40 .. 49 B, 54 B, 56 B3 B, 44 .. 51 .. 64 76 B. 64 .. 124 .. 109 277 The columns V, U, E give the proportions of the standard colours which are equivalent to 100 of the given colour; and the sum of V, U, E gives a co- efficient, which gives a general idea of the brightness. It will be seen that the first admixture of yellow diminishes the brightness of the blue. The negative values of U indicate that a mixture of V, U, and E cannot be made equivalent to the given colour. The experiments from which these results were taken had ON THE THEORY OP COMPOUND COLOURS. 245 the negative values transferred to the other side of the equation. They were all made by means of the colour-top, and were verified by repetition at different times. It may be necessary to remark, in conclusion, with reference to the mode of registering visible colours in terms of three arbitrary standard colours, that it proceeds upon that theory of three primary elements in the sensation of colour, which treats the investigation of the laws of visible colour as a branch of human physiology, incapable of being deduced from the laws of Ught itself, as set forth in physical optics. It takes advantage of the methods of optics to study vision itself; and its appeal is not to physical principles, but to our consciousness of our own sensations. [From the Report of the British Association, 1856.] XIV. On an Instrument to illustrate Poinsdt's Theory of Rotation. In studying the rotation of a solid body according to Poins&t's method, we have to consider the successive positions of the instantaneous axis of rotation with reference both to directions fixed in space and axes assumed in the moving body. The paths traced out by the pole of this axis on the invariable plane and on the central ellipsoid form interesting subjects of mathematical investigation. But when we attempt to follow with our eye the motion of a rotating body, we find it difficult to determine through what point of the body the instantaneous axis passes at any time, — and to determine its path must be still more difficult. I have endeavoured to render visible the path of the instantaneous axis, and to vary the circumstances of motion, by means of a top of the same kind as that used by Mr Elliot, to illustrate precession*. The body of the instrument is a hoUow cone of wood, rising from a ring, 7 inches in diameter and 1 inch thick. An iron axis, 8 inches long, screws into the vertex of the cone. The lower extremity has a point of hard steel, which rests in an agate cup, and forms the support of the instrument. An iron nut, three ounces in weight, is made to screw on the axis, and to be fixed at any point ; and in the wooden ring are screwed four bolts, of three ounces, working horizontally, and four bolts, of one ounce, workmg vertically. On the upper part of the axis is placed a disc of card, on which are drawn four concentric rings. Each ring is divided into four quadrants, which are coloured red, yellow, green, and blue. The spaces between the rings are white. When the top is in motion, it is easy to see in which quad- rant the instantaneous axis is at any moment and the distance between it and the axis of the instrument; and we observe, — Ist. That the instantaneous axis travels in a closed curve, and returns to its original position in the body. 2ndly. * Transactions of the Royal Scottish Society of Arts, 1855. ON AN INSTRUMENT TO ILLUSTRATE POINSOT'S THEORY OF ROTATION. 247 That by working the vertical bolts, we can make the axis of the instrument the centre of this closed curve. It will then be one of the principal axes of inertia. 3rdly. That, by working the nut on the axis, we can make the order of colours either red, yellow, green, blue, or the reverse. When the order of colours is in the same direction as the rotation, it indicates that the axis of the instrument is that of greatest moment of inertia. 4thly. That if we screw the two pairs of opposite horizontal bolts to different distances from the axis, the path of the instantaneous pole will no longer be equidistant from the axis, but will describe an ellipse, whose longer axis is in the direction of the mean axis of the instrument. 5thly. That if we now make one of the two horizontal axes less and the other greater than the vertical axis, the instantaneous pole will separate from the axis of the instrument, and the axis will inchne more and more till the spinning can no longer go on, on account of the obliquity. It is easy to see that, by attending to the laws of motion, we may produce any of the above effects at pleasure, and illustrate many different propositions by means of the same instrument. [From the Transactions of the Royal Society of Edinburgh, Vol. xxi. Part iv.] XV. On a Dynamical Top, for exhibiting the phenoynena of the motion of a system of invariable form about a fixed point, with some suggestions as to the Earth's motion. (Read 20th April, 1857.) To those who study the progress of exact science, the common spinning-top is a symbol of the labours and the perplexities of men who had successfully threaded the mazes of the planetary motions. The mathematicians of the last age, searching through nature for problems worthy of their analysis, found in this toy of their youth, gmple occupation for their highest mathematical powers. No illustration of astronomical precession can be devised more perfect than that presented by a properly balanced top, but yet the motion of rotation has intricacies far exceeding those of the theory of precession. Accordingly, we find Euler and D'Alembert devoting their talent and their patience to the estabhshment of the laws of the rotation of solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poias6t has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligible propositions supersede equations. In the practical department of the subject, we must notice the rotatory machine of Bohnenberger, and the nautical top of Troughton. In the first of these instruments we have the model of the Gyroscope, by which Foucault has been able to render visible the effects of the earth's rotation. The beautiful experiments by which Mr J. Elliot has made the ideas of precession so familiar to us are performed with a top, similar in some respects to Troughton's, though not borrowed from his. ON A DYNAMICAL TOP. 249 The top which I have the honour to spin before the Society, differs from that of Mr Elliot in having more adjustments, and in being designed to exhibit far more comphcated phenomena. The arrangement of these adjustments, so as to produce the desired effects, depends on the mathematical theory of rotation. The method of exhibiting the motion of the axis of rotation, by means of a coloured disc, is essential to the success of these adjustments. This optical contrivance for rendering visible the nature of the rapid motion of the top, and the practical methods of applying the theory of rotation to such an instrument as the one before us, are the grounds on which I bring my instrument and experiments before the Society as my own. I propose, therefore, in the first place, to give a brief outline of such parts of the theory of rotation as are necessary for the explanation of the phenomena of the top. I shall then describe the instrument with its adjustments, and the effect of each, the mode of observing of the coloured disc when the top is in motion, and the use of the top in illustrating the mathematical theory, with the method of making the different experiments. Lastly, I shall attempt to explain the nature of a possible variation in the earth's axis due to its figure. This variation, if it exists, must cause a periodic inequahty in the latitude of every place on the earth's surface, going through its period in about eleven months. The amount of variation must be very small, but its character gives it importance, and the necessary observations are already made, and only require reduction. On the Theory of Rotation. The theory of the rotation of a rigid system is strictly deduced from the elementary laws of motion, but the complexity of the motion of the particles of a body freely rotating renders the subject so intricate, that it has never been thoroughly understood by any but the most expert mathematicians. Many who have mastered the lunar theory have come to erroneous conclusions on this sub- ject ; and even Newton has chosen to deduce the disturbance of the earth's axis from his theory of the motion of the nodes of a free orbit, rather than attack the problem of the rotation of a solid body. VOL. L 32 250 ON A DYNAMICAL TOP. The method by which M. Poins6t has rendered the theory more manageable, is by the liberal introduction of "appropriate ideas," chiefly of a geometrical character, most of which had been rendered familiar to mathematicians by the writings of Monge, but which then first became illustrations of this branch of dynamics. If any further progress is to be made in simplifying and arranging the theory, it must be by the method which Poins6t has repeatedly pointed out as the only one which can lead to a true knowledge of the subject, — that of proceeding from one distinct idea to another, instead of trusting to symbols and equations. An important contribution to our stock of appropriate ideas and methods has lately been made by Mr R. B. Hayward, in a paper, " On a Direct Method of estimating Velocities, Accelerations, and all similar quantities, with respect to axes, moveable in any manner in Space." [Trans. Cambridge Phil. Soc. Vol. x. Part i.) * In this communication I intend to confine myself to that part of the subject which the top is intended to Ulustrate, namely, the alteration of the position of the axis in a body rotating freely about its centre of gravity. I shall, therefore, deduce the theory as briefly as possible, from two considera- tions only, — the permanence of the original angular momentum in direction and magnitude, and the permanence of the original vis viva. * The mathematical difl]culties of the theory of rotation arise chiefly from the want of geometrical illustrations and sensible images, by which we might fix the results of analysis in our minds. It is easy to understand the motion of a body revolving about a fixed axle. Every point in the body describes a circle about the axis, and returns to its original position after each complete revolution. But if the axle itself be in motion, the paths of the different points of the body will no longer be circular or re-entrant. Even the velocity of rotation about the axis requires a careful definition, and the proposition that, in all motion about a fixed point, there is always one line of particles forming an instantaneous axis, is usually given in the form of a very repulsive mass of calculation. Most of these difficulties may be got rid of by devoting a little attention to the mechanics and geometry of the problem before entering on the discussion of the equations. Mr Hayward, in his paper already referred to, has made great use of the mechanical conception of Angular Momentum. * 7th May, 1857. The paragraphs marked thus have been rewritten since the paper was read. ON A DYNAMICAL TOP. 251 Definition. — The Angular Momentum of a particle about an aods is mea- sured by the product of the mass of the particle, its velocity resolved in the normal plane, and the perpendicular from the axis on the direction of motion. * The angular momentum of any system about an axis is the algebraical sum of the angular momenta of its parts. As the rate of change of the linear momentum of a particle measures the moving force which acts on it, so the rate of change of angular raomentum measures the moment of that force about an axis. All actions between the parts of a system, being pairs of equal and opposite forces, produce equal and opposite changes in the angular momentum of those parts. Hence the whole angular momentum of the system is not affected by these actions and re-actions. * When a system of invariable form revolves about an axis, the angular velocity of every part is the same, and the angular momentum about the axis is the product of the angular velocity and the moment of inertia about that axis. * It is only in particular cases, however, that the whole angular momentum can be estimated in this way. In general, the axis of angular momentum differs from the axis of rotation, so that there wUI be a residual angular momentum about an axis perpendicular to that of rotation, unless that axis has one of three positions, called the principal axes of the body. By referring everything to these three axes, the theory is greatly simplified. The moment of inertia about one of these axes is greater than that about any other axis through the same point, and that about one of the others is a miai- mvmi. These two are at right angles, and the third axis is perpendicular to their plane, and is caUed the mean axis. * Let A, B, C be the moments of inertia about the principal axes through the centre of gravity, taken in order of magnitude, and let Wi Wj W3 be the angular velocities about them, then the angular momenta will be Aw^, Bo)„ and Cwg. Angular momenta may be compounded Hke forces or velocities, by the law of the "parallelogram," and since these three are at right angles to each other, their resultant is jA'o),' + B'co,' + C'co,' = H (1), and this must be constant, both in magnitude and direction in space, since no external forces act on the body. 32—2 2 5a ON A DYNAMICAL TOP. We shall call this axis of angular momentum the invariable axis. It is perpendicular to what has been called the invariable plane. Poins6t calls it the axis of the couple of impulsion. The direction-cosines of this axis in the body are, , Aco^ jBwj C(03 ^^^' ^ = ^' """^-H- Since I, m and n vary during the motion, we need some additional condition to determine the relation between them. We find this in the property of the vis viva of a system of invariable form in which there is no friction. The vis viva of such a system must be constant. We express this in the equation Ao>,' + B(o,'+Cco,'=V (2). Substituting the values of Wj, a^, (o^ in terms of I, m, n, V rn? n" _ V 1 1 1 F and this equation becomes aH' + }fm^ + c^n^ = e' (3), and the equation to the cone, described by the invariable axis within the body, is (a?-e')x' + {}f-e')f + {(f-e')z' = Q (4). The intersections of this cone with planes perpendicular to the principal axes are found by putting x, y, or z, constant in this equation. By giving e various values, all the different paths of the pole of the invariable axis, corresponding to different initial circumstances, may be traced. *In the figures, I have supposed a^ = 100, 6^=107, and c^ = 110. The first figure represents a section of the various cones by a plane perpendicular to the axis of x, which is that of greatest moment of inertia. These sections are ellipses having their major axis parallel to the axis of h. The value of e' corresponding to each of these curves is indicated by figures beside the curve. The ellipticity increases with the size of the ellipse, so that the section corresponding to e'—107 would be two parallel straight lines (beyond the bounds of the figure), after which the sections would be hyperbolas. ON A DYNAMICAL TOP. 253 *The second figure represents the sections made by a plane, perpendicular to the mean axis. They are all hyperbolas, except when 6" = 107, when the section is two intersecting straight lines. The third figure shows the sections perpendicular to the axis of least moment of inertia. From e^=110 to e^=107 the sections are eUipses, 6^=107 gives two parallel straight lines, and beyond these the curves are hyperbolas. *The fourth and fifth figures show the sections of the series of cones made by a cube and a sphere respectively. The use of these figures is to exhibit the connexion between the different curves described about the three principal axes by the invariable axis during the motion of the body. *We have next to compare the velocity of the invariable axis with respect to the body, with that of the body itself round one of the principal axes. Since the invariable axis is fixed in space, its motion relative to the body must be equal and opposite to that of the portion of the body through which it passes. Now the angular velocity of a portion of the body whose direction - cosines are I, m, n, about the axis of oo is Oil ' /7 \ J— j3 - j—j, [Icoi + mw2 + no),). Substituting the values of tUj, a^, a^, in terms of I, m, n, and taking account of equation (3), this expression becomes Changing the sign and putting I = -^ we have the angular velocity of the invariable axis about that of x always positive about the axis of greatest moment, negative about that of least moment, and positive or negative about the mean axis according to the value of e^ The direction of the motion in every case is represented by the arrows in the figures. The arrows on the outside of each figure indicate the direction of rotation of the body. *If we attend to the ciirve described by the pole of the invariable axis 254 ON A DYNAMICAL TOP. on the sphere in fig. 5, we shall see that the areas described by that point, if projected on the plane of yz, are swept out at the rate ' a' Now the semi-axes of the projection of the spherical ellipse described by the pole are je^ — a? , le' — a' Dividing the area of this ellipse by the area described during one revo- lution of the body, we find the number of revolutions of the body during the description of the ellipse — a' Jlf — a? Jd' — a' The projections of the spherical ellipses upon the plane of yz are all similar ellipses, and described in the same number of revolutions; and in each ellipse so projected, the area described in any time is proportional to the number of revolutions of the body about the axis of x, so that if we measure time by revolutions of the body, the motion of the projection of the pole of the invariable axis is identical with that of a body acted on by an attractive central force varying directly as the distance. In the case of the hyperbolas in the plane of the greatest and least axis, this force must be supposed repulsive. The dots in the figures 1, 2, 3, are intended to indicate roughly the progress made by the invariable axis during each revolution of the body about the axis of x, y and z respectively. It must be remembered that the rotation about these axes varies with their inclination to the invariable axis, so that the angular velocity diminishes as the inclination increases, and there- fore the areas in the eUipses above mentioned are not described with uniform velocity in absolute time, but are less rapidly swept out at the extremities of the major axis than at those of the minor. *When two of the axes have equal moments of inertia, or h = c, then the angular velocity to^ is constant, and the path of the invariable axis is circular, the number of revolutions of the body during one circuit of the invariable axis, being a? If -a?' ON A DYNAMICAL TOP. 255 The motion is in the same direction as that of rotation, or in the opposite direction, according as the axis of x is that of greatest or of least moment of inertia. *Both in this case, and in that in which the three axes are unequal, the motion of the invariable axis in the body may be rendered very slow by diminishing the difference of the moments of inertia. The angular velocity of the axis of x about the invariable axis in space is O), 'a\\-iy which is greater or less than Wj, as ^ is greater or less than a\ and, when these quantities are nearly equal, is very nearly the same as ^ will be the direction of the ray after emergence, cutting the axis in F^, (unless x = y. ON THE GENERAL LAWS OE OPTICAL INSTRUMENTS. 275 when aj}^ becomes parallel to the axis). The point F^ may be found, by remembering that ATpbi — Bft^, AjX^ = xAia^, Bfi^^yBfi^. We find — A,F, = c, ■y-x Let g^ be the point at which the emergent ray is at the same distance from the axis as the incident ray, draw gJJ^ perpendicular to the axis, then we have y-x Similarly, if a^^F-^ be a ray, which, after emergence, becomes parallel to the axis ; and g^G.^ a line perpendicular to the axis, equal to the distance of the parallel emergent ray, then A,F, = c,^L^, F,G, = ^^. x—y x—y Definitions. I. The point F.^, the focus of incident rays when the emergent rays are parallel to the axis, is called the first principal focus of the instrument. II. The plane G^^ at which incident rays through F.^ are at the same distance from the axis as they are after emergence, is called the first princi- pal plane of the instrument. Ffi-^ is called the first focal length. III. The point F^, the focus of emergent rays when the incident rays are parallel, is called the second principal focus. IV. The plane G^^, at which the emergent rays are at the same distance from the axis, as before incidence, is called the second principal plane, and Ffi^ is called the second focal length. When x = y, the ray is parallel to the axis, both at incidence and emerg- ence, and there are no such points as F and G. The instrument is then called a telescope. x{=y) is called the linear magnifying power and is denoted c by I, and the ratio - is denoted by n, and may be called the elongation. In the more general case, in which x and y are different, the principal foci and principal planes afford the readiest means of finding the position of images. 35—2 276 ON THE GENERAL LAWS OE OPTICAL INSTRUMENTS. Prop. IV. Given the principal foci and principal planes of an instrument, to find the relations of the foci of the incident and emergent pencils. Let F^, F, (fig. 3) be the principal foci, G„ G, the principal planes, Q, the focus of incident light, QJ^^ perpendicular to the axis. Through Q^ draw the ray Q^g^F^. Since this ray passes through F^ it emerges parallel to the axis, and at a distance from it equal to G^g^. Its direction after emergence is therefore Q,g^ where G,g, = G,g^. Through Q^ draw Q{y^ parallel to the axis. The corresponding emergent ray wiU pass through F^, and will cut the second principal plane at a distance G{y^=G^y^, so that F^y„ is the direction of this ray after emergence. Since both rays pass through the focus of the emergent pencil, Q^, the point of intersection, is that focus. Draw Q^P^ perpendicular to the axis. Then P^Q^ = G{y^ = G^y^, and G^g^^G,g^ = F^Q^. By similar triangles F.F^Q, and F,G,g, P,F, : F,G, :: P,Q, : {G,g,^) P,Q,. And by similar triangles F^P^Q^ and F^G^y^ We may put these relations into the concise form P,F,_P,Q,_GJ', Ffi, PA F,P,' and the values of F^P^ and P^ are F,P, = ^^^^^^ and P.Q.^^P.Q,. These expressions give the distance of the image from F^ measured along the axis, and also the perpendicular distance from the axis, so that they serve to determine completely the position of the image of any point, when the princi- pal foci and principal planes are known. Prop. V. To find the focus of emergent rays, when the instrument is a telesGO'pe. Let Q-^ (fig. 4) be the focus of incident rays, and let Q-Pj}>., be a ray parallel to the axis ; then, since the instrument is telescopic, the emergent ray Qjx})^ will be parallel to the axis, and Q^P^^^l . Q^P-^. ON THE GENERAL LAWS OP OPTICAL INSTRUMENTS. 277 Let QajB, be a ray through B^, the emergent ray will be Q^a^B^, and Now PS^PS,J_lPR^^P^^PjB, AJB^ A,a, l.A.a, A,a, A,B,' BA AM, so that ^5-^ = ^~fr = *ij a constant ratio. Cor. If a point C be taken on the axis of the instrument so that ^^^-A,B,-A,B, AA = 13^ A^., then CB, = n.CB,. Def. The point C is called the centre of the telescope. It appears, therefore, that the image of an object in a telescope has its dimensions perpendicular to the axis equal to I times the corresponding dimen- sions of the object, and the distance of any part from the plane through C equal to n times the distance of the corresponding part of the object. Of course all longitudinal distances among objects must be multiplied by n to obtain those of their images, and the tangent of the angular magnitude of an object as seen from a given point in the axis must be multiplied by - to obtain that of the image of the object as seen from the image of the given point. The quantity - is therefore called the angular magnifying power, and is denoted by m. Prop. VI. To find the principal foci and principal planes of a combina- tion of two instruments having a common axis. Let /, /' (fig. 5) be the two instruments, G^F-^F^G, the principal foci and planes of the first, G^F^F^G^ those of the second, T^(f>^(j),V, those of the com- bination. Let the ray g^^g^g^ pass through both instruments, and let it be parallel to the axis before entering the first instrument. It wiU therefore pass through F, the second principal focus of the first instrument, and through g, so that G^^ = G^i^Ti- On emergence from the second instrument it wiU pass through ^^ the focus conjugate to F^, and through g^ in the second principal plane, so that 278 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. G„'g^ = G^g^. (f>^ is by definition the second principal focus of the combination of instruments, and if r^yj be the second principal plane, then T^y^ — G^g^. We have now to find the positions of (f)^ and r^. By Prop. IV., we have F'G' G'F' Or, the distance of the principal focus of the combination, from that of the second instrument, is equal to the product of the focal lengths of the second instrument, divided by the distance of the second principal focus of the first instrument from the first of the second. From this we get G'F' G'F or tr2^2 = -p-p-, . Now, by the pairs of similar triangles (f)G^'g^', ^T^y^ and Ffir^g', Ffi^g^, V^r G;gr G(gr g:f,- Multiplying the two sides of the former equation respectively by the first and last of these equal quantities, we get GF G'F' Or, the second focal distance of a combination is the product of the second focal lengths of its two components, divided by the distance of their consecutive principal foci. If we call the focal distances of the first instrument /, and /„ those of the second /; and //, and those of the combination f^, % and put FJF^ = d, then the positions of the principal foci are found from the values and the focal lengths of the combination from '^'~ d ' '''~ d ' ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 279 When d = 0, all these values become infinite, and the compound instrument becomes a telescope. Prop. VII. To find the linear magnifying power, the elongation, and the centre of the instrument, when the combination becomes a telescope. Here (fig. 6) the second principal focus of the first instrument coincides at F with the first of the second. (In the figure, the focal distances of both instru- ments are taken in the opposite direction from that formerly assumed. They are therefore to be regarded as negative.) In the first place, i^/ is conjugate to F^, for a pencil whose focus before incidence is F^ wOl be parallel to the axis between the instruments, and will converge to F^' after emergence. Also if (?i^i be an object in the first principal plane, G,g^ will be its first image, equal to itself, and if Hh be its final image G,F - /, ' FG' f Now the linear magnifying power is j^ — , and the elongation is -p-j-i , because F^ and H are the images of F■^ and G^ respectively ; therefore 1=--^, and n=-^. If The angular magnifying power = m = -— — 4^ • The centre of the telescope is at the point C, such that 1 —n When n becomes 1 the telescope has no centre. The effect of the instrument is then simply to alter the position of an object by a certain distance measured along the axis, as in the case of refraction through a plate of glass bounded by parallel planes. In certain cases this constant distance itself disappears, as in the case of a combination of three convex lenses of which the focal lengths are 280 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 4, 1, 4 and the distances 4 and 4. This combination simply inverts every object without altering its magnitude or distance along the axis. The preceding theory of perfect instruments is quite independent of the mode in which the course of the rays is changed within the instrument, as we are supposed to know only that the path of every ray is straight before it enters, and after it emerges from the instrument. We have now to con- sider, how far these results can be applied to actual instruments, in which the course of the rays is changed by reflexion or refraction. We know that such instruments may be made so as to fulfil approximately the conditions of a perfect instrument, but that absolute perfection has not yet been obtained. Let us inquire whether any additional general law of optical instruments can be deduced from the laws of reflexion and refraction, and whether the imper- fection of instruments is necessary or removeable. The following theorem is a necessary consequence of the known laws of reflexion and refraction, whatever theory we adopt. If we multiply the length of the parts of a ray which are in different media by the indices of refraction of those media, and call the sum of these products the reduced path of the ray, then : I. The extremities of all rays from a given origin, which have the same reduced path, lie in a surface normal to those rays. II. When a pencil of rays is brought to a focus, the reduced path from the origin to the focus is the same for every ray of the pencil. In the undulatory theory, the " reduced path " of a ray is the distance through which light would travel in space, during the time which the ray takes to traverse the various media, and the surface of equal "reduced paths" is the wave-surface. In extraordinary refraction the wave-surface is not always normal to the ray, but the other parts of the proposition are true in this and all other cases. From this general theorem in optics we may deduce the following propo- sitions, true for all instruments depending on refraction and reflexion. Prop. VIII. In any optical instrument depending on refraction or reflex- ion, if aitti, fei/Si (fig. 7) be two objects and ajx^, hJS^ their images, A^B,^ the distance of the objects, A^B^ that of the images, fi^ the index of refraction of ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 281 the medium in whicli the objects are, (jl^ that of the medium in which the images are, then ^' A,B, -f"^ A A ' approximately, when the objects are small. Since a^ is the image of a^, the reduced path of the ray afiTa^ will be equal to that of a^^a^, and the reduced paths of the rays a^^^a^ and afi^a, will be equal. Also because hfi^ and 62^3 are conjugate foci, the reduced paths of the rays \a}>^ and \a}).„ and of /S^a^A and ^^aS^ will be equal. So that the reduced paths «!&! + &i«2 = « A + At*2 b^a^ + aj)^ = ^iC''2 + ^2^2 ^i^a + afi^ = fi^a^ + a^iSj . •. a,\ + OjjSi + a,b^ + a,^^ = a^^ + aj)^ + afi^ + afi^, these being still the reduced paths of the rays, that is, the length of each ray multiplied by the index of refraction of the medium. If the figure is symmetrical about the axis, we may write the equation fll («A - «l^l) = j^2 («2^2 - «2&2)j where a^^, &c. are now the actual lengths of the rays so named. Now a^," = A,B,' + 1 {a,a, + lSiY> so that afi^ — afi^ = a^a^y.'bfi^, . -, . a^a^ X 61/81 and /..(aA-aA)=/x.^^^ + ^^^^. I f\ 7 \ ^2 2 2r^2 Similarly /u.^ (a^^^ - a^Oa) = ^2 ^o _^^^ • OiOiXftA «2a2 X 62^, So that the equation //.i r, , — r=H'i ^ 1 ^a ' VOL. L 36 282 ON THE QENEEAL LAWS OF OPTICAL INSTRUMENTS. is true accurately, and since when the objects are small, the denominators are nearly 24i-Bi and 2^2-^2, the proposition is proved approximately true. Using the expressions of Prop. III., this equation becomes 1 xy Now by Prop. III., when x and y are different, the focal lengths /"i and f^ are x — y' '^y~^ therefore 4 — -^—^ — — by the present theorem. So that in any instrument, not a telescope, the focal lengths are directly as the indices of refraction of the media to which they belong. If, as in most cases, these media are the same, then the two focal distances are equal. When x = y, the instrument becomes a telescope, and we have, by Prop. V., I — X, and ti= -; and therefore by this theorem IM^ n' We may iind / experimentally by measuring the actual diameter of the image of a known near object, such as the aperture of the object glass. If O be the diameter of the aperture and O' that of the circle of light at the eye-hole (which is its image), then From this we find the elongation and the angular magnifying power n = — l\ and m = — y . When fii = ii.„, as in ordinary cases, m = j = — , which is Gauss' rule for deter- mining the magnifying power of a telescope. ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 283 Prop. IX. It is impossible, by means of any combination of reflexions and refractions, to produce a perfect image of an object at two different distances, unless tbe instrument be a telescope, and l = n=-, m=l. It appears from tbe iavestigation of Prop. VIII. that the results there obtained, if true when the objects are very small, will be incorrect when the objects are large, unless a^fij^ + aj)^ : a^^ + ap^ :: A^B^ : AJB^, and it is easy to prove that this cannot be, unless all the lines in the one figure are proportional to the corresponding lines in the other. In this way we might show that we cannot in general have an astigmatic, plane, undistorted image of a plane object. But we can prove that we cannot get perfectly focussed images of an object in two positions, even at the expense of curvature and distortion. We shall first prove that if two objects have perfect images, the reduced path of the ray joinuig any given points of the two objects is equal to that of the ray joining the corresponding points of the images. Let a^ (fig. 8) be the perfect image of a^ and /S^ of /3i. Let A^a^ = a^, B^,-^b„ A,a, = a„ B,^,^h„ A,B,^c„ AJB.^c^. Draw aJD^ parallel to the axis to meet the plane B^, and aJD^ to the plane of 5, Since everything is symmetrical about the axis of the instrument we shall have the angles D,BJ3,:^D^,/3,--0, then in either figure, omitting the suffixes, = c^ + a' + V- 2ab cos 9. It has been shown in Prop. YIII. that the difierence of the reduced paths of the rays aA, ^A in the object must be equal to the difference of the reduced paths of ah„ a A "i the image. Therefore, since we may assume any value for 9 II, J{a^ + 6,= + c^ - 2a A cos 9) - fx, J {a,' + &/ + c/ - 2a A cos 9) 36—2 284 ON THE GENERAL LAWS OE OPTICAL INSTRUMENTS. 33 constant for all values of 6. This can be only when and fJ^iJip^A) =H-z'j{c('2^2), which shows that the constant must vanish, and that the lengths of lines joining corresponding points of the objects and of the images must be inversely as the indices of refraction before incidence and after emergence. Next let ABC, DEF (fig. 9) represent three points in the one object and three points in the other object, the figure being drawn to a scale so that all the lines in the figure are the actual lines multiplied by fx^. The lines of the figure represent the reduced paths of the rays between the corresponding points of the objects. Now it may be shown that the form of this figure cannot be altered with- out altering the length of one or more of the nine fines joining the points ABC to DEF. Therefore since the reduced paths of the rays in the image are equal to those in the object, the figure must represent the image on a scale of yi,^ to 1, and therefore the instrument must magnify every part of the object alike and elongate the distances parallel to the axis in the same proportion. It is therefore a telescope, and m=l. If /A, = /x.2, the image is exactly equal to the object, which is the case in reflexion in a plane mirror, which we know to be a perfect instrument for all distances. The only case in which by refraction at a single surface we can get a perfect image of more than one point of the object, is when the refracting surface is a sphere, radius r, index ^, and when the two objects are spherical surfaces, concentric with the sphere, their radii being - , and r ; and the two images also concentric spheres, radii fir, and r. In this latter case the image is perfect, only at these particular distances and not generally. I am not aware of any other case in which a perfect image of an object can be formed, the rays being straight before they enter, and after they emerge from the instrument. The only case in which perfect astigmatism for all pencils has hitherto been proved to exist, was suggested to me by the consideration VOL. I. PLATE IV. 0= > ^ Co 53 «=t« \l ] ^to M-tA c> \^ > A. ^^ faoe page 284 Cambridge University Press ON THE GENERAL LAWS OE OPTICAL INSTRUMENTS. 285 of the structure of the crystalline lens in fish, and was published in one of the problem-papers of the Oamhridge and Dublin Mathematical Journal. My own method of treating that problem is to be found in that Journal, for February, 1854. The case is that of a medium whose index of refraction varies with the distance from a centre, so that if )u,„ be its value at the centre, a a given line, and r the distance of any point where the index is /a, then a' a -Vr^ The path of every ray within this medium is a circle in a plane passing through the centre of the medium. Every ray from a point in the medium, distant h from the centre, will a^ converge to a point on the opposite side of the centre and distant from it -rr ■ It will be observed that both the object and the image are included in the variable medium, otherwise the images would not be perfect. This case therefore forms no exception to the result of Prop. IX., in which the object and image are supposed to be outside the instrument. Aberdeen, 12th Jan., 1858. 2.U [From the Proceedings of the Royal Society of Edinburgh, Vol. iv.] XVIII. On Theories of the Constitution of Saturn's Rings. The planet Saturn is surrounded by several concentric flattened rings, which appear to be quite free from any connection with each other, or with the planet, except that due to gravitation. The exterior diameter of the whole system of rings is estimated at about 176,000 miles, the breadth from outer to inner edge of the entire system, 36,000 miles, and the thickness not more than 100 miles. It is evident that a system of this kind, so broad and so thin, must depend for its stability upon the dynamical equilibrium between the motions of each part of the system, and the attractions which act on it, and that the cohesion of the parts of so large a body can have no effect whatever on its motions, though it were made of the most rigid material known on earth. It is therefore necessary, in order to satisfy the demands of physical astronomy, to explain how a material system, presenting the appearance of Saturn's Rings, can be maintained in permanent motion consistently with the laws of gravitation. The principal hypotheses which present themselves are these — I. The rings are solid bodies, regular or irregular. II. The rings are fluid bodies, Hquid or gaseous. III. The rings are composed of loose materials. The results of mathematical investigation applied to the first case are, — 1st. That a uniform ring cannot have a permanent motion. 2nd. That it is possible, by loading one side of the ring, to produce stability of motion, but that this loading must be very great compared with the whole mass of the rest of the ring, being as 82 to 18. ON THEORIES OF THE CONSTITUTION OF SATURN's RINGS. 287 3rd. That this loading must not only be very great, but very nicely adjusted; because, if it were less than '81, or more than -83 of the whole, the motion would be unstable. The mode in which such a system would be destroyed would be by the collision between the planet and the inside of the ring. And it is evident that as no loading so enormous in comparison with the ring actually exists, we are forced to consider the rings as fluid, or at least not solid ; and we find that, in the case of a fluid ring, waves would be gene- rated, which would break it up into portions, the number of which would depend on the mass of Saturn directly, and on that of the ring inversely. It appears, therefore, that the only constitution possible for such a ring is a series of disconnected masses, which may be fluid or solid, and need not be equal. The complicated internal motions of such a ring have been investigated, and found to consist of four series of waves, which, when combined together, will reproduce any form of original distui'bance with all its co;nsequences. The motion of one of these waves was exhibited to the Society by means of a small mechanical model made by Ramage of Aberdeen, This theory of the rings, being indicated by the mechanical theory as the only one consistent with permanent motion, is further confirmed by recent obser- vations on the inner obscure ring of Saturn. The limb of the planet is seen through the substance of this ring, not refracted, as it would be through a gas or fluid, but in its true position, as would be the case if the light passed through interstices between. the separate particles composing the ring. As the whole investigations are shortly to be published in a separate form, the mathematical methods employed were not laid before the Society. XIX. On the Stability of the motion of Saturn's Rings. [An Essay, -which obtained the Adams Prize for the year 1856, in the University of Cambridge.] ADVEETISEMENT. The Subject of the Prize was announced in the following terms : — The University having accepted a fund, raised by several members of St John's CoUegp, for the purpose of foimding a Prize to be called the Adams Prize, for the best Essay on some subject of Pure Mathematics, Astronomy, or other branch of Natural Philosophy, the Prize to be given once in two years, and to be open to the competition of all persons who heme at any time been admitted to a degree in this University : — The Examiners give Notice, that the following is the subject for the Prize to be adjudged in 1857:— The Motions of Saturn's Rings. *#* The problem may be treated on the supposition that the system of Rings is exactly or very approximately concentric with Saturn and symmetrically disposed about the plane of his Equator, and different hypotheses may be made respecting the physical constitution of the Rings. It may be supposed (1) that they are rigid: (2) that they are fluid, or in part aeriform: (3) that they consist of masses of matter not mutually coherent. The question will be considered to be answered by ascertainiag on these hypotheses severally, whether the conditions of mechanical stability are satisfied by the mutual attractions and motions of the Planet and the Rings. It is desirable that an attempt should also be made to determine on which of the above hypotheses the appearances both of the bright Rings and the recently discovered dark Ring may be most satisfactorily explained; and to indicate any causes to which a change of form such as is supposed from a comparison of modem with the earlier observations to have taken place, may be attributed. E. GUEST, rice-Chancellor. J. CHALLIS. S. PARKINSON. W. THOMSON. Ma^ch 23, 1855. ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 289 CONTENTS. Nature of the Problem 290 Zaplaae's investigation of the Equilihrium of a Ring, and its minimum density 292 His proof that the plane of the Rings will follow that of SaturtHs Equator — that a solid uniform Ring is necessarily unstable ................ 29.3 Further investigation required — Tlieory of an Irregular Solid Ring leads to the result that to ensure stability the irregularity must he enormous 294 Theory of a fluid or discontinuous Ring resolves itself into, the investigation of a series of waves . . 295 PAET I. ON THE MOTION OF A EIGID BODY OF ANY FORM ABOUT A SPHERE. Equations of Motion 296 Problem I. To find the conditions under which a uniform motion of the Ring is possible . . . 298 Problem II. To fmd the equations of the jnotion when slightly disturbed 299 Problem III. To reduce the three simultaneous equations of motion to the form of a single linear equation 300 Problem IV. To determine whether the motion of the Ring is stable or unstable, by means of the relations of the coefficients A, B, C 301 Problem V. To find tlie centre of gravity, the radius of gyration, and the variations of the potential near the centre of a circular ring of small but variable section 302 Problem VI. To determine the condition of stability of the motion in terms of the coefficients f, g, h, which indicate the distribution of mass in the ring 306 Results. \st, a uniform, ring is unstable. 2nd, a ring varying in section according to t/ie law of sines is unstable. 3rd, a uniform ring loaded with a heavy particle may be stable, provided the mxiss of the particle be between 'SISSBS and '8279 of the whole. Case in which the ring is to the particle as 18 -4352/x2R 318 9. Solution of the biquadratic equation when the mass of the Ring is small ; and complete expressions for the nation of each satellite 319 10. Each satellite moves (relatively to the ring) in an ellipse 321 VOL. I. 37 323 326 328 329 330 335 290 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 11. Eax:h satellite moves absolutely in space in a curve which is marly an ellipse for the large values of n, and a spiral of many marly circular coils when n is small 321 12. The form of the ring at a given instant is a series of undulations 322 13. These undulations travel rouyvi the ring with velocity-^ relative to ths ring, and u,-~ absolutely 323 14 General Solution of the Problem— Given the position and motion of every satellite at any om time, to calculate the position and motion of any satellite at any other time, provided that the condition of stability is fulfilled 15. Calculation of the effect of a periodic external disturbing force 16. TreatmeTU of disturbing forces in general 17. Theory of free waves and forced waves 18. Motion of tU ring when the coyiditiotis of stability are not fulfilled. Two different ways in which the ring may be broJcen up 19. Motion of a ring of unequal satellites 20. Motion of a ring composed of a cloud of scattered particles 336 21. Calculation of the forces arising from the displacements of such a system 337 22. Application to the case of a ring of this hind. The mean density must be excessively small, which is inconsistent with its moving as a whole • ■ • '^'^° 23. On the forces arising from inequalities in a thin stratum of gravitating incompressible fluid of indefiiiite extent .......■■••••■••" 24. Application to the case of a flattened fluid ring, moving with uniform angular velocity. Such a ring will be broken up into portions which may continue to revolve as a ring of satellites ... 344 ON THE MUTUAL PERTURBATIONS OF TWO KINGS. 25. Application of the general theory of free and forced waves 345 26. To determine tlie attractions betioeen the rings 346 27. To form the equatioTts of motio7i 348 28. Method of determining tlie reaction of the forced wave on tlie free wave which produced it . . 349 29. Gases in which the peHurbations increase i'ndefimitely 351 30. Application to the theory of an indefinite number of concentric rings 352 31. On the effect of long-continued disturbances on a system of rings v. 352 32. On the effect of collisions among tlie parts of a revolving system 354 33. On the effect of internal friction in a fluid ring 354 Becapitulatimi of tlie Theory of tlie Motion of a Rigid Ring. Reasom for rejecting the hypothesis of rigidity 356 Recapitulation of the Tlicory of a Ring of Equal Satellites ......... 360 Description of a working model shewing the motions of such a system ....... 363 Tlieory of Rings of various constitutions ........-•••• 36v Mutual action of Two Rings 370 Case of many coTicentric Rings, so that the positions of S and R are now determined. Let BRB' be a straight line through R, fixed with respect to the substance of the ring, and let BRK = , ON THE STABILITY OF THE MOTION OP SATURN's RINGS. 2.97 This determines tlie angular position of the ring, so that from the values of r, 6, and ^ the configuration of the system may be deduced, as far as relates to the plane of reference. We have next to determine the forces which act between the ring and the sphere, and this we shall do by means of the potential function due to the ring, which we shall call V. The value of V for any point of space S, depends on its position relatively to the ring, and it is found from the equation \ r where dm is an element of the mass of the ring, and r is the distance of that element from the given point, and the summation is extended over every element of mass belonging to the ring. V will then depend entirely upon the position of the point S relatively to the ring, and may be expressed as a function of r, the distance of S from R, the centre of gravity of the ring, and ^, the angle which the line SR makes with the line RB, fixed in the ring. A particle P, placed at S, will, by the theory of potentials, experience a dV .... . 1 dV moving force P -^ in the direction which tends to increase r, and P - -r-r in a tangential direction, tending to increase (, in the case of uniform motion, and let r=r, +')\, 0=cot+e„ when the motion is slightly disturbed, where r^, 6^, and <^i are to be treated as small quantities of the first order, and their powers and products are to be dV dV neglected. We may expand -jy and -j-r by Taylor's Theorem, dV_dV cTT ,dn^, dr ~dr "^ dr '^'^ drd'P'' dv_dv drv^ .^A d~d'^drdcj>''''^d(f>' "^^ 38—2 300 ON THE STABILITY OF THE MOTION OF SATUBN S RINGS. where the values of the differential coefficients on the right-hand side of the equations are those in which r„ stands for r, and ^^ for ^. and taking account of equations (9) and (10), we may write these equations. Substituting these values in equations (6), (7), (8), and retaining all small quantities of the first order while omitting their powers and products, we have the following system of linear equations in r^, 6-^, and ^i, i2(2r„a,§ + r/^^) + (J? + ^)(ilfn + i\r^,) =0 (11), R (^-o.%-2r,co^^-{R + S){Iyr, + Mi>:) = Q (12), Prob. III. To reduce the three simultaneous equations of motion to the form of a single linear equation. d Let us write n instead of the symbol -r- , then arranging the equations in terms of r^, 0^, and , =0 (16). Here we have three equations to determine three quantities r^, 6^, (f)^ ; but it is evident that only a relation can be determined between them, and that in the process for finding their absolute values, the three quantities will vanish together, and leave the following relation among the coefficients. ON THE STABILITY OP THE MOTION OE SATURN S RINGS. 301 - {2i2r„coJi + {R + S)M} {2Rr,mn} {R¥n' - SN] + {Rre - Rco' -{R + S) L} {RhV} {{R + S)N} + {SM) {Rr,V) {R + S) M-{SM) {2Rr,con) {R + S) N \=0 (17). + {2Rr,ain + {R + S) 31} {RJtfnf} {{R + S)M} - {Rn' - R(o' -{R + S)} {Rr.'n'} {R¥n' - SN} By multiplying up, and arranging by powers of n and dividing by Rn^, this equation becomes An* + Bn'+C=0 1 (18), where A = R'r:¥, B = ZRr:i^(a'-R{R + S)Lr,'¥-R{{R + S)¥ + Sr']N ■ (19). C= R{{R + S)¥- QSr,"} o>' + {R + S) {{R + S)B + Sr/\ {LN- M') . Here we have a biquadratic equation in n which may be treated as a d quadratic in n^, it being remembered that n stands for the operation -j- . Prob. IV. To determine whether the motion of the ring is stable or unstable, by means of the relations of the coefficients A, B, C. The equations to determine the forms of r^, 0^, and e" cos [at + a), which indicates a periodic disturbance, whose amplitude continually increases till it disarranges the system. (5) If n be of the form —b±\/ — la, a negative quantity and an im- possible one, the corresponding term of the solution is i>e-**cos (at + a), which indicates a periodic disturbance whose amplitude is constantly diminishing. It is manifest that the first and fourth cases are inconsistent with the permanent motion of the system. Now since equation (18) contains only even powers of n, it must have pairs of equal and opposite roots, so that every root coming under the second or fifth cases, implies the existence of another root belonging to the first or fourth. If such a root exists, some disturbance may occur to produce the kind of derangement corresponding to it, so that the system is not safe unless roots of the first and fourth kinds are altogether excluded. This cannot be done without excluding those of the second and fifth kinds, so that, to insure stability, all the four roots must be of the third kind, that is, pure impossible quantities. That this may be the case, both values of n' must be real and negative, and the conditions of this are— 1st. That A, B, and C should be of the same sign, 2ndly. That Br'>iAC. When these conditions are fulfilled, the disturbances will be periodic and consistent with stability. When they are not both fulfilled, a small disturbance may produce total derangement of the system. Prob. V. To find the centre of gravity, the radius of gyration, and the variations of the potential near the centre of a circular ring of small but variable section. Let a be the radius of the ring, and let 6 be the angle subtended at the centre between the radius through the centre of gravity and the line through a given point in the ring. Then if fx be the mass of unit of length of the ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 303 ring near the given point, /x will be a periodic function of d, and may there- fore be expanded by Fourier's theorem in the series, /i = 2^{l + 2/cos^ + f(7Cos2^ + fAsin2^ + 2*cos(3^ + a)-|-&c.} (21), where /, g, h, &c. are arbitrary coefficients, and E is the mass of the ring. (1) The moment of the ring about the diameter perpendicular to the prime radius is Rr^ = fia? cos Odd = Raf, therefore the distance of the centre of gravity from the centre of the ring, (2) The radius of gyration of the ring about its centre in its own plane is evidently the radius of the ring =a, but if h be that about the centre of gravity, we have (3) The potential at any point is found by dividing the mass of each element by its distance from the given point, and integrating over the whole mass. Let the given point be near the centre of the ring, and let its position be defined by the co-ordinates r' and ^/r, of which r' is small compared with a. The distance {p) between this point and a point in the ring is r The other terms contain powers of — higher than the second. a We have now to determine the value of the integral. Jo V=\ ^acW; P and in multiplying the terms of (/x) by those of (-) , we need retain oioly those which contain constant (Quantities, for all those which contain sines or 304 ON THE STABILITY OF THE MOTION OE SATUEN S KINGS. cosines of multiples of {^ — 6) wiU vanish when integrated between the limits. In this way we find V=^{l+f''^cosr}j + i''-^{l+gcos2i, + hshi2rp)} (22). U/ it U/ T The other terms containing higher powers of — . In order to express V in terms of r, and (f)^, as we have assumed in the former investigation, we must put r cos \jj= -n + in^j', ?-'sini/>= -r„^i, V=^{^-f'^ + i%{^+9)+i^Ai>. + ir<}>n3-9)}- .(23). From which we find dr 2a R (24). These results may be confirmed by the following considerations applicable to any circular ring, and not involving any expansion or integration. Let af be the distance of the centre of gravity from the centre of the ring, and let the ring revolve about its centre with velocity w. Then the force necessary to keep the ring in that orbit will be —Rafoi^. But let /S be a mass fixed at the centre of the ring, then if a" every portion of the ring will be separately retained in its orbit by the attrac- tion of S, so that the whole ring will be retained in its orbit. The resultant attraction must therefore pass through the centre of gravity, and be -Raf(o'=-RS^,', therefore dr a^ ' ON THE STABILITY OF THE MOTION OF SATUEN's RINGS. 305 The equation _ + _+„ + 4^^ = is true for any system of matter attracting according to the law of gravitation. If we bear in mind that the expression is identical in form with that which measures the total efflux of fluid from a differential element of volume, where -7- , -j— , -J- are the rates at which the fluid passes through its sides, we may easily form the equation for any other case. Now let the position of a point in space be determined by the co-ordinates r, <^ and 2, where z is measured perpendicularly to the plane of the angle <^. Then by choosing the directions of the axes x, y, z, so as to coincide with those of the radius vector r, the per- pendicular to it in the plane of <^, and the normal, we shall have dx = dr, dy — rd(j), dz = dz, dV^dV dV^ldV dV^dV dx dr ' dy r dcj)' dz dz The quantities of fluid passing through an element of area in each direction are (^V ,j, dVl., dV ,, , -7- rd'^ dz' ^ ^' which is necessarily equivalent to the former expression. d^V Now at the centre of the ring -j^- may be found by considering the attrac- tion on a point just above the centre at a distance z, — =-R ^ dz {a' + z')^' d'V B , „ -^=--3,whenz = 0. Also we know - -^ =--, and r = a/, , . . . . d'V 1 d'V ^R /^.x so that m any circular nng "^ + ^= d^~ a^ an equation satisfied by the former values of L and N. VOL. I. ^^ 306 ON THE STABILITY OP THE MOTION OP SATURN 'S RINGS. By referring to the original expression for tlie variable section of the ring, it appears that the effect of the coefficient f is to make the ring thicker on one side and thinner on the other in a uniformly graduated manner. The effect of g is to thicken the ring at two opposite sides, and diminish its section in the parts between. The coefficient li indicates an inequality of the same kind, only not symmetrically disposed about the diameter through the centre of gravity. Other terms indicating inequalities recurring three or more times in the circumference of the ring, have no effect on the values of L, M and N. There is one remarkable case, however, in which the irregiilarity consists of a single heavy particle placed at a point on the circumference of the ring. Let P be the mass of the particle, and Q that of the uniform ring on which it is fixed, then R = P-\-Q, f=- ••• 9 = -§ = ¥ (27). Prob. VI. To determine the conditions of stability of the motion in terms of the coefficients yj g, h, which indicate the distribution of mass in the ring. The quantities which enter into the differential equation of motion (18) are R, S, ¥, r„, w^, L, M, N. We must observe that S is very large compared with R, and therefore we neglect R in those terms in which it is added to S, and we put S = aW, ^-^3(1+^), ON THE STABILITY OP THE MOTION OF SATURN's RINGS. 307 Substituting these values in equation (18) and dividing by E'a^p, we obtain (l-/>^ + (l-|/= + i/^5r)7iV + (|-6/'-i^'-iA^ + 2/V)a.^ = (28). The condition of stabiHty is that this equation shall give both values of n' negative, and this renders it necessary that all the coefficients should have the same sign, and that the square of the second should exceed four times the product of the first and third. (1) Now if we suppose the ring to be uniform, / g and h disappear, and the equation becomes n^ + nV + | = (29), which gives impossible values to n' and indicates the instability of a uniform ring. (2) If we make g and ^ = 0, we have the case of a ring thicker at one side than the other, and varying in section according to the simple law of sines. We must remember, however, that f must be less than -J, in order that the section of the ring at the thinnest part may be real. The equation becomes (l_/^)^^ + (l_|/^)nV + (f-6/>^ = (30). The condition that the third term should be positive gives /'<-375. The condition that n^ should be real gives 7 1/' - 1 1 2/= + 32 negative, which requires/^ to be between "37445 and 1'2. The condition of stability is therefore that /' should lie between •37445 and -375, but the construction of the ring on this principle requires that /' should be less than -25, so that it is impossible to reconcile this form of the ring with the conditions of stability. (3) Let us next take the case of a uniform ring, loaded with a heavy particle at a point of its circumference. We have then g=2f, h = 0, and the equation becomes {l-f^)n* + (l-^f + ^f)nW + {i-^r + ef')co^=0 (31). 39—2 308 ON THE STABILITY OF THE MOTION OE SATURN'S EINGS. Dividing each term by 1 — /, we get (l+/)7i^ + (l+/-f/>V + |{3(l+/)-8/^} 0,^ = (32). The first condition gives /less than '8279. The second condition gives f greater than •815865. Let us assume as a particular case between these limits /= "82, which makes the ratio of the mass of the particle to that of the ring as 82 to 18, then the equation becomes 1-82 ?i'+ -8114 nV+-9696cu' = (33), which gives V — l7i= ±"5916(u or ± •3076ft). These values of n indicate variations of r^, 6^, and cf)^, which are com- pounded of two simple periodic inequalities, the period of the one being 1'69 revolutions, and that of the other 3 •251 revolutions of the ring. The relations between the phases and amplitudes of these inequalities must be deduced from equations (14), (15), (16), in order that the character of the motion may be completely determined. Equations (14), (15), (16) may be written as follows: {into + Jioy') J + 2fn% +/(3 -g) oy, = (34), {n'-lco-^{Z-\-g))'^-2fo^ne,-^fho^'4>, = Q (35), -//^o,'^ + 2(l-/^)n^^, + {2(l-/>^-/=(3-5r)ft,^},^, = (36). By eliminating one of the variables between any two of these equations, we may determine the relation between the two remaining variables. Assuming one of these to be a periodic function of t of the form A cos vt, and remem- bering that n stands for the operation -^ , we may find the form of the other. Thus, eliminating 6^ between the first and second equations, {^» + ina,= (5-^)-t-W}^-i->={(3-sr)a)-Vi4<^i = (37). ON THE STABILITY OP THE MOTION OF SATURn's RINGS, 309 Assuming — = -4 sin vt, and ^ (3 - 5^) ^cos [vt -^) + ^fhoyvQ sin {vt - /3). Equating vt to 0, and to - , we get the equations {v^ - ^1^0)= (5 - g)} A =foi'Q {(3 -g)wcosfi- ^hv sin /3}, - hco' A =fco'Q {(3 -g)o)S,m/3 + ^hv cos /8}, from wliicli to determine Q and yS. In all cases in whicli tlie mass is disposed symmetrically about the diameter through the centre of gravity, ^ = and the equations may be greatly simplified. Let ^1 = P cos (vi — a), then the second equation becomes {v" + i&j' (3 + g)} ^ sin vt = 2Pfo)v sin {vt - a), whence a = 0, P = ^' + if/+9) ^ (38). 2jo)v The first equation becomes 4:Ao)v cos vt - 2P/i'' cos vt + Qf{3-g) a!' cos {vt -/3)-0, whence ^ = 0, Q = ^1:=^^A (39). In the numerical example in which a heavy particle was fixed to the cir- cumference of the ring, we have, when /= "82, v_j-5916 P_j3-21 Q^f-1-229 a," 1-3076' A \5-72' A \- 797' so that if we put cot = 0^ = the mean anomaly, ^ = ^sin(-5916^o-a)+-5sin(-3076^„-/3) (40), ^, = 3-21^ cos (-5916 ^„-a) + 5-72 5 cos (-3076 ^„-/3) (41), «^,= -1-229^ cos(-5916^„-a)-5-797Pcos (-3076 ^„-/S) ... (42). These three equations serve to determine r^, 6^ and ^, when the original motion is given. They contain four arbitrary constants A, B, a, fi. Now since 310 ON THE STABILITY OF THE MOTION OF SATUEN S RINGS. the original values r^, 6^, the angle of position measured on that plane, and t, the distance from it. The equations of motion will be dt %-'^-^ .(13). dr d4,^<^_rp dt dt ^ df~ dl=-^7^+^ If we substitute the value of C in the third equation and remember that r is nearly = 1, we find n"^S+-RJ. (14). As this expression is necessarily positive, the value of n is always real, and the disturbances normal to the plane of the ring can always be propa- 40—2 316 ON THE STABILITY OF THE MOTION OE SATURN S RINGS. gated as waves, and therefore can never be the cause of instability. We therefore confine our attention to the motion in the plane of the ring as deduced from the two former equations. Putting r-l+p and ^ = w« + s + cr, and omitting powers and products of p, (T and their differential coefficients, dt df (15). Substituting the values of p and cr as given above, these equations become (o'-S-- RK+ U + 2S--RL + 7f)A cos (ms + nt + a) p. p + {2o)n +- EM )B cos {r/is + nt + 13) = (16), H- 1 1 {2oin + -RM)A sin [ms + nt + a) + [n^ + -EN) Bsin{ms + nt + /3) = 0....{17). p p Putting for (ms + n^) any two different values, we find from the second equation (17) a=i8 (18), and {2a)n + -E3f)A + {n'+-EN)B = (19), P p and from the first (16) [or + 2S-- EL + n') A + {2om+- EM) B = (20), p p and w'-S~-EK^O (21). p ^ ' Eliminating A and B from these equations, we get n'-{Sa)'-2S + -E{L-N)}7f -Aa>~EMn + (co' + 2S--EL)-EN-~R'M'=.0 (22) p ^ p ' p p \ I' a biquadratic equation to determine n. For every real value of n there are terms in the expressions for p and cr of the form A cos {ms + nt + a). ON THE STABILITY OP THE MOTION OF SATUEN's RINGS. 317 For every pure impossible root of the form ±J — In' there are terms of the forms ^e*"'' cos {ms + a). Although the negative exponential coefficient indicates a continually diminishing displacement which is consistent with stability, the positive value which neces- sarily accompanies it indicates a continually increasing disturbance, which would completely derange the system in course of time. For every mixed root of the form ±J — In +n, there are terms of the form yle*"'* cos {ms + nt + a). If we take the positive exponential, we have a series of m waves travelling with velocity — and increasing in amplitude with the coefficient e"*""'*- The negative exponential gives us a series of m waves gradually dying away, but the negative exponential cannot exist without the possibility of the positive one having a finite coefficient, so that it is necessary for the stability of the motion that the four values of n be all real, and none of them either impossible quantities or the sums of possible and impossible quantities. We have therefore to determine the relations among the quantities K, L, M, N, R, S, that the equation n'-{S+'^R{^K+L-N)]n^ - 4a. - RMn+{^S+ - R {K- L)} ^RN-\ RM^' =U=0 may have four real roots. 7. In the first place, U is positive, when w is a large enough quantity, whether positive or negative. It is also positive when n = 0, provided S be large, as it must be, com- pared with ^RL, -RM and - RN. If we can now find a positive and a negative value of n for which U is negative, there must be four real values of n for which U^O, and the four roots will be real. 318 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. Now if we put n= ±'Ji'JS, U=-^S' + i-R {7N± 4 ^/2M- L-3K)S+\b' {KN-LN^ M% which is negative if 5^ be large compared to R. So that a ring of satellites can always be rendered stable by increasing the mass of the central body and the angular velocity of the ring. The values of L, M, and N depend on m, the number of undulations in the ring. When m = ^, the values of L and N will be at their maximum and M— 0. If we determine the relation between S and R in this case so that the system may be stable, the stability of the system for every other displacement will be secured. 8. To find the mass which must be given to the central body in order that a ring of satellites may permanently revolve round it. We have seen that when the attraction of the central body is sufficiently great compared with the forces arising from the mutual action of the satellites, a permanent ring is possible. Now the forces between the satellites depend on the manner in which the displacement of each satellite takes place. The con- ception of a perfectly arbitrary displacement of all the satellites may be rendered manageable by separating it into a number of partial displacements depending on periodic functions. The motions arising from these small displacements will take place independently, so that we have to consider only one at a time. Of aU these displacements, that which produces the greatest disturbing forces is that in which consecutive satellites are oppositely displaced, that is, when m = ^, for then the nearest satellites are displaced so as to increase as much as possible the effects of the displacement of the satellite between them. If we make ju. a large quantity, we shall have S^H^^-^^^^ = ^(l + 3- + 5- + &c.) = <(l-0518), i = ^, -5259, M=0, N^2L, /f very small ON THE STABILITY OF THE MOTION OF SATUEN's RINGS. 319 Let - RL = X, then the equation of motion will be n'-{S-x)v?-^2x{3S-x)^U=0 (23). The conditions of this equation having real roots are S>x (24), {S-xY>%x{ZS-x) (25). The last condition gives the equation iS'-26/S'a; + 9x=>0, whence 5'>26-642a;, or/S<0-351x (26). The last solution is inadmissible because S must be greater than x, so that the true condition is /S>25"649a3, > 25-649 ii2^ -5259, S>-A352fi'E (27). So that if there were 100 satellites in the ring, then 5'>4352 E is the condition which must be fulfilled in order that the motion arising from every conceivable displacement may be periodic. If this condition be not fulfilled, and if S be not sufficient to render the motion perfectly stable, then although the motion depending upon long undu- lations may remain stable, the short undulations will increase in amplitude till some of the neighbouring satellites are brought into collision. 9. To determine the nature of the motion when the system of satellites is of small mass compared with the central body. The equation for the determination of 7i is U=n'-\)t + s + (r = (ot + s — 2 --4 sin (ms + nt + a). n ^ VOL. I. 41 322 ON THE STABILITY OF THE MOTION OF SATURN S KINGS. When n is nearly equal to +& e (41), And putting m^= , we have another equation Li y4i sin aj + ^2 sin Oj + ^3 sin Oj + yli sin 04 = J5J sin e (42), ON THE STABILITY OF THE MOTION OP SATURN's RINGS. 325 Differentiating (32) with respect to t, we get two other equations -A^n^sm a-&G. = Fco8f (43), A^n^ cos a + &c. = Fsxa.f (44). Bearing in mind that B^, B^, &c. are connected with A^, A„ &c. by equa- tion (33), and that B is therefore proportional to A, we may write B^Afi, where ^= f— . ^ being thus a function of n and a known quantity. The value of o- then becomes at the epoch cr = ^^1 sin {ms + aj + &c. = 6^ cos [ms + g), from which we obtain the two equations ^^1 sin Oi + &c. = G* cos ^f (45), ^i)8i cos Oj + &c. = —G&ing (46). Differentiating with respect to t, we get the remaining equations A-^P^n^ cosai + &c. =11 cosh (47), ^^i^i sin Oj + &c. = ^sin y^ (48). We have thus found eight equations to determine the eight quantities A^, &c. and a^, &c. To solve them, we may take the four in which .^icosaj, &c. occur, and treat them as simple equations, so as to find At^goso^, &c. Then taking those in which ^isinoj, &c. occur, and determining the values of those quantities, we can easily deduce the value of A^ and aj, &c. from these. We now know the amplitude and phase of each of the four waves whose index is m. All other systems of waves belonging to any other index must be treated in the same way, and since the original disturbance, however arbitrary, can be broken up into periodic functions of the form of equations (37 — 40), our solution is perfectly general, and applicable to every possible disturbance of a ring fulfilling the condition of stability (27). 326 ON THE STABILITY OI" THE MOTION 01" SATURN S RINGS. 15. We come next to consider the effect of an external disturbing force, due either to the irregularities of the planet, the attraction of satellites, or the motion of waves in other rings. All disturbing forces of this kind may be expressed in series of which the general term is A cos {.vt + ins + a), where v is an angular velocity and m a whole number. Let P co8{ms + vt+p) be the central part of the force, acting inwards, and Q sin {ms + vt + q) the tangential part, acting forwards. Let p = A cos (ms + vt + a) and a- — Bsin [ms + vt + yS), be the terms of p and ra) ■^{if-^- RN)Bw!x{ms + vt + fi)-\-QBvii{ms + vt + q) = (50). Making ms + vt=^Q in the first equation and - in the second, {^oi''--R{2K+L) + v']A co3a + {2(ov + -RM)Bcosfi-Pcoap = (51). {2o}V + - RM) A cos a + {v' + -RN)B cos 13 + Q cos q = (52). Then if we put U' = v*-{oy' + -R{2K+L-N')}v'-4.-RMv + {3co'--R{2K+L)}-RN-\r'3P (53), fj- p. p. ON THE STABILITY OF THE MOTION OP SATURN's RINGS. 327 \^'e shall find the value of A cos a and B cos ^ ; ylcosa = ^, P cos j) + ' Jt, ^cosg- (54). 2(ov + ~RM v'+3o,'--R{K+L) BGoal3= JJ7 Pcosp ^7 Qcosq (55). Substituting sines for cosines in equations (51), (52), we may find the values of A sin a and B sin ^. ■ Now U' is precisely the same function of v that U is of n, so that if v coincides with one of the four values of n, U' will vanish, the coefiicients A and B will become infinite, and the ring will be destroyed. The disturbing force is supposed to arise from a revolving body, or an undulation of any kind V which has an angular velocity — — relatively to the ring, and therefore an absolute angular velocity = w . If then the absolute angular velocity of the disturbing body is exactly or nearly equal to the absolute angular velocity of any of the free waves of the ring, that wave will increase till the ring be destroyed. The velocities of the free waves are nearly ^fl+i\ ea + i /3-BN, CO-- Js^-RN, andwfl--") (56). When the angular velocity of the disturbing body is greater than that of the first wave, between those of the second and third, or less than that of the fourth, U' is positive. When it is between the first and second, or between the third and fourth, V is negative. Let us now simplify our conception of the disturbance by attending to the central force only, and let us put p = 0, so that P is a maximum when ms + vt is a multiple of 2ti. We find in this case a = 0, and ;8 = 0. Also v' + -RN ^- ^P (57), 2(av+-RM B= ^ P.. (58). 328 ON THE STABILITY OF THE MOTION OE SATUEN'S KINGS. When U' is positive, A will be of the same sign as P, that is, the parts of the ring wUl be furthest from the centre where the disturbing force towards the centre is greatest. When U' is negative, the contrary will be the case. When V is positive, B will be of the opposite sign to A, and the parts of the ring furthest from the centre wiU be most crowded. When v is negative, the contrary wUl be the case. Let us now attend only to the tangential force, and let us put ^ = 0, We find in this case also a = 0, ^ — 0, 2(ov+-EM A = ^ Q (59), v''+3co'--E{K+L) B= ^, Q ...(60). The tangential displacement is here in the same or in the opposite direc- tion to the tangential force, according as U' is negative or positive. The crowding of satellites is at the points farthest from or nearest to Saturn according as i) is positive or negative. 16. The effect of any disturbing force is to be determined in the following manner. The disturbing force, whether radial or tangential, acting on the ring may be conceived to vary from one satellite to another, and to be different at different times. It is therefore a perfectly arbitrary function of s and t. Let Fourier's method be applied to the general disturbing force so as to divide it up into terms depending on periodic functions of s, so that each term is of the form F {t) cos (ms + a), where the function of t is still perfectly arbitrary. But it appears from the general theory of the permanent motions of the heavenly bodies that they may all be expressed by periodic functions of t arranged in series. Let vt be the argument of one of these terms, then the corresponding term of the disturbance wiU be of the form P cos {ttis + vt + a). This term of the disturbing force indicates an alternately positive and negative action, disposed in m waves round the ring, completing its period ON THE STABILITY OF THE MOTION OF SATURn'S EINGS. 329 Ojj. relatively to each particle in the time — , and travelHng as a wave among the particles with an angular velocity , the angular velocity relative to fixed V space being of course w . The whole disturbing force may be split up into m V m terms of this kind. l7. Each of these elementary disturbances wiU produce its own wave in the ring, independent of those which belong to the ring itself. This new wave, due to external disturbance, and following different laws from the natural waves of the ring, is called the forced wave. The angular velocity of the forced wave is the same as that of the disturbing force, and its maxima and minima coin- cide with those of the force, but the extent of the disturbance and its direction depend on the comparative velocities of the forced wave and the four natural waves. When the velocity of the forced wave lies between the velocities of the two middle free waves, or is greater than that of the swiftest, or less than that of the slowest, then the radial displacement due to a radial disturbing force is in the same direction as the force, but the tangential displacement due to a tangential disturbing force is in the opposite direction to the force. The radial force therefore in this case produces a positive forced wave, and , the tangential force a negative forced ivave. When the velocity of the forced wave is either between the velocities of the first and second free waves, or between those of the third and fourth, then the radial disturbance produces a forced wave in the contrary direction to that in which it acts, or a negative wave, and the tangential force produces a positive wave. The coefficient of the forced wave changes sign whenever its velocity passes through the value of any of the velocities of the free waves, but it does so by becoming infinite, and not by vanishing, so that when the angular velocity very nearly coincides with that of a free wave, the forced wave becomes very great, and if the velocity of the disturbing force were made exactly equal to that of a free wave, the coefficient of the forced wave would become infinite. In such a case we should have to readjust our approximations, and to find whether such a coincidence might involve a physical impossibility. VOL. L 42 330 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. The forced wave which, we have just investigated is that which would main- tain itself in the ring, supposing that it had been set agoing at the commence- ment of the motion. It is in fact the form of dynamical equilibrium of the ring under the influence of the given forces. In order to find the actual motion of the ring we must combine this forced wave with all the free waves, which go on independently of it, and in this way the solution of the problem becomes perfectly complete, and we can determine the whole motion under any given initial circumstances, as we did in the case where no disturbing force acted. For instance, if the ring were perfectly uniform and circular at the instant when the disturbing force began to act, we should have to combine with the constant forced wave a system of four free waves so disposed, that at the given epoch, the displacements due to them should exactly neutralize those due to the forced wave. By the combined effect of these four free waves and the forced one the whole motion of the ring would be accounted for, beginning from its undisturbed state. The disturbances which are of most importance in the theory of Saturn's rings are those which are produced in one ring by the action of attractive forces arising from waves belonging to another ring. The effect of this kind of action is to produce in each ring, besides its own four free waves, four forced waves corresponding to the free waves of the other ring. There will thus be eight waves in each ring, and the corresponding waves in the two rings will act and react on each other, so that, strictly speak- ing, every one of the waves will be in some measure a forced wave, although the system of eight waves wUl be the free motion of the two rings taken together. The theory of the mutual disturbance and combined motion of two concentric rings of satellites requires special consideration. 18. On the motion of a ring of satellites when the conditions of stability are not fulfilled. "We have hitherto been occupied with the case of a ring of satellites, the stability of which was ensured by the smallness of mass of the satellites com- pared with that of the central body. We have seen that the statically unstable condition of each satellite between its two immediate neighbours may be com- pensated by the dynamical effect of its revolution round the planet, and a planet of sufiicient mass can not only direct the motion of such satellites round its ON THE STABILITY OF THE MOTION OF SATURn's RINGS. 331 own body, but can likewise exercise an influence over tbeir relations to each other, so as to overrule their natural tendency to crowd together, and distribute and preserve them in the form of a ring. We have traced the motion of each satellite, the general shape of the disturbed ring, and the motion of the various waves of disturbance round the ring, and determined the laws both of the natural or free waves of the ring, and of the forced waves, due to extraneous disturbing forces. We have now to consider the cases in which such a permanent motion of the ring is impossible, and to determine the mode in which a ring, originally regular, will break up, in the different cases of instability. The equation from which we deduce the conditions of stability is — + ■1.3(0'-- R {2K+L)\ ~RN - \r'M'=0. The quantity, which, in the critical cases, determines the nature of the roots of this equation, is N. The quantity M in the third term is always small compared with L and iV" when m is large, that is, in. the case of the dangerous short waves. We may therefore begin our study of the critical cases by leaving out the third term. The equation then becomes a quadratic in ri', and in order that all the values of n may be real, both values of ?i^ must be real and positive. The condition of the values of n'' being real is co' + (o'-R{AK+2L-UN) + \R'{2K+L + NY>0 (61), which shews that o^ must either be about 14 times at least smaller, or about 14 times at least greater, than quantities like - RN. That both values of n^ may be positive, we must have w' + -R{2K+L-N)>0 ^ hco'--R{2K+L)\-RN>0 (62). 42—2 332 ON THE STABILITY OF THE MOTION OE SATURN S RINGS. We must therefore take the larger value of w', and also add the condition that N be positive. We may therefore state roughly, that, to ensure stability, , the coefficient of tangential attraction, must He between zero and -^o)\ If the quantity be negative, the two s7naU values of n will become pure impossible quantities. If it exceed -^a)\ all the values of n will take the form of mixed impossible quantities. If we write x for - RN, and omit the other disturbing forces, the equation becomes U==n'-{co'-x)n' + 3a)'x^0 (63), whence n' = i{a'-x)±i>Joi*-lia)'x + x' (64). If X be small, two of the values of n are nearly +w, and the others are small quantities, real when x is positive and impossible when x is negative. 2 If X be greater than (7— y48)27-856 (277^) (78), and if L be as much as -JiV, then ft)'>25-649 (273-^) (79), so that it is not important whether we calculate the value of L or not. The condition of stability is, that the average density must not exceed a certain value. Let us ascertain the relation between the maximum density of the ring and that of the planet. Let b be the radius of the planet, that of the ring being unity, then the mass of Saturn is ^nVk' = o:y' if ¥ be the density of the planet. If we assume that the radius of the ring is twice that of the planet, as Laplace has done, then h = ^ and ¥ x = 334-2 to 307-7 (80), so that the density of the ring cannot exceed -g^ of that of the planet. Now Laplace has shewn that if the outer and inner parts of the ring have the same angular velocity, the ring will not hold together if the ratio of the density of the planet to that of the ring exceeds 1'3, so that in the first place, our ring cannot have uniform angular velocity, and in the second place, Laplace's ring cannot preserve its form, if it is composed of loose materials acting on each other only by the attraction of gravitation, and moving with the same angular velocity throughout. 23. On the forces arising from inequalities of thickness in a thin stratum of fluid of indefinite extent. The forces which act on any portion of a continuous fluid are of two kinds, the pressures of contiguous portions of fluid, and the attractions of all portions of the fluid whether near or distant. In the case of a thin stratum of fluid, not ON THE STABILITY OF THE MOTION OE SATURN's EINGS. 339 acted on by any external forces, the pressures are due mainly to the component of the attraction which is perpendicular to the plane of the stratum. It is easy to shew that a fluid acted on by such a force will tend to assume a position of equilibrium, in which its free surface is plane ; and that any irregu- larities will tend to equalise themselves, so that the plane surface will be one of stable equilibrium. It is also evident, that if we consider only that part of the attraction which is parallel to the plane of the stratum, we shall find it always directed towards the thicker parts, so that the effect of this force is to draw the fluid from thinner to thicker parts, and so to increase irregularities and destroy equilibrium. The normal attraction therefore tends to preserve the stability of equilibrium, while the tangential attraction tends to render equilibrium unstable. According to the nature of the irregularities one or other of these forces will prevail, so that if the extent of the irregularities is small, the normal forces will ensure stability, while, if the inequalities cover much space, the tangential forces will render equilibrium unstable, and break up the stratum into beads. To fix our ideas, let us conceive the irregularities of the stratum split up into the form of a number of systems of waves superposed on one another, then, by what we have just said, it appears, that very short waves will disap- pear of themselves, and be consistent with stability, while very long waves will tend to increase in height, and will destroy the form of the stratum. In order to determine the law according to which these opposite effects take place, we must subject the case to mathematical investigation. Let us suppose the fluid incompressible, and of the density Tc, and let it be originally contained between two parallel planes, at distances +c and — c from that of (xy), and extending to infinity. Let us next conceive a series of imaginary planes, parallel to the plane of {yz), to be plunged into the fluid stratum at inflnitesimal distances from one another, so as to divide the fluid into imaginary slices perpendicular to the plane of the stratum. Next let these planes be displaced parallel to the axis of x according to this law— that if x be the original distance of the plane from the origin, and f its displacement in the direction of x, $=Acos'mx (81). 43—2 340 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. According to this law of displacement, certain alterations will take place in the distances between consecutive planes; but since the fluid is incompressible, and of indefinite extent in the direction of y, the change of dimension must occur in the direction of z. The original thickness of the stratum was 2c. Let its thickness at any point after displacement be 2c + 2^, then we must have (2c + 20(l+§) = 2c (82), 7^ or ^= — c --r-=GmA sin ma; (83). Let us assume that the increase of thickness 2^ is due to an increase of t, at each surface ; this is necessary for the equilibrium of the fluid between the imaginary planes. We have now produced artificially, by means of these planes, a system of waves of longitudinal displacement whose length is — and ampHtude A ; and we have found that this has produced a system of waves of normal displace- ment on each surface, having the same length, with a height =:^cniA. In order to determine the forces arising from these displacements, we must, in the first place, determine the potential function at any point of space, and this depends partly on the state of the fluid before displacement, and partly on the displacement itself We have, in all cases — d'V d'V d'V ^ d^ + W^d^^~ ^^ ^^^^- Within the fluid, p = k; beyond it, p = 0. Before displacement, the equation is reduced to W = -4"P (85)- Instead of assuming V=0 at infinity, we shall assume F=0 at the origin, and since in this case all is symmetrical, we have within the fluid V,= - ^-nhz' ; '^ = - irtlz dz dV at the bounding planes V= - 2TrJcd' ; -j- = T iirhc dz dV beyond them V^ = 2ir^c ( T 2z + c) ; -^ = ^ AttJcc ^ ~ ' dz J .(86); ON THE STABILITY OF THE MOTION OP SATURN'S RINGS. 341 the upper sign being understood to refer to the boundary at distance +c, and the lower to the boundary at distance — c from the origin. Having ascertaiaed the potential of the tmdisturbed stratum, we find that of the disturbance by calculating the effect of a stratimi of density k and thickness ^, spread over each surface according to the law of thickness already found. By supposing the coeflS.cient A small enough, (as we may do in calcu- lating the displacements on which stability depends), we may diminish the absolute thickness indefinitely, and reduce the case to that of a mere " super- ficial density," such as is treated of in the theory of electricity. We have here, too, to regard some parts as of negative density ; but we must recollect that we are dealing with the difference between a disturbed and an undisturbed system, which may be positive or negative, though no real mass can be negative. Let us for an instant conceive only one of these surfaces to exist, and let us transfer the origin to it. Then the law of thickness is t, = mcA sin?na3 (83), and we know that the normal component of attraction at the surface is the same as if the thickness had been uniform throughout, so that on the positive side of the surface. Also, the solution of the equation d'V dyv_ da? ^ dz'~ ' consists of a series of terms of the form Ce'" sin ix. Of these the only one with which we have to do is that in which i= —m. Applying the condition as to the normal force at the surface, we get V=2Trkce-""Asuimx (87), for the potential on the positive side of the surface, and V=2TThce'^'Aammx (88), on the negative side. 342 ON THE STABILITY OF THE MOTION OF SATURN S BINGS. Calculating the potentials of a pair of such surfaces at distances +c and -c from the plane of xy, and calling V the sum of their potentials, we have for the space between these planes V' = 2irhGA sin ma;e-'"= (e™' + e"'^) ) beyond them F; = 27r^c^ sinma;e^'"'(e™ + e-'~) | the upper or lower sign of the index being taken according as z is positive or negative. These potentials must be added to those formerly obtained, to get the potential at any point after displacement. We have next to calculate the pressure of the fluid at any point, on the supposition that the imaginary planes protect each slice of the fluid from the pressure of the adjacent slices, so that it is in equilibrium under the action of the forces of attraction, and the pressure of these planes on each side. Now in a fluid of density Tc, in equilibrium under forces whose potential is V, we have always — so that if we know that the value of p is p^ where that of F is V„, then at any other point p=p, + 'k{V-V,). Now, at the free surface of the fluid, p = Q, and the distance from the free surface of the disturbed fluid to the plane of the original surface is ^, a small quantity. The attraction which acts on this stratum of fluid is, in the first place, that of the undisturbed stratum, and this is equal to d-n-^c, towards that stratum. The pressure due to this cause at the level of the original surface will be 47rFc^, and the pressure arising from the attractive forces due to the displacements upon this thin layer of fluid, will be small quantities of the second order, which we neglect. We thus find the pressure when « = c to be, _Po = ^TrMd^mA sin vrix. The potential of the undisturbed mass when z = c is and the potential of the disturbance itself for the same value of z, is F; = 2TrlcA sin mx (1 + e"'™'). ON THE STABILITY OP THE MOTION OF SATURN's RINGS. 343 So that we find the general value of p at any other point to be p = 2tt¥ (e' - z') + ^-rrtcA sin mx {2cm - 1 - e"™" + e"" (e™^ + £-»«)} ... (90). This expression gives the pressure of the fluid at any point, as depending on the state of constraint produced by the displacement of the imaginary planes. The accelerating effect of these pressures on any particle, if it were allowed to move parallel to x, instead of being confined by the planes, would be _1 djy k dx' The accelerating effect of the attractions in the same direction is dV dx ' so that the whole acceleration parallel to x is X=: —^TrhmcA cosmcc {2wc — e"^™" — 1) (91). It is to be observed, that this quantity is independent of z, so that every particle in the slice, by the combined effect of pressure and attraction, is urged with the same force, and, if the imaginary planes were removed, each slice would move parallel to itself without distortion, as long as the absolute dis- placements remained small. We have now to consider the direction of the resultant force X, and its changes of magnitude. We must remember that the original displacement is A cos mx, if therefore (2mc — e"^™*— 1) be positive, X wUl be opposed to the displacement, and the equihbrium will be stable, whereas if that quantity be negative, X will act along with the displacement and increase it, and so constitute an unstable condition. It may be seen that large values of mc give positive results and small ones negative. The sign changes when 2mc= 1-147 (92), which corresponds to a wave-length \ = 2c^ = 2c{b-m) (93). The length of the complete wave in the critical case is 5*471 times the thickness of the stratum. Waves shorter than this are stable, longer waves are unstable. 344 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. The quantity 2mc{2mc-e-'^-l), has a minimum when 2mc=*607 (94), and the wave-length is 10 "3 5 3 times the thickness of the stratum. In this case 2mc{2mG-e-'""'-l)= -'509 (95), and X=-509nkAcosmx (96). 24. Let us now conceive that the stratum of fluid, instead of being infinite in extent, is limited in breadth to about 100 times the thickness. The pressures and attractions will not be much altered by this removal of a distant part of the stratum. Let us also suppose that this thin but broad strip is bent round in its own plane into a circular ring whose radius is more than ten times the breadth of the strip, and that the waves, instead of being exactly parallel to each other, have their ridges in the direction of radii of the ring. We shall then have transformed our stratum into one of Saturn's Rings, if we suppose those rings to be liquid, and that a considerable breadth of the ring has the same angular velocity. Let us now investigate the conditions of stability by putting x= — 2'rrkmc [2mc — e"^" — 1) into the equation for n. We know that x must lie between and — — to 1 o y ensure stabHity. Now the greatest value of x in the fluid stratum is '50977^. Taking Laplace's ratio of the diameter of the ring to that of the planet, this gives 42 '5 as the minimum value of the density of the planet divided by that of the fluid of the ring. Now Laplace has shewn that any value of this ratio greater than 1'3 is inconsistent with the rotation of any considerable breadth of the fluid at the same angular velocity, so that our hypothesis of a broad ring with uniform velocity is untenable. But the stability of such a ring is impossible for another reason, namely, that for waves in which 2mc> 1-147, x is negative, and the ring will be destroyed by these short waves in the manner described at page (333). When the fluid ring is treated, not as a broad strip, but as a filament of circular or elliptic section, the mathematical difficulties are very much increased, ON THE STABILITY OP THE MOTION OF SATURN's RINGS. 345 but it may be shown that in this case also there will be a maximum value of X, which will requu-e the density of the planet to be several times that of the ring, and that in all cases short waves will give rise to negative values of X, inconsistent with the stability of the ring. It appears, therefore, that a ring composed of a continuous liquid mass cannot revolve about a central body without being broken up, but that the parts of such a broken ring may, under certain conditions, form a permanent ring of satellites. On the Mutual Perturbations of Two Rings. 25. We shall assume that the difference of the mean radii of the rings is small compared with the radii themselves, but large compared with the distance of consecutive satellites of the same ring. We shall also assume that each ring separately satisfies the conditions of stabihty. We have seen that the effect of a disturbing force on a ring is to produce a series of waves whose number and period correspond with those of the dis- turbing force which produces them, so that we have only to calculate the coefficient belonging to the wave from that of the disturbing force. Hence in investigating the simultaneous motions of two rings, we may assume that the mutually disturbing waves travel with the same absolute angular velocity, and that a maximum in one corresponds either to a maximum or a minimum of the other, according as the coefficients have the same or opposite signs. Since the motions of the particles of each ring are affected by the disturbance of the other ring, as well as of that to which they belong, the equations of motion of the two rings will be involved in each other, and the final equation for determining the wave-velocity will have eight roots instead of four. But as each of the rings has four free waves, we may suppose these to originate forced waves in the other ring, so that we may consider the eight waves of each ring as consisting of four free waves and four forced ones. In strictness, however, the wave-velocity of the "free" waves will be affected by the existence of the forced waves which they produce in the other ring, so that none of the waves are really " free " in either ring independently, though the whole motion of the system of two rings as a whole is free. VOL. L 44 346 ON THE STABILITY OP THE MOTION OF SATUEN's RINGS, We shall find, however, that it is best to consider the waves first as free, and then to determine the reaction of the other ring upon them, which is such as to alter the wave-velocity of both, as we shall see. The forces due to the second ring may be separated into three parts. 1st. The constant attraction when both rings are at rest. 2nd. The variation of the attraction on the first ring, due to its own disturbances. 3rd. The variation of the attraction due to the disturbances of the second ring. The first of these affects only the angular velocity. The second affects the waves of each ring independently, and the mutual action of the waves depends entirely on the third class of forces. 26. To determine the attractions between two rings. Let R and a be the mass and radius of the exterior ring, R' and a' those of the interior, and let all quantities belonging to the interior ring be marked with accented letters. (Fig. 5.) 1st. Attraction between the rings when at rest. Since the rings are at a distance small compared with their radii, we may calculate the attraction on a particle of the first ring as if the second were an infinite straight line at distance a' — a from the first. The mass of unit of length of the second ring is - — > , and the accelerating effect of the attraction of such a filament on an element of the first rina" is o 7?' — 77 A inwards (97), TTa{a — a) ^ ' The attraction of the first ring on the second may be found by transposing accented and unaccented letters. In consequence of these forces, the outer ring will revolve faster, and the inner ring slower than would otherwise be the case. These forces enter into the constant terms of the equations of motion, and may be included in the value of K. ON THE STABILITY OF THE MOTION OP SATUEN's RINGS. 347 2nd. Variation due to disturbance of first ring. If we put a(l+p) for a in the last expression, we get the attraction when the first ring is displaced. The part depending on p is - rra'^a-aj P ^™"^^ (9«)- This is the only variation of force arising from the displacement of the first ring. It affects the value of L in the equations of motion. 3rd. Variation due to waves in the second ring. On account of the waves, the second ring varies in distance from the first, and also in mass of unit of length, and each of these alterations produces variations both in the radial and tangential force, so that there are four things to be calculated : 1st. Eadial force due to radial displacement. 2nd. Eadial force due to tangential displacement, 3rd. Tangential force due to radial displacement. 4th. Tangential force due to tangential displacement. 1st. Put a'{l+p') for a', and we get the term in p' ■ — 7 V-7 \i p' inwards = \'p', say (99). Tra {a —af '^ r ./ 2nd. By the tangential displacement of the second ring the section is da ds' reduced in the proportion of 1 to 1 — -r, , and therefore there is an alteration of the radial force equal to inwards = —p! -j-, say (100). Tra' {a — a') ds' ds' 3rd. By the radial displacement of the second ring the direction of the filament near the part in question is altered, so that the attraction is no longer radial but forwards^ and the tangential part of the force is R 7x^'=+/^'^'f«^^^^ds (101). ira {a-a) ds ^^ ds 44—2 348 ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 4th. By the tangential displacement of the second ring a tangential force arises, depending on the relation between the length of the waves and the distance between the rings. a — a r+"a;sinpa; , _ If we make m — >- =p, and m j ax - n, TV the tangential force is -na ia-a'f ^°"' = ''''^' ^^^^^" We may now write down the values of X. fi, and v by transposing accented and unaccented letters. Comparing these values with those of \', /, and /, it will be seen that the following relations are approximately true when a is nearly equal to a' : ^'^ _/i = !^ = :^* (104). X II' V Ra 27. To form the equations of motion. *The original equations were Putting p = ^ cos {ms + nt), cr = 5 sin (ms + nt), p = A' cos (ms + nt), :mB = Q\ .^^^. {2oyn+M)A + (n' + N)B-i,:mA' + vB' = Q] ^ '' The corresponding equations for the second ring may be found by trans- posing accented and unaccented letters. We should then have four equations to determine the ratios of A, B, A', B', and a resultant equation of the eighth degree to determine n. But we may make use of a more convenient method, since X', /*', and v are small. Eliminating B we find An' -A{oi' + IK+L -N)n'- iAmMn + AN (3ft>^)\ _ {-\'A' + fi'mB')n' + {fi'mA'-v'B') 2conj~^ ^^'^' * [The analysis in this article is somewhat unsatisfactory, the equations of motion employed being those which were applicable in the case of a riiig of radiusi unity. Ed.] ON THE STABILITY OF THE MOTION OF SATUEN's EIN6S. 349 Putting B = ^A, A' = xA, R=^'A' = ^xA, we have n*-{(x^^{ + 2K) + L-N]n^-A(aMn + Z(>i'N\_J^_ ^^4w'-2G>=n + &c (108), -^= -X'n" + iJi'm^n' + 211711(071-21^' fi'con (109), T dn X'n — u'mB'n — 2u,'m&) + 2v'0' = <"——> and q = — -j- Similarly in the second ring we should have -'=/-2'^ (114); X and since the corresponding waves in the two rings must have the same abso- lute angular velocity, w = '!a-', or p-qx=p' -q' - (115)- 350 ON THE STABILITY OP THE MOTION OE SATURN S RINGS. This is a quadratic equation in x, the roots of which are real when {p-y)'+4g'2' is positive. "When this condition is not fulfilled, the roots are impossible, and the general solution of the equations of motion will contain exponential factors, indicating destructive oscillations in the rings. Since q and g' are small quantities, the solution is always real whenever jp and J)' are considerably different. The absolute angular velocities of the two pairs of reacting waves, are then nearly p -\ — ^^, , and »' — i^, , ^ p—p ^ p—p instead of p and p', as they would have been if there had been no reaction of the forced wave upon the free wave which produces it. When p and p' axe equal or nearly equal, the character of the solution will depend on the sign of qc[. "We must therefore determine the signs of q and q' in such cases. Putting ^' = —r, we may write the values of q and q' 7v ?=;^ \-ir2iim -^-- -4z/'- - n ' \n 11/ n n m" in^ — 2oy^ . (Ji ft) \ -WW , A + 2iim 7 — 4^ — - n ' \n n n n (116). ^ m'" An"-2(o" Eeferring to the values of the disturbing forces, we find that \' _ /*' _ ^' _ ^'* \ fx, V Ha' Hence q^n_^n--2a/^ Ra q n' in'-2oi' ' Ra' ^^^''• Since qq' is of the same sign as —, , we have only to determine whether ,,2 '2 2?^ — — , and 2n'~ — , are of the same or of different signs. If these quantities are of the same sign, qq' is positive, if of different signs, qq is negative. ON THE STABILITY OF THE MOTION OF SATUKN's RINGS. 351 Now there are four values of n, whicli give four corresponding values of In n O) «i= —CD + Scc, 2^1 is negative, 2 n^= — a small quantity, 2n^ is positive, .2 «s = + a small quantity, 2^3 is negative, 2 «4 = a) — &c., 2^4 is positive. The quantity with which we have to do is therefore positive for the even orders of waves and negative for the odd ones, and the corresponding quantity in the other ring obeys the same law. Hence when the waves which act upon each other are either both of even or both of odd names, qq' will be positive, but when one belongs to an even series, and the other to an odd series, qq' is negative. 29. The values of p and p' are, roughly, D.= w • + &c. - (118). i>i = W +^-&C., p,= 0} + &C., p, = (0-&,C., p,= o> - — + &C, i>i' = '>2ft), but cu'<36), no such coincidence is possible. For p^ is always less than p^, it is greater than Pi when m = 1 or 2, and less than Pi when m is 3 or a greater number. There are of course an infinite number of ways in which this noncoincidence might be secured, but it is plain that if a number of concentric rings were placed at small intervals from each other, such coincidences must occur accurately or approximately between some pairs of rings, and if the value of {p—p'Y is brought lower than ~Aqq, there will be destructive interference. This investigation is applicable to any number of concentric rings, for, by the principle of superposition of small displacements, the reciprocal actions of any pair of rings are independent of aU the rest. 31. On the effect of long-continued disturbances on a system of rings. The result of our previous investigations has been to point out several ways in which disturbances may accumulate till collisions of the difierent par- ticles of the rings take place. After such a collision the particles will still continue to revolve about the planet, but there will be a loss of energy in the system during the collision which can never be restored. Such collisions however will not affect what is called the Angular Momentum of the system about the planet, which will therefore remain constant. ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 353 Let M be the mass of the system of rings, and Sm that of one ring whose radius is ?-, and angular velocity o} = S^r-^. The angular momentum of the ring is half its vis viva is -^wV^Sm = ^Sr~^ Sm. The potential energy due to Saturn's attraction on the ring is — Sr'^Sm. The angular momentum of the whole system is invariable, and is S^%{r^Bm) = A (119). The whole energy of the system is the sum of half the vis viva and the potential energy, and is -iSt{r-'Sm) = JS (120). A is invariable, while U necessarily diminishes. We shall find that as E diminishes, the distribution of the rings must be altered, some of the outer rings moving outwards, while the inner rings move inwards, so as either to spread out the whole system more, both on the outer and on the inner edge of the system, or, without affecting the extreme rings, to diminish the density or number of the rings at the mean distance, and increase it at or near the inner and outer edges. Let us put X = r^, then A = 8^"% (xdm) is constant. Now let x^= J^,j J , S {dm) and aj = »j + x\ then we may write -^ = t{r-'Sm)=t{x-'dm), S = tdvi{xr-2--,+ S^,-kc.), = ^t{dm)-^,t{x'dm) + §,t{x%n)-ko (121). Xj J(^i *^i l,(^x'dm)-&c [i.2 VOL. I. 45 354 ON THE STABILITY OF THE MOTION OF SATUEN'S RINGS. Now t{dm) = M a constant, t{xdm) = 0, and t{x"Bm) is a quantity which increases when the rings are spread out from the mean distance either way, x' being subject only to the restriction t {xdm) = 0. But t {x"dm) may increase without the extreme values of x' being increased, provided some other values be increased. 32. In fact, if we consider the very innermost particle as moving in an elHpse, and at the further apse of its orbit encountering another particle belonging to a larger orbit, we know that' the second particle, when at the same distance from the planet, moves the faster. The result is, that the interior satellite will receive a forward impulse at its further apse, and will move in a larger and less eccentric orbit than before. In the same way one of the outermost particles may receive a backward impulse at its nearer apse, and so be made to move in a smaller and less eccentric orbit than before. When we come to deal with collisions among bodies of unknown number, size, and shape, we can no longer trace the mathematical laws of their motion with any distinctness. All we can now do is to collect the results of our investi- gations and to make the best use we can of them in forming an opinion as to the constitution of the actual rings of Saturn which are still in existence and apparently in steady motion, whatever catastrophes may be indicated by the various theories we have attempted. 33. To find the Loss of Energy due to internal friction in a broad Fluid Ring, the parts of which revolve about the Planet, each with the velocity of a satellite at the same distance. Conceive a fluid, the particles of which move parallel to the axis of x with a velocity u, u being a function of z, then there will be a tangential pres- sure on a plane parallel to xy du ., „ = ju,--T- on unit 01 area due to the relative sliding of the parts of the fluid over each other. In the case of the ring we have (o = S^r'^. The absolute velocity of any particle is wr. That of a particle at distance (r + Sr) is cur + -7- (cor) 8r, ON THE STABILITY OF THE MOTION OP SATUKN's RINGS. 355 If the angular velocity had been uniform, there would have been no sliding, and the velocity would have been (or + ciiBr. The sliding is therefore d(t) tv and the friction on unit of area perpendicular to r is jjlv -j- . The loss of Energy, per unit of area, is the product of the sliding by the friction, do) or, ixr dr Br in unit of time. The loss of Energy in a part of the King whose radius is r, breadth 8r, and thickness c, is Sr. In the case before us it is ^vfiScr'^ Sr. If the thickness of the ring is uniform between r = a and 7' = b, the whole loss of Energy is in unit of time. Now half the vis viva of an elementary ring is irpcrSr r^co^ = irpcSBr, and this between the limits r = a and r = h gives irpcS {a — b). The potential due to the attraction of S is twice this quantity with the sign changed, so that E= —irpcS {a — b), ^^^ -^ = ^''''^{1-1)' 1 dE_ _o /f .i E dt~ ^ p ab' 45—2 356 ON THE STABILITY OF THE MOTION OF SATURN S EINGS. Now Professor Stokes finds a/^ = 0-0564 for water, ^ P and =0"116 for air, taking the unit of space one English, inch, and the unit of time one second. We may take a=88,209 miles, and & = 77,636 for the ring A; and a = 75,845, and 6 = 58,660 for the ring B. We may also take one year as the unit of time. The quantity representing the ratio of the loss of energy in a year to the whole energy is El[t= 60,880,000,000,000 ^""^ ^^^ """^ ^' ^^^ 39,540,000,000,000 ^"" *^" ™^ ^' showing that the effect of internal friction in a ring of water moving with steady motion is inappreciably small. It cannot be from this cause therefore that any decay can take place in the motion of the ring, provided that no waves arise to disturb the motion. Recapitulation of the Theory of the Motion of a Rigid Ring. The position of the ring relative to Saturn at any given instant is defined by three variable quantities. 1st. The distance between the centre of gravity of Saturn and the centre of gravity of the ring. This distance we denote by r. 2nd. The angle which the line r makes with a fixed line in the plane of the motion of the ring. This angle is called 6. 3rd. The angle between the line r and a line fixed with respect to the ring so that it coincides with r when the ring is in its mean position. This is the angle ^. The values of these three quantities determine the position of the ring so far as its motion in its own plane is concerned. They may be referred to as the radius vector, longitude, and angle of libration of the ring. The forces which act between the ring and the planet depend entirely upon their relative positions. The method adopted above consists in determining the ON THE STABILITY OF THE MOTION OF SATURn's RINGS. 357 potential (F) of the ring at the centre of the planet in terms of r and (ft. Then the work done by any displacement of the system is measured by the change of VS during that displacement. The attraction between the centre of gravity dV of the Ring and that of the planet is -S-~j-, and the moment of the couple dV tending to turn the ring about its centre of gravity is S-rj. It is proved in Problem V, that if a be the radius of a circular ring, r, = af the distance of its centre of gravity from the centre of the circle, and R the mass of the ring, then, at the centre of the ring, -y- = :/, -tt = 0. civ a/ dm d^V 7? It also appears that -y-T = i~i (1 +S'). "which is positive when ^ > — 1, d^V R and that Tj-^ =i— /^ (3— g'), which is positive when g<3. d^'V . . . If -pr is positive, then the attraction between the centres decreases as the distance increases, so that, if the two centres were kept at rest at a given d^V . distance by a constant force, the equilibrium would be unstable. If -t-tz is positive, then the forces tend to increase the angle of libration, in whichever direction the libration takes place, so that if the ring were fixed by an axis through its centre of gravity, its equilibrium round that axis would be unstable. In the case of the uniform ring with a heavy particle on its circumference whose weight = '82 of the whole, the direction of the whole attractive force of the ring near the centre will pass through a point lying in the same radius as the centre of gravity, but at a distance from the centre = |^ a. (Fig. 6.) If we call this point O, the line SO will indicate the direction and position of the force acting on the ring, which we may call F. It is evident that the force F, acting on the ring in the line OS, will tend to turn it round its centre of gravity R and to increase the angle of libration KRO. The direct action of this force can never reduce the angle of libration to zero again. To understand the indirect action of the force, we must recollect that the centre of gravity [R) of the ring is revolving about Saturn in the direction of the arrows, and that the ring is revolving about its centre of gravity 358 ON THE STABILITY 0¥ THE MOTION OF SATURN S EINGS, with nearly the same velocity. If the angular velocity of the centre of gravity about Saturn were always equal to the rotatory velocity of the ring, there would be no Hbration. Now suppose that the angle of rotation of the ring is in advance of the longitude of its centre of gravity, so that the line RO has got in advance of SRK by the angle of libration KRO. The attraction between the planet and the ring is a force F acting in SO. We resolve this force into a couple, whose moment is F'RN, and a force F acting through R the centre of gravity of the ring. The couple affects the rotation of the ring, but not the position of its centre of gravity, and the force RF acts on the centre of gravity without affecting the rotation. Now the couple, in the case represented in the figure, acts in the positive direction, so as to increase the angular velocity of the ring, which was already greater than the velocity of revolution of R about S, so that the angle of libration would increase, and never be reduced to zero. The force RF does not act in the direction of S, but behind it, so that it becomes a retarding force acting upon the centre of gravity of the ring. Now the effect of a retarding force is to cause the distance of the revolving body to decrease and the angular velocity to increase, so that a retarding force increases the angular velocity of R about S. The effect of the attraction along SO in the case of the figure is, first, to increase the rate of rotation of the ring round R, and secondly, to increase the angular velocity of R about S. If the second effect is greater than the first, then, although the line RO increases its angular velocity, SR will increase its angular velocity more, and will overtake RO, and restore the ring to its original position, so that SRO wUl be made a straight line as at first. If this accelerat- ing effect is not greater than the acceleration of rotation about R due to the couple, then no compensation will take place, and the motion will be essentially unstable. If in the figure we had drawn ^ negative instead of positive, then the couple would have been negative, the tangential force on R accelerative, r would have increased, and in the cases of stability the retardation of 9 would be greater than that of {0 + (j>), and the normal position would be restored, as before. ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 359 The object of the investigation is to find the conditions under which this compensation is possible. It is evident that when SRO becomes straight, there is still a difference of angular velocities between the rotation of the ring and the revolution of the centre of gravity, so that there will be an oscillation on the other side, and the motion wUl proceed by alternate oscillations without limit. If we begin with r at its mean value, and negative, then the rotation of the ring will be retarded, r will be increased, the revolution of r wiU be more retarded, and thus )[§)-^o,(a + ^r + Y + k'^} = H. If now we suppose ^ and t] to be infinitely «mall, the last of these equations becomes (a^ + F)^ + 2a,a?+a J=0 (a). 376 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. If p and q denote the components parallel and perpendicular to OG of tlie attraction' of the body on a unit of matter at 8, we have X=pcos- qsm=p(p + q, since q and <^ are each infinitely small ; and if we put V= potential at 8, and " df ' ^ dv" "^ d^dv ■■ then p =/- ct^ -jn, q = -^r)- y^, X =/- a^ - yrj, Y=f(j) -^n- it If we make these substitutions for X and Y, and take into account that M /=co^c.| C&), the first and second equations of motion become S-^-l-«*f^^-«f-|("f+")=» <* §+2»|-"''+"^-|(^'+'f)='' w- Combining equations (a), (c), and {d), by the same method as that adopted in the text, we find that the differential equation in ^, 17, or <^, is of the form . d*u , _d^M „ where A = F, 0=0,' {k^ - 3a^) + 0^^^ {(«>= + k') (a + /3) - 4a^/3} + {a^ + F)|^, («^ _ y). In comparing this result with that obtained in the Essay, we must put r„ for a, R for M, B+8 for 8, L for a, iVr-/ for ^, ilfr„ for 7. VOL. I. PLATE V. To face page 376 Cambridge Vniverdty Press. [From the Philosophical Magazine for January and July, I860.] XX. Illustrations of the Dynamical Theory of Gases* - PART I. On the Motions and Collisions of Perfectly Elastic Spheres. So many of the properties of matter, especially when in the gaseous form, can be deduced from the hypothesis that their minute parts are in rapid motion, the velocity increasing with the temperature, that the precise nature of this motion becomes a subject of rational curiosity. Daniel BemouiLli, Herapath, Joule, Kronig, Clausius, &c. have shewn that the relations between pressure, temperature, and density in a perfect gas can be explained by supposing the particles to move with uniform velocity ia straight lines, striking against the sides of the containing vessel and thus producing pressure. It is not necessary to suppose each particle to travel to any great distance in the same straight line ; for the effect in producing pressure will be the same if the particles strike against each other ; so that the straight line described may be very short. M. Clausius has determined the mean length of path in terms of the average distance of the particles, and the distance between the centres of two particles when coUision takes place. We have at present no means of ascertaining either of these distances ; but certain phenomena, such as the internal friction of gases, the conduction of heat through a gas, and the diffusion of one gas through another, seem to indicate the possibility of determining accurately the mean length of path which a particle describes between two successive collisions. (_ In order to lay the foundation of such investigations on strict mechanical principles, I shall demonstrate the laws of motion of an indefinite number of small, hard, and perfectly elastic spheres acting on one another only during impact. * Eead at the Meeting of the British Association at Aberdeen, September 21, 1859. VOL. I. \ '^8 378 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. If the properties of such a system of bodies are found to correspond to those of gases, an important physical analogy will be established, which may lead to more accurate knowledge of the properties of matter. If experiments on gases are inconsistent with the hypothesis of these propositions, then our theory, though consistent with itself, is proved to be incapable of explaining the phenomena of gases. In either case it is necessary to follow out the consequences of the hypothesis.'^ Instead of saying that the particles are hard, spherical, and elastic, we may if we please say that the particles are centres of foi'ce, of which the action is insensible except at a certain small distance, when it suddenly appears as a repulsive force of very great intensity. It is evident that either assumption will lead to the same results. For the sake of avoiding the repetition of a long phrase about these repulsive forces, I shall proceed upon the assumption of perfectly elastic spherical bodies. If we suppose those aggregate molecules Avhich move together to have a bounding surface which is not spherical, then the rotatory motion of the system will store up a certain proportion of the whole vis viva, as has been shewn by Clausius, and in this way we may account for the value of the specific heat being greater than on the more simple hypothesis. On the Jlotion and Collision of Perfectly Elastic Spheres. Prop. I. Two spheres moving in opposite directions with velocities inversely as their masses strike one another ; to determine their motions after impact. Let F and Q be the position of the centres at impact ; AP, BQ the directions and magnitudes of the velocities before impact ; Pa, Qh the same after impact ; then, resolving the velocities parallel and per- pendicular to PQ the line of centres, we find that the velocities parallel to the line of centres are exactly I'eversed, while those perpendicular to that line are unchanged. Compounding these velocities again, we find that the velocity of each ball is the same before and after impact, and that the directions before and after impact lie in the same plane with the Hne of centres, and make equal angles with it. Illustrations of the dynamical theory of gases. 379 Prop. II. To find the probability of the direction of the velocity after impact lying between given limits. In order that a collision may take place, the line of motion of one of the balls must pass the centre of the other at a distance less than the sum of their radii; that is, it must pass through a circle whose centre is that of the other ball, and radius (s) the sum of the radii of the balls. Within this circle every position is equally probable, and therefore the probability of the distance from the centre being between r and r + dr is 2rdr Now let (f> be the angle APa between the original direction and the direction after impact, then APN^^tp, and r = s sin ^(f>, and the probability becomes ■J sin d(f). The area of a spherical zone between the angles of polar distance (j) and + d(f) is 273- sio (jidij) ; therefore if w be any small area on the surface of a sphere, radius unity, the probability of the direction of rebound passing through this area is Oi 4:TT ' so that the probability is independent of {'^) = C'e'''\ If we make A positive, the number of particles will increase with the velocity, and we should find the whole number of particles infinite. We there- fore make A negative and equal to — ^ , so that the number between x and x + dx is NCe~<^'dx. Integrating from a;=— oo toa;=+co,we find the whole number of particles, 1 -5' f{x) is therefore 7=e ""'. Whence we may draw the following conclusions : — 1st. The number of particles whose velocity, resolved in a certain direction, lies between x and x + dx is N^e'^^'dx (1). aVTr 2nd. The number whose actual velocity lies between v and v + dv is N-rrv'e''^'dv (2). a Vtt 3rd. To find the mean value of v, add the velocities of all the particles together and divide by the number of particles ; the result is mean velocity = 7= (3)- 4th, To find the mean value of v\ add all the values together and divide by iV, mean value of 'y' = |a' (4), This is greater than the square of the mean velocity, as it ought to be. 382 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. It appears from this proposition that the velocities are distributed among the particles according to the same law as the errors are distributed among the observations in the theory of the " method of least squares." The velocities range from to oo , but the number of those having great velocities is com- paratively small. In addition to these velocities, which are in all directions equally, there may be a general motion of translation of the entire system of particles which must be compounded with the motion of the particles relatively to one another. We may call the one the motion of translation, and the other the motion of agitation. Prop. V. Two systems of particles move each according to the law stated in Prop. IV. ; to find the number of pairs of particles, one of each system, whose relative velocity lies between given limits. Let there be N particles of the first system, and N' of the second, then NN' is the whole number of such pairs. Let us consider the velocities in the direction of x only; then by Prop. IV. the number of the first kind, whose velocities are between x and x + dx, is 1 -^ iV — j=e '■■' dx. The number of the second kind, whose velocity is between x + y and x + y + dy, is N'-^e' I'' dy, where ^ is the value of a for the second system. The number of pairs which fulfil both conditions is NN' -y^e'^--' P' ' dxdy. Now x may have any value from - oo to + oo consistently with the difference of velocities being between y and y + dy; therefore integrating between these limits, we find ''''' jTwr.''^' '" <^) for the whole number of pairs whose difference of velocity lies between y and y + dy. ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 383 This expression, wbich. is of the same form with (1) if we put NN' for N, a^ + /8^ for a^ and y for x, shews that the distribution of relative velocities is regulated by the same law as that of the velocities themselves, and that the mean relative velocity is the square root of the sum of the squares of the mean velocities of the two systems. Since the direction of motion of every particle in one of the systems may be reversed without changing the distribution of velocities, it follows that the velocities compounded of the velocities of two particles, one in each system, are distributed according to the same formula (5) as the relative velocities. Prop. VI. Two systems of particles move in the same vessel ; to prove that the mean vis viva of each particle will become the same in the two systems. Let P be the mass of each particle of the first system, Q that of each particle of the second. Let p, q be the mean veloci- ties in the two systems before impact, and let p', q' be the mean velocities after one impact. Let OA = p and OB = q, and let A OB be a right angle ; then, by Prop, v., AB wiU be the mean relative velocity, OG will be the mean velocity of the centre of gravity ; and drawing aGh at right angles to OG, and making aG — AG and hG = BG, then Oa will be the mean velocity of P after impact, compounded oi OG and Ga, and Oh will be that of Q after impact. Jp" + q\ Now AB = Jf + ^^, AG = jP^Jf + ^, BG=^^ OG: JPY + Q v. 3 V 3 m If v = M these expressions coincide. Clausius in applying this result and putting u, v for the mean velocities assumes that the mean relative velocity is given by expressions of the same form, so that when the mean velocities are each equal to u the mean relative velocity would be, ^u. This step is, however, open to objection, and in fact if we take the expressions given above for the mean velocity, treating u and v as the velocities of two particles which may have any values between and 00 , to calculate the mean relative velocity we should proceed as follows : Since the number of particles with velocities between u and u + du is iV ^ / u'e '^' du, the mean relative velocity is a vt ao3- / '*»'« ^«"Vmm + s — )dudv+-T^r^] juVe \a'^p/ {v+-^~)dudv. " ~ Jo Jv \ 3 uj a'ji'-n- Jo Jo \ 3 v/ 16 2 This expression, when reduced, leads to -j= •Ja' + 13', which is the result in the text. Ed.] VTT 49—2 388 ILLUSTRATIONS 01" THE DYNAMICAL THEORY OF GASES. The probability of not being struck by a particle of the other kind in the same distance is e "' Therefore the probability of not being struck by any particle before reaching a distance x is and if I., be the mean distance for a" particle of the first kind, ^^-y27r5,W, + ,7^1 + Js'W, '.(12). Similarly, if \ be the mean distance for a particle of the second kind, j=j27TS,W, + w Jl+'^,s"N, (13). The mean density of the particles of the first kind is N,M^ = p^, and that of the second NJ!d^ = p^. If we put ^W2^, i'-V^.^. C-7^£ D^S% (U). J =Ap, + Bp„ j = Cp, + Dp, (15), and B M^^vJ C M,v,~v,' ^^^f- Prop. XII. To find the pressure on unit of area of the side of .the vessel due to the impact of the particles upon it. Let -^= number of particles in unit of volume; M= mass of each particle ; V = velocity of each particle ; I = mean path of each particle ; then the number of particles in unit of area of a stratum dz thick is ^dz (17). The number of colUsions of these particles in unit of time is ^^2 7 (18). ILLUSTEATIONS OF THE DYNAMICAL THEORY OE GASES. 389 The number of particles which after collision reach a distance between nl and {n + dn) I is Nje-^'dzdn (19). The proportion of these which strike on unit of area at distance z is nl — z , . • , -2nr ■ (20)' the mean velocity of these in the direction of z is nl + z , , ^-^ (21). Multiplying together (19), (20), and (21), and M, we find the momentum at impact MN ^3 {nH' - z') e-« dz dn. Integrating with respect to z from to nl, we get iMNv" we~" dn. Integrating with respect to n from to oo , we get for the momentum in the direction of z of the striking particles ; for the momentum of the particles after impact is the same, but in the opposite direction ; so that the whole pressure on unit of area is twice this quantity, or This value of p is independent of I the length of path. In applying this result to the theory of gases, we put MN— p, and v^ = 3^, and then p = hp, which is Boyle and Mariotte's law. By (4) we have if^ia\ .: a'^2k (23). We have seen that, on the hypothesis of elastic particles moving in straight lines, the pressure of a gas can be explained by the assumption that the square of the .velocity is proportional directly to the absolute temperature, and inversely to the specific gravity of the gas at constant temperature, so that at the same 390 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. pressure and temperature the value of NMif is the same for all gases. But we found in Prop. VI. that when two sets of particles communicate agitation to one another, the value of Mif is the same in each. From this it appears that N, the number of particles in unit of volume, is the same for all gases at the same pressure and temperature. This result agrees with the chemical law, that equal volumes of gases are chemically equivalent. We have next to determine the value of I, the mean length of the path of a particle between consecutive collisions. The most direct method of doing this depends upon the fact, that when different strata of a gas slide upon one another with different velocities, they act upon one another with a tan- gential force tending to prevent this sliding, and similar in its results to the friction between two solid surfaces sliding over each other in the same way. The explanation of gaseous friction, according to our hypothesis, is, that particles having the mean velocity of translation belonging to one layer of the gas, pass out of it into another layer having a different velocity of translation ; and by striking against the particles of the second layer, exert upon it a tangential force which constitutes the internal friction of the gas. The whole friction between two portions of gas separated by a plane surface, depends upon the total action between all the layers on the one side of that surface upon all the layers on the other side. Prop. XIII. To find the internal friction in a system of moving particles. Let the system be divided into layers parallel to the plane of xy, and let the motion of translation of each layer be u in the direction of x, and let u = A+ Bz. We have to consider the mutual action between the layers on the positive and negative sides of the plane xy. Let us first determine the action between two layers dz and dz', at distances z and —z' on opposite sides of this plane, each unit of area. The number of particles which, starting frohi dz in unit of time, reach a distance between nl and {n-\-dn)l is by (19), N -J e~" dz dn. The number of these which have the ends of their paths in the layer dz' is V N —t; e~" dz dz' dn. 2nr The mean velocity in the direction of x which each of these has before impact is A+ Bz, and after impact A + Bz' ; and its mass ia M, so that a mean ILLUSTEATIONS OF THE DYNAMICAL THEORY OF GASES, 391 momentum =MB{z-z') is communicated by eacli particle. The whole action due to these collisions is therefore NMB ~ {z - z') e-» dz dz' dn. We must first integrate with respect to z' between z' = and z' = z-nl; this gives INMB 2^ {nH' - z') e-" dz dn for the action between the layer dz and all the layers below the plane xy. Then integrate from z = to z = nl, ^MNBlvTv'e-"' dn. Integrate from ?i = to w = co , and we find the whole friction between unit of area above and below the plane to be where /x is the ordinary coefficient of internal friction, ''"^'■'''"iTl ^ <'''• where p is the density, I the mean length of path of a particle, and v the 1 ., 2a „ lik mean velocity v = —r= = 2 / — , '=^?Vs (^^)- Now Professor Stokes finds by experiments on air, 7; ^ = •116. P If we suppose \/A; = 930 feet per second for air at 60°, and therefore the mean velocity ^ = 1505 feet per second, then the value of I, the mean distance travelled over by a particle between consecutive collisions, =43:7^5-0^^^ ^^ ^°- inch, and each particle makes 8,077,200,000 collisions per second. A remarkable result here presented to us in equation (24), is that if this explanation of gaseous friction be true, the coefficient of friction is independent of the density. Such a consequence of a mathematical theory is very startling, and the only experiment I have met with on the subject does not seem to confirm it. We must next compare our theory with what is known of the diffiision of gases, and the conduction of heat through a gas. 392 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. PART II. * On THE Process of Diffusion of two or more kinds of moving particles AMONG ONE ANOTHER. We have shewn, in the first part of this paper, that the motions of a system of many small elastic particles are of two kinds : one, a general motion of translation of the whole system, which may be caUed the motion in mass; and the other a motion of agitation, or molecular motion, in virtue of which velocities in all directions are distributed among the particles according to a certain law. In the cases we are considering, the collisions are so frequent that the law of distribution of the molecular velocities, if disturbed in any way, will be re-established in an inappreciably short time ; so that the motion will always consist of this definite motion of agitation, combined with the general motion of translation. When two gases are in communication, streams of the two gases might run freely in opposite directions, if it were not for the collisions which take place between the particles. The rate at which they actually interpenetrate each other must be investigated. The diffusion is due partly to the spreading of the particles by the molecular agitation, and partly , to the actual motion of the two opposite currents in mass, produced by the pressure behind, and resisted * [The metliods and results of this paper have been criticised by Clausius in a memoir published in Poggendorff's Annalen, Vol. cxv., and in the Philosophical Magazine, Vol., xxiii. His main objec- tion is that the various circumstances of the strata, discussed in the paper, have not been sufficiently- represented in the equations. In particular, if there be a series of strata at different temperatures perpendicular to the axis of x, then the proportion of molecules whose directions form vdth the axis of X angles whose cosines lie between jj, and /x + dix is not ^dfj, as has been assumed by Maxwell throughout his work, but ^Rdfj. where H is a. factor to be determined. In discussing the steady conduction of heat through a gas Clausius assumes that, in addition to the velocity attributed to the molecule according to Maxwell's theory, we must also suppose a velocity normal to the stratum and depending on the temperature of the stratum. On this assumption the factor R is investigated along with other modifications, and an expression for the assumed velocity is determined from the consideration that when the flow of heat is steady there is no movement of the mass. Clausius combining his own results with those of Maxwell points out that the expression contained in (28) of the paper involves as a result the motion of the gas. He also disputes the accuracy of ex- pression (59) for the Conduction of Heat. In the introduction to the memoir published in the Fhil. Trans., 1866, it will be found that Maxwell expresses dissatisfaction with his former theory of the Diffusion of Gases, and admits the force of the objections made by Clausius to his expression for the Conduction of Heat. Ed.] ILLUSTRATIONS OF THE DYNAMICAL THEOHY OF GASES. 393 by the collisions of the opposite stream. When the densities are equal, the diffusions due to these two causes respectively are as 2 to 3. Prop. XIV. In a system of particles whose density, velocity, <&c. are functions of x, to find the quantity of matter transferred across the plane of yz, due to the motion of agitation alone. If the number of particles, their velocity, or their length of path is greater on one side of this plane than on the other, then more particles will cross the plane in one direction than in the other ; and there will be a transference of matter across the plane, the amount of which may be calculated. Let there be taken a stratum whose thickness is dx, and area unity, at a distance x from the origin. The number of collisions taking place in this stratum in unit of time will be Njdx. ' ^^ The proportion of these which reach a distance between nl and {n^dn)l before they strike another particle is e~" dn. The proportion of these which pass through the plane yz is nl + x 2nl when X is between —nl and 0, 7)1 — IT and —J- when x is between and + nl ; the sign being negative in the latter case, because the particles cross the plane in the negative direction. The mass of each particle \s, M ; so that the quantity of matter which is projected from the stratum dx, crosses the plane yz in a positive direction, and strikes other particles at distances between nl and (n + dn) I is ^^^^'^p^dxe-^dn (26), where x must be between +nl, and the upper or lower sign is to be taken according as x is positive or negative. In integrating this expression, we must remember that N, v, and I are functions of x, not vanishing with x, and of which the variations are very small between the limits x= —nl and £c= +nl. VOL. I. 50 394 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. As we may have occasion to perform similar integrations, we may state here, to save trouble, that if U and r are functions of x not vanishing with x, whose variations are very small between the limits a; = + r and x= —r. /: "±£r.- and the only way in which the equation of equilibrium of the plug can generally subsist is when Z, = Xj and Z, = l^. This implies that A = C and B = D. Now we know that v^B = v^C. Let K=^ —^, then we sha^ have A = C=^Kv,\ B.^D = LKv,\. (40), 11 1 and j- = j- = K {v,p, + v,p;)i-^ (41). The diffusion is due partly to the motion of translation, and partly to that of agitation. Let us find the part due to the motion of translation- The equation of motion of one of the gases through the plug is found by adding the forces due to pressures to those due to resistances, and equating these to the moving force, which in the case of slow motions may be neglected altogether. The result for ^he first is ^Pl f A 1 D 7 -^1 \ '^Pi n 7 ^1 + ^^^^^^' i^^ (P^x-F.)+^-^].^' = (42). Making use of the simplifications w« have just discovered, thjs becomes whence U=-f ^ (">■ + "■'i'.) , (44). Kv^% (i>xV, +_p,Vi) + —j^- 400 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. whence the rate of diiFusion due to the motion of translation may be found ; for (?. = ^, and^,= -^ .......(45). To find the diffusion due to the motion of agitation, we must find the value of g,. _ L d Pj f dv Siimlarly, q,^ + ;^^{\+KLv,{p,+p,)} (47). The whole diffusions are ^i + g'j and Q^ + q^. The values of q^ and q^ have a term not following Graham's law of the square roots of the specific gravities, but following the law of equal volumes. The closer the material of the plug, the less will this term affect the result. Our assumptions that the porous plug acts like a system of fixed particles, and that Graham's law is fulfilled more accurately the more compact the material of the plug, are scarcely sufiiciently well verified for the foundation of a theory of gases ; and even if we admit the original assumption that they are systems of moving elastic particles, we have not very good evidence as yet for the relation among the quantities A, B, C, and D. Prop. XX. To find the rate of diffusion between two vessels connected by a tube. When diffusion takes place through a large opening, such as a tube con- necting two vessels, the question is simplified by the absence of the porous diffusion plug; and since the pressure is constant throughout the apparatus, the volumes of the two gases passing opposite ways through the tube at the same time must be equal. Now the quantity of gas which passes through the tube is due partly to the motion of agitation as in Prop. XIV., and partly to the mean motion of translation as in Prop. XV. ILLUSTRATIONS OF THE DYNAMICAL THEORY OP GASES. 401 Let U3 suppose the volumes of the two vessels to be a and h, and the length of the tube between them c, and its trans- verse section s. Let a be filled with the first gas, /^ "" ^ /'^ and h with the second at the commencement of the experiment, and let the pressure throughout the apparatus be P. Let a volume y of the first gas pass from a to h, and a volume y' of the second pass from h to a; then if p^ and p., represent the pressures in a due to the first and second kinds of gas, and p\ and p\ the same in the vessel h, Since there is stIU equilibrium, Pi+P2=Pi+p\, which gives y — y '^"^^ Pi +P2 = P —p'^ +p'2 (49). The rate of diffusion wiU be +-/ for the one gas, and — -^ for the other, measured in volume of gas at pressure P. Now the rate of diffusion of the first gas will be d di-' F F ^^"^' and that of the second, _dy_^ -^'^Tx(P^^^)+P^^^ dt F We have also the equation, derived from Props. XVI. and XVII. , ^{ApA{M, + M,) + BpAM,- CpAM} + Bp,p,vM{V,- ^.) = (52). From these three equations we can eliminate V^ and V^, and find -^ in terms of p and ~ , so that we may write l=/(^"t) (=^)- VOL. I. 51 402 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. Since the capacity of the tube is small compared with that of the vessels, we may consider -^ constant through the whole length of the tube. We may then solve the differential equation in p and x; and then making p^p^ when x = 0, and p=p\ when cc = c, and substituting for p^ and p\ their values in terms of y, we shall have a differential equation in y and t, which being solved, will give the amount of gas diffused in a given time. The solution of these equations would be difficult unless we assume rela- tions among the quantities A, B, C, D, which are not yet sufficiently estab- lished in the case of gases of different density. Let us suppose that in a particular case the two gases have the same density, and that the four quan- tities A, B, C, D are all equal. The volume diffused, owing to the motion of agitation of the particles, is then 3 P dx ^^' and that due to the motion of translation, or the interpenetration of the two gases in opposite streams, is s dp M P dx V ' The values of v are distributed according to the law of Prop. IV., so that 2a 1 . 2 the mean value of v is -7=- , and that of - is -7=- , that of h being: ial The J TV V Jtra ^ ^ diffusions due to these two causes are therefore in the ratio of 2 to 3, and their sum is dt~ ^ sj TT P dx ^^^!- dy If we suppose -j- constant throughout the tube, or, in other words, if we regard the motion as steady for a short time, then -J- will be constant and equal to ^-^ — — ; or substituting from (48), l=-*V?SJ«» + ^)!'-«*} (")^ whence y = ---^[i-e~" ^»*"»" ) (56). ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 403 By choosing pairs of gases of equal density, and ascertaining the amount of diffusion in a given time, we might determine the value of I in this expres- sion. The diffusion of nitrogen into carbonic oxide or of deutoxide of nitrogen into carbonic acid, would be suitable cases for experiment. The only existing experiment which approximately fulfils the conditions is one by Graham, quoted by Herapath from Brande's Quarterly Journal of Science, Vol. xviii. p. 76. A tube 9 inches long and 0'9 inch diameter, communicated with the atmosphere by a tube 2 inches long and 0"12 inch diameter; 152 parts of defiant gas being placed in the tube, the quantity remaining after four hours was 99 parts. In this case there is not much difference of specific gravity between the gases, and we have a = 9x(0"9)^- cubic inches, h=co, c = 2 inches, and s = {0'12y-r square inches; '=N/Ii?'°^''«-i-H..C-^) m .: Z = 0-00000256 inch ^sggVoo i^^h (58). Prop. XXI. To Jind the amount of energy which crosses unit of area in unit of time when the velocity of agitation is greater on one side of the area than on the other. The energy of a single particle is composed of two parts, — the vis viva of the centre of gravity, and the vis viva of the various motions of rotation round that centre, or, if the particle be capable of internal motions, the vis viva of these. We shall suppose that the whole vis viva bears a constant proportion to that due to the motion of the centre of gravity, or JE = i/3Mv', where yS is a coefficient, the experimental value of which is 1-634. Substituting ^ for M in Prop. XIV., we get for the transference of energy across unit of area in unit of time, Jq=-i^Jimv'J^vl), 51—2 404 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. where J is the mechanical equivalent of heat in foot-pounds, and q is the transfer of heat in thermal units. Now MN=p, and Z = ^, so that MNl = ^ ^5= -if I (-)■ Also, if T is the absolute temperature, 1 dT_2dv_ T dx V dx' .: Jq=-^fiplv^'^ (60), where p must be measured in dynamical units of force. Let J =772 foot-pounds, p = 2116 pounds to square foot, l = zoiyooo inch, i) = 1505 feet per second, T=522 or 62" Fahrenheit; then r-T 40000a; (61), where q is the flow of heat in thermal units per square foot of area ; and T' and T are the temperatures at the two sides of a stratum of air x inches thick. In Prof. Eankine's work on the Steam-engine, p. 259, values of the thermal resistance, or the reciprocal of the conductivity, are given for various substances as computed from a Table of conductivities deduced by M. Peclet from experi- ments by M. Despretz : — Eesistance. Gold, Platinum, Silver 0-0036 Copper 0-0040 Iron , 0-0096 Lead 0-0198 Brick 0-3306 Air hy our calculation 40000 It appears, therefore, that the resistance of a stratum of air to the con- duction of heat is about 10,000,000 times greater than that of a stratum of ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 405 copper of equal thickness. It would be almost impossible to establish the value of the conductivity of a gas by direct experiment, as the heat radiated from the sides of the vessel would be far greater than the heat conducted through the air, even if currents could be entirely prevented''". PAET in. ON THE COLLISION OF PERFECTLY ELASTIC BODIES OF ANY FORM. When two perfectly smooth spheres strike each other, the force which acts between them always passes through their centres of gravity ; and therefore their motions of rotation, if they have any, are not affected by the colhsion, and do not enter into our calculations. But, when the bodies are not spherical, the force of compact will not, in general, be in the line joining their centres of gravity ; and therefore the force of impact will depend both on the motion of the centres and the motions of rotation before impact, and it will affect both these motions after impact. In this way the velocities of the centres and the velocities of rotation will act and react on each other, so that finally there wiU be some relation established between them ; and since the rotations of the particles about their three axes are quantities related to each other in the same way as the three velocities of their centres, the reasoning of Prop. IV. will apply to rotation as well as velocity, and both will be distributed according to the law cix a n/tt * [Clausius, in the memoir cited in the last foot-note, has pointed out two oversights in this calculation. In the first place the numbers have not been properly reduced to English measure, and have still to be multiplied by 4356, the ratio of the English pound to the kilogramme. The numbers have, further, been calculated with one hour as the unit of time, whereas Maxwell has used them as if a second had been the unit. Taking account of these circumstances and using his own expression for the conduction which differs from (59) only in having ^\ in place of | on the right-hand side, Clausius finds that the resistance of a stratum of air to the conduction of heat is 1400 times greater than that of a stratum of lead of the same thickness, or about 7000 times greater than that of copper. Ed.] 406 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. Also, by Prop. V., if x be tbe average velocity of one set of particles, and y that of another, then the average value of the sum or difference of the velocities is Jaf + 2/' ; from which it is easy to see that, if in each individual case u = ax-{-})y + cz, where x, y, z are independent quantities distributed according to the law above stated, then the average values of these quantities will be connected by the equation u^ = a V + &y + c^'^^' Prop. XXII. Two 'perfectly elastic bodies of any form strike each other : given their motions before impact, and the line of impact, to find their motions after impact. Let M^ and M^ be the centres of gravity of the two bodies. M^X^, M^Y^, and -Mi^i the principal axes of the first ; and MJC^, JfaF^ and M^^ those of the second. Let / be the point of impact, and R^IR^ the line of impact. Let the co-ordinates of / with respect to M.^ be .Xi^/A) and with respect to M^ let them be x^.f^. Let the direction-cosines of the line of impact RfR^ be l^mji^ with respect to M^, and l2m.j)i^ with respect to M^. Let M, and M^ be the masses, and A.B^C, and A^B„C^ the moments of inertia of the bodies about their principal axes. Let the velocities of the centres of gravity, resolved in the direction of the principal axes of each body, be U^, Vi, W^, and U^, V^, W^, before impact, and U\, F„ W\, and U\, V\, W'^ after impact. Let the angular velocities round the same axes be ^1, g-i, r^, and p^, q^, r^, before impact, and p\, q\, r\, and p\_, q\, r\, after impact. ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 407 Let R be the impulsive force between the bodies, measured by the momentum it produces in each. Then, for the velocities of the centres of gravity, we have the following equations : U\^U, + ^, U\=U,-§^ (62), with two other pairs of equations in V and W. The equations for the angular velocities are p\ =P, + -J (2/i«i - Zi^i). P\ =P^ - J- (2/2«2 - z,in,) (63), with two other pairs of equations for q and r. The condition of perfect elasticity is that the whole vis viva shall be the same after impact as before, which gives the equation M^ ( U\ - U\) + M, ( U'\ - U\) + A, {p'\ -p\) + A, {p'% -p\) + &c. = 0. . . . (64). The terms relating to the axis of x are here given; those relating to y and z may be easily written down. Substituting the values of these terms, as given by equations (62) and (63), and dividing by R, we find I^IJJ\+ U,)-k{U\+ U,) + {y,n,-z,m,){p\+p,)-{y,n,-z,m,) (p',,+p,) + &c. = 0...(65). Now if V, be the velocity of the striking-point of the first body before impact, resolved along the line of impact, Vi = IJJ^ + {y^n^ - Zi^i) Pi + &c. ; and if we put v, for the velocity of the other striking-point resolved along the same line, and v\ and v\ the same quantities after impact, we may write, equation (65), v^ + v\-v,-v^ = Q (6b), or v^ — Vr, = v\ — v\ (67), which shows that the velocity of separation of the striking-points resolved in the line of impact is equal to that of approach. 408 ILLUSTRATIONS OF THE DYNAMICAL THEORY OP GASES. Substituting the values of the accented quantities in equation (65) by means of equations (63) and (64), and transposing terms in R, we find 2{U,\- U,l, +p, {y,n, - z,m,) -p, {y,n, - zm,)} + &c. ^-^[M^K^ A ^ A, +*' ^^'^' the other terms being related to y and z as these are to x. From this equation we may find the value of R ; and by substituting this in equations (63), (64), we may obtain the values of all the velocities after impact. We may, for example, find the value of U\ from the equation I' ^ I! ^ {yjh-zfn^ , {y.n,-z,m.;)' , ,_ l M, + 2 £/,Z, - 2pi (y^Wi - z^m,) + 2j5, (y^^i^ - z,m,) - &c. (69). Prop. XXIII. To find the relations hettoeen the average velocities of trans- lation and rotation after many collisions among many bodies. Taking equation (69), which applies to an Individual collision, we see that U\ is expressed as a linear function of Uj^, ZZ,, p^, p^. Sec, all of which are quantities of which the values are distributed among the different particles according to the law of Prop. lY. It follows from Prop. V., that if we square every term of the equation, we shall have a new equation between the average values of the difierent quantities. It is plain that, as soon as the required relations have been established, they will remain the same after collision, so that we may put U{^= U^ in the equation of averages. The equation between the average values may then be written {M,U:'-M„U:)^ + {M,U,^-A,p^) ^^'''' ~ "''^''^^ + (-M; £/-/ - A.p^) ^y-^"""- ~2^''^^ + &c. = 0. Now since there are collisions in every possible way, so that the values of I, m, n, &c. and x, y, z, &c. are infinitely varied, this equation cannot subsist unless M, U,' = M, £/•/ = A,p,' = A,p: = &c. The final state, therefore, of any number of systems of moving particles of any form is that in which the average vis viva of translation along each of the ILLUSTKATIONS OF THE DYNAMICAL THEORY OF GASES. 409 three axes is the same in all the systems, and equal to the average vis viva of rotation about each of the three principal axes of each particle. Adding the vires vivce with respect to the other axes, we find that the whole vis viva of translation is equal to that of rotation in each system of particles, and is also the same for different systems, as was proved in Prop. VI. This result (which is true, however nearly the bodies approach the spherical form, provided the motion of rotation is at all affected by the collisions) seems decisive against the unqualified acceptation of the hypothesis that gases are such systems of hard elastic particles. For the ascertained fact that y, the ratio of the specific heat at constant pressure to that at constant volume, is equal to 1-408, requires that the ratio of the whole vis viva to the vis viva of translation should be whereas, according to our hypothesis, /3 = 2 . We have now followed the mathematical theory of the collisions of hard elastic particles through various cases, in which there seems to be an analogy with the phenomena of gases. We have deduced, as others have done already, the relations of pressure, temperature, and density of a single gas. We have also proved that when two different gases act freely on each other (that is, when at the same temperature), the mass of the single particles of each is inversely proportional to the square of the molecular velocity ; and therefore, at equal temperature and pressure, the number of particles in unit of volume is the same. We then offered an explanation of the internal friction of gases, and deduced from experiments a value of the mean length of path of a particle between successive collisions. We have applied the theory to the law of diffusion of gases, and, fi:om an experiment on defiant gas, we have deduced a value of the length of path not very different from that deduced from experiments on friction. Using this value of the length of path between collisions, we found that the resistance of air to the conduction of heat is 10,000,000 times that of copper, a result in accordance with experience. Finally, by establishing a necessary relation between the motions of trans- lation and rotation of all particles not spherical, we proved that a system of such particles could not possibly satisfy the known relation between the two specific heats of all gases. VOL. I. 52 [From the Philosophical Transactions, mdccclx.] XXI. On the Theory of Compound Colours, and the Relations of the Colours of the Spectrum. Communicated by Professor Stokes, Sec. R.S. (Received January 5, — Read March 22, 1860.) § I. Introduction. According to Newton's analysis of light''', every colour in nature is pro- duced by tbe mixture, in various proportions, of the different kinds of light into which white light is divided by refraction. By means of a prism we may analyse any coloured light, and determine the proportions in which the diiferent homogeneous rays enter into it; and by means of a lens we may recombine these rays, and reproduce the original coloured light. Newton has also shewnt how to combine the different rays of the spectrum so as to form a single beam of light, and how to alter the proportions of the different colours so as to exhibit the result of combining them in any arbitrary manner. The number of different kinds of homogeneous light being infinite, and the proportion in which each may be combined being also variable indefinitely, the results of such combinations could not be appreciated by the eye, unless the chromatic effect of every mixture, however complicated, could be expressed in some simpler form. Colours, as seen by the human eye of the normal type, can all be reduced to a few classes, and expressed by a few well-known names; and even those colours which have different names have obvious relations among them- selves. Every colour, except purple, is similar to some colour of the spectrum!, * Optics, Book I. Part 2, Prop. 7. t Lectiones Opticce, Part 2, § 1, pp. 100 to 105; and Optics, Book i. Part 2, Prop. 11. X Optics, Book I. Part 2, Prop. 4. ON THE THEOEY OP COMPOUND COLOURS. 411 although less intense; and all purples may be compounded of blue and red, and diluted with white to any required tint. Brown colours, which at first sight seem different, are merely red, orange or yellow of feeble intensity, more or less diluted with white. It appears therefore that the result of any mixture of colours, however complicated, may be defined by its relation to a certain small number of well-known colours. Having selected our standard colours, and determined the relations of a given colour to these, we have defined that colour completely as to its appearance. Any colour which has the same relation to the standard colours, will be identical in appearance, though its optical constitution, as revealed by the prism, may be very different. We may express this by saying that two compound colours may be chro- matically identical, but optically different. The optical properties of light are those which have reference to its origin and propagation through media, till it falls on the sensitive organ of vision ; the chromatical properties of light are those which have reference to its power of exciting certain sensations of colour, perceived through the organ of vision. The investigation of the chromatic relations of the rays of the spectrum must therefore be founded upon observations of the apparent identity of com- pound colours, as seen by an eye either of the normal or of some abnormal type ; and the results to which the investigation leads must be regarded as partaking of a physiological, as well as of a physical character, and as indicating certain laws of sensation, depending on the constitution of the organ of vision, which may be different in different individuals. We have to determine the laws of the composition of colours in general, to reduce the number of standard colours to the smallest possible, to discover, if we can, what they are, and to ascertain the relation which the homogeneous light of different parts of the spectrum bears to the standard colours. § II. History of the Theory of Compound Colours. The foundation of the theory of the composition of colours was laid by Newton*. He first shews that, by the mixture of homogeneal light, colours may be produced which are "like to the colours of homogeneal light as to the appearance of colour, but not as to the immutability of colour and consti- * Optics, Book I. Part 2, Props. 4, 5, 6. 52—2 412 ON THE THEORY OE COMPOUND COLOURS. tution of light." Red and yellow give an orange colour, which is chromatically similar to the orange of the spectrum, but optically different, because it is resolved into its component colours by a prism, while the orange of the spectrum remains unchanged. When the colours to be mixed lie at a distance from one another in the spectrum, the resultant appears paler than that intermediate colour of the spectrum which it most resembles ; and when several are mixed, the resultant may appear white. Newton* is always careful, however, not to call any mixture white, unless it agrees with common white light in its optical as well as its chromatical properties, and is a mixture of all the homogeneal colours. The theory of compound colours is first presented in a mathematical form in Prop. 6, "' In a mixture of primary colours, the quantity a,nd quality of each being given, to know the colour of the compound." He divides the circumference of a circle into seven parts, proportional to the seven musical intervals, in accordance with his opinion about the proportions of the colours in the spectrum. At the centre of gravity of each of these arcs he places a little circle, whose area is proportional to the number of rays of the corre- sponding colour which enter into the given mixture. The position of the centre of gravity of all these circles indicates the nature of the resultant colour. A radius drawn through it points out that colour of the spectrum which it most resembles, and the distance from the centre determines the fulness of its colour. "With respect to this construction, Newton says, " This rule I conceive accurate enough for practice, though not mathematically accurate." He gives no reasons for the different parts of his rule, but we shall find that his method of finding the centre of gravity of the component colours is completely con- firmed by my observations, and that it involves mathematically the theory of three elements of colour ; but that the disposition of the colours on the circumference of a circle was only a provisional arrangement, and that the true relations of the colours of the spectrum can only be determined by direct observation. Young t appears to have originated the theory, that the three elements of colour are determined as much by the constitution of the sense of sight as by anything external to us. He conceives that three different sensations may be excited by light, but that the proportion in which each of the three is excited depends on the nature of the hght. He conjectures that these primary sensa- * 7th and 8th Letters to Oldenburg. + Young's Lectures on Natural Philosojjhy, Kelland's Edition, p. 345, ov Quarto, 1807, Vol i. p. 441 ; see also Young in Philosophical Transactions, 1801, or Works in Quarto, Vol ii. p. 617. ON THE THEORY OF COMPOUND COLOURS. 413 tions correspond to red, green, and violet. A blue ray, for example, though homogeneous in itself, he conceives capable of exciting both the green and the violet sensation, and therefore he would call blue a compound colour, though the colour of a simple kind of light. The quality of any colour depends, according to this theory, on the ratios of the intensities of the three sensations which it excites, and its brightness depends on the sum of these three intensities. Sir David Brewster, in his paper entitled " On a New Analysis of Solar Light, indicating three Primary Colours, forming Coincident Spectra of equal length*," regards the actual colours of the spectrum as arising from the inter- mixture, in various proportions, of three primary kinds of light, red, yellow, and blue, each of which is variable in intensity, but uniform in colour, from one end of the spectrum to the other ; so that every colour in the spectrum is really compound, and might be shewn to be so if we had the means of separating its elements. Sir David Brewster, in his researches, employed coloured media, which, according to him, absorb the three elements of a single prismatic colour in different degrees, and change their proportions, so as to alter the colour of the light, without altering its refrangibility. In this paper I shall not enter into the very important questions affecting the physical theory of light, which can only be settled by a careful inquiry into the phenomena of absorption. The physiological facts, that we have a threefold sensation of colour, and that the three elements of this sensation are affected in different proportions by light of different refrangibilities, are equally true, whether we adopt the physical theory that there are three kinds of light corresponding to these three colour-sensations, or whether we regard light of definite refrangibility as an undulation of known length, and therefore variable only in intensity, but capable of producing different chemical actions on different substances, of being absorbed in different degrees by different media, and of exciting in different degrees the three different colour-sensations of the human eye. Sir David Brewster has given a diagram of three curves, in which the base-line represents the length of the spectrum, and the ordinates of the curves represent, by estimation, the intensities of the three kinds of light at each point of the spectrum. I have employed a diagram of the same kind to express the * Transactions of the Royal Society of Edinburgh, Vol. xii. p. 1 23. 414 ON THE THEORY OF COMPOUND COLOURS. results arrived at in this paper, the ordinates being made to represent the intensities of each of the three elements of colour, as calculated from the experiments. The most complete series of experiments on the mixture of the colours of the spectrum, is that of Professor Helmholtz*, of Konigsberg. By using two shts at right angles to one another, he formed two pure spectra, the fixed lines of which were seen crossing one another when viewed in the ordinary way by means of a telescope. The colours of these spectra were thus combined in every possible way, and the effect of the combination of any two could be seen separately by drawing the eye back from the eye-piece of the telescope, when the compound colour was seen by itself at the eye-hole. The proportion of the components was altered by turning the combined slits round in their own plane. One result of these experiments was, that a colour, chromatically identical with white, could be formed by combining yellow with indigo. M. Helmholtz was not then able to produce white with any other pair of simple colours, and considered that three simple colours were required in general to produce white, one from each of the three portions into which the spectrum is divided by the yellow and indigo. Professor Grassmannf shewed that Newton's theory of compound colours implies that there are an infinite number of pairs of complementary colours in the spectrum, and pointed out the means of finding them. He also shewed how colours may be represented by lines, and combined by the method of the parallelogram. In a second memoirj, M. Helmholtz describes his method of ascertaining these pairs of complementary colours. He formed a pure spectrum by means of a slit, a prism, and a lens ; and in this spectrum he placed an apparatus having two parallel slits which were capable of adjustment both in position and breadth, so as to let through any two portions of the spectrum, in any proportions. Behind this slit, these rays were united in an image of the prism, which was received on paper. By arranging the slits, the colour of this image may be reduced to white, and made identical with that of paper illuminated with white light. The wave-lengths of the component colours were then measured by observing the angle of diffraction through a grating. It was found that the * Poggendorffs Annalen, Band lxxxvii. (Philosophical Magazine, 1852, December). t Ibid. Band lxxxix. (Philosophical Magazine, 1854, April). J Jhid. Band xciv. ON THE THEORY OF COMPOUND COLOURS. 41 & colours from red to green-yellow (X = 2082) were complementary to colours ranging from green-blue (X=1818) to violet, and that the colours between green-yellow and green-blue have no homogeneous complementaries, but must be neutralized lay mixtures of red and violet. M. Helmholtz also gives a provisional diagram of the curve formed by the spectrum on Newton's diagram, for which his experiments did not furnish him Tvith the complete data. Accounts of experiments by myself on the mixture of artificial colours by rapid rotation, may be found in the Transactions of the Royal Society of Edinburgh, Vol. xxi. Pt. 2 (1855); in an appendix to Professor George Wilson's work on Colour- Blindness ; in the Report of the British Association for 1856, p. 12 ; and in the Philosophical Magazine, July 1857, p. 40. These experiments shew that, for the normal eye, there are three, and only three, elements of colour, and that in the colour-blind one of these is absent. They also prove that chromatic observations may be made, both by normal and abnormal eyes, with such accuracy, as to warrant the employment of the results in the calcu- lation of colour-equations, and in laying down colour-diagrams by Newton's rule. The first instrument which I made (in 1852) to examine the mixtures of the colours of the spectrum was similar to that which I now use, but smaller, and it had no constant light for a term of comparison. The second was 6^ feet long, made in 1855, and shewed tivo combinations of colour side by side. I have now succeeded in making the mixture much more perfect, and the comparisons more exact, by using white reflected light, instead of the second compound colour. An apparatus in which the light passes through the prisms, and is reflected back again in nearly the same path by a concave mirror, was shewn by me to the British Association in 1856. It has the advantage of being portable, and need not be more than half the length of the other, in order to produce a spectrum of equal length. I am so well satisfied with the working of this form of the instrument, that I intend to make use of it in obtaining equations from a greater variety of observers than I could meet with when I vras obliged to use the more bulky instrument. It is difiicult at first to get the observer to believe that the compound light can ever be so adjusted as to appear to his eyes , identical with the white light in contact with it. He has to learn what adjustments are necessary to produce the requisite alteration under all circumstances, and he must never be satisfied tiU the two parts of the field are identical in colour and illumination. To do this thoroughly, implies 416 ON THE THEORY OF COMPOUND COLOURS. not merely good eyes, but a power of judging as to the exact nature of the difierence between two very pale and nearly identical tints, whether they differ in the amount of red, green, or blue, or in brightness of illumination. In the following paper I shall first lay down the mathematical theory of Newton's diagram, with its relation to Young's theory of the colour-sensation. I shall then describe the experimental method of mixing the colours of the spectrum, and determining the wave-lengths of the colours mixed. The results of my experiments will then be given, and the chromatic relations of the spectrum exhibited in a system of colour-equations, in Newton's diagram, and in three curves of intensity, as in Brewster's diagram. The differences between the results of two observers will then be discussed, shewing on what they depend, and in what way such differences may affect the vision of persons otherwise free from defects of sight. § III. Mathematical TJieory of Newton's Diagram of Colours. Newton's diagram is a plane figure, designed to exhibit the relations of colours to each other. Every point in the diagram represents a colour, simple or compound, and we may conceive the diagram itself so painted, that every colour is found at its corresponding point. Any colour', differing only in quantity of illumination from one of the colours of the diagram, is referred to it as a unit, and is measured by the latio of the illumination of the given colour to that of the corresponding colour in the diagram. In this way the quantity of a colour is estimated. The resultant of mixing any two colours of the diagram is found by dividing the line joining them inversely as the quantity of each ; then, if the sum of these quantities is unity, the resultant will have the illumination as well as the colour of the point so found ; but if the sum of the components is different from unity, the quantity of the resultant will be measured by the sum of the components. This method of determining the position of the resultant colour is mathe- matically identical with that of finding the centre of gravity of two weights, and placing a weight equal to their sum at the point so found. We shall therefore speak of the resultant tint as the sum of its components placed at their centre of gravity. ON THE THEORY OF COMPOUND COLOURS. 417 By compounding this resultant tint with some other colour, we may find the position of a mixture of three colours, at the centre of gravity of its components ; and by taking these components in different proportions, we may obtain colours corresponding to every part of the triangle of which they are the angular points. In this way, by taking any three colours we should be able to construct a triangular portion of Newton's diagram by painting it with mixtures of the three colours. Of course these mixtures must be made to correspond with optical mixtures of light, not with mechanical mixtures of pigments. Let us now take any colour belonging to a point of the diagram outside this triangle. To make the centre of gravity of the three weights coincide with this point, one or more of the weights must be made negative. This, though following fi:om mathematical principles, is not capable of direct physical inter- pretation, as we cannot exhibit a negative colour. The equation between the three selected colours, x, y, z, and the new colour u, may in the first case be written u = x + y + z (1), X, y, 2 being the quantities of colour required to produce u. In the second case suppose that z must be made negative, u = x + y — z (2). As we cannot reahze the term —2 as a negative colour, we transpose it to the other side of the equation, which then becomes u-\-z = x-\-y (3), which may be interpreted to mean, that the resultant tint, u-\-z, is identical with the resultant, x + y. We thus find a mixture of the new colour with one of the selected colours, which is chromatically equivalent to a mixture of the other two selected colours. When the equation takes the form u = x-y-z (4), two of the components being negative, we must transpose them thus, u + y + z = x (5), which means that a mixture of certain proportions of the new colour and two of the three selected, is chromatically equivalent to the third. We may thus in all cases find the relation between any three colours and a fourth, and exhibit VOL. L 53 418 ON THE THEORY OP COMPOUND COLOURS. this relation in a form capable of experimental verification ; and by proceeding in this way we may map out the positions of all colours upon Newton's diagram. Every colour in natirre will then be defined by the position of the corresponding coloior in the diagram, and by the ratio of its illumination to that of the colour in the diagram. § IV. Method of representing Colours by Straight Lines drawn from a Point. To extend our ideas of the relations of colours, let us form a new geome- trical conception by the aid of solid geometry. Let us take as origin any point not in the plane of the diagram, and let us draw lines through this point to the diflPerent points of the diagram ; then the direction of any of these lines will depend upon the position of the point of the diagram through which it passes, so that we may take this line as the representative of the corresponding colour on the diagram. In order to indicate the quantity of this colour, let it be produced beyond the plane of the diagram in the same ratio as the given colour exceeds in mumiriation the colour on the diagram. In this way every colour in nature will be represented by a Hne drawn through the origin, whose direction indicates the quality of the colour, while its length indicates its quantity. Let us find the resultant of two colours by this method. Let be the origin and AB be a section of the plane of the diagram by that of the paper. Let OP, OQ be fines representing colours, A, B the corresponding points in the diagram; then the quantity of P will be -— =«, \JJL and that of Q will be -Q^ = q- The resultant of these wiU be represented in the diagram by the point C, where AC: CB :: q : p, and the quantity of the resultant will he p + q, so that if we produce DC to R, so that OR = {p + q)OC, the line OR will represent the resultant of OP and OQ in direction and magnitude. It is easy to prove, from this construction, that OR is the diagonal of the parallelogram of which OP and OQ are two sides. It appears therefore that if colours are represented in quantity and quality by the magnitude and direction of straight lines, the rule for the composition of colours is identical ON THE THEORY OF COMPOUND COLOURS. 419 with that for the composition of forces in. mechanics. This analogy has been well brought out by Professor Grassmann in Poggendorff's Annalen, Bd. lxxxix. We may conceive an arrangement of actual colours in space founded upon this construction. Suppose each of these radiating lines representing a given colour to be itself illuminated with that colour, the brightness increasing from zero at the origin to unity, where it cuts the plane of the diagram, and becoming continually more intense in proportion to the distance from the origin. In this way every colour in nature may be matched, both in quality and quantity, by some point in this coloured space. If we take any three lines through the origin as axes, we may, by co-ordi- nates parallel to these Hnes, express the position of any point in space. That point will correspond to a colour which is the resultant of the three colours represented by the three co-ordinates. This system of co-ordinates is an illustration of the resolution of a colour into three components. According to the theory of Young, the human eye is capable of three distinct primitive sensations of colour, which by their composition in various proportions, produce the sensations of actual colour in aU their varieties. Whether any kinds of light have the power of exciting these primitive sensations separately, has not yet been determined. If colours corresponding to the three primitive sensations can be exhibited, then aU colours, whether produced by light, disease, or imagination, are com- pounded of these, and have their places within the triangle formed by joining the three primaries. If the colours of the pure spectrum, as laid down on the diagram, form a triangle, the colours at the angles may correspond to the primitive sensations. If the curve of the spectrum does not reach the angles of the circum- scribing triangle, then no colour in the spectrum, and therefore no colour in nature, corresponds to any of the three primary sensations. The only data at present existing for determining the primary colours, are derived from the comparison of observations of colour-equations by colour-bhnd, and by normal eyes. The colour-blind equations differ from the others by the non-existence of one of the elements of colour, the relation of which to known colours can be ascertained. It appears, from observations made for me by two colour-blind persons*, that the elementary sensation which they do not possess is a red approaching to crimson, lying beyond both vermilion and carmine. These * Transactions of the Royal Society of Edimhwrgh, Vol. xxi. Pt. 2, p. 286. 53—2 420 ON THE THEORY OF COMPOUND COLOUES. observations are confirmed by those of Mr Pole, and by others which I have obtained since. I have hopes of being able to procure a set of colour-blind equations between the colours of the spectrum, which will indicate the missing primary in a more exact manner. The experiments which I am going to describe have for their object the determination of the position of the colours of the spectrum upon Newton's diagram, from actual observations of the mixtures of those colours. They were conducted in such a way, that in every observation the judgment of the observer was exercised upon two parts of an illuminated field, one of which was so adjusted as to be chromatically identical with the other, which, during the whole series of observations, remained of one constant intensity of white. In this way the effects of subjective colours were entirely got rid of, and all the observa- tions were of the same kind, and therefore may claim to be equally accurate ; which is not the case when comparisons are made between bright colours of different kinds. The chart of the spectrum, deduced from these observations, exhibits the colours arranged very exactly along two sides of a triangle, the extreme red and violet forming doubtful portions of the third side. This result greatly simplifies the theory of colour, if it does not actually point out the three primary colours themselves. § V. Description of an Instrument for making definite Mixtures of the Colours of the Spectrum. The experimental method which I have used consists in forming a combi- nation of three colours belonging to different portions of the spectrum, the quantity of each being so adjusted that the mixture shall be white, and equal in intensity to a given white. Fig. 1, Plate VI. p. 444, represents the instrument for making the observations. It consists of two tubes, or long boxes, of deal, of rectangular section, joined together at an angle of about 100°. The part ^^ is about five feet long, seven inches broad, and four deep ; KN is about two feet long, five inches broad, and four deep ; BD is a partition parallel to the side of the long box. The whole of the inside of the instrument is painted black, and the only openings are at the end AC, and at E. At the angle there is a lid, which is opened when the optical parts have to be adjusted or cleaned. ON THE THEORY OP COMPOUND COLOURS. 421 At ^ is a fine vertical slit; Z is a lens; at P there are two equilateral prisms. The slit E, the lens L, and the prisms P are so adjusted, that when light is admitted at j& a pure spectrum is formed at AB, the extremity of the long box. A mirror at M is also adjusted so as to reflect the light from E along the narrow compartment of the long box to BC. See Fig. 3. At AB is placed the contrivance shewn in Fig. 2, Plate I. A'B' is a rect- angular frame of brass, having a rectangular aperture of 6 x 1 inches. On this frame are placed six brass sliders, X, Y, Z. Each of these carries a knife-edge of brass in the plane of the surface of the frame. These six moveable knife-edges form three slits, X, Y, Z, which may be so adjusted as to coincide with any three portions of the pure spectrum formed by light from E. The intervals behind the sliders are closed by hinged shutters, which allow the sliders to move without letting hght pass between them. The inner edge of the brass frame is graduated to twentieths of an inch, so that the position of any slit can be read ofi". The breadth of the slit is ascertained by means of a wedge-shaped piece of metal, six inches long, and tapering to a point from a breadth of half an inch. This is gently inserted into each slit, and the breadth is determined by the distance to which it enters, the divisions on the wedge corresponding to the 200th of an inch difference in breadth, so that the unit of breadth is '005 inch. Now suppose light to enter at E, to pass through the lens, and to be refracted by the two prisms at P; a pure spectrum, shewing Fraunhofer's lines, is formed at AB, but only that part is allowed to pass which faUs on the three slits X, Y, Z. The rest is stopped by the shutters. Suppose that the portion falling on X belongs to the red part of the spectrum ; then, of the white light entering at E, only the red will come through the slit X. If we were to admit red light at X it would be refracted to E, by the principle in Optics, that the course of any ray may be reversed. If, instead of red light, we were to admit white light at X, still only red light would come to E ; for aU other light would be either more or less refracted, and would not reach the slit at E. Applying the eye at the slit E, we should see the prism P uniformly iUuminated with red Hght, of the kind corresponding to the part of the spectrum which falls on the slit X when light is admitted at E. Let the slit Y correspond to another portion of the spectrum, say the green ; then, if white light is admitted at Y, the prism, as seen by an eye at E, wiU. be uniformly illuminated with green Hght ; and if white Hght be admitted at X 422 ON THE THEOBY OF COMPOUND COLOURS. and Y simultaneously, tlie colour seen at H will be a compound of red and green, the proportions depending on the breadth of the slits and the intensity of the hght which enters them. The third sht Z, enables us to combine any three kinds of light in any given proportions, so that an eye at U shall see the face of the prism at F uniformly illuminated with the colour resulting from the combination of the three. The position of these three rays in the spectrum is found by admitting the light at H, and comparing the position of the slits with the position of the principal fixed lines ; and the breadth of the slits is determined by means of the wedge. At the same time white light is admitted through BC to the mirror of black glass at M, whence it is reflected to E, past the edge of the prism at P, so that the eye at E sees through the lens a field consisting of two portions, separated by the edge of the prism; that on the left hand being compounded of three colours of the spectrum refracted by the prism, while that on the right hand is white light reflected from the mirror. By adjusting the slits properly, these two portions of the field may be made equal, both in colour and brightness, so that the edge of the prism becomes almost invisible. In making experiments, the instrument was placed on a table in a room moderately lighted, with the end AB turned towards a large board covered with white paper, and placed in the open air, so as to be uniformly illuminated by the sun. In this way the three shts and the mirror M were all illuminated with white light of the same intensity, and all were afiected in the same ratio by any change of illumination ; so that if the two halves of the field were rendered equal when the sun was under a cloud, they were found nearly correct when the sun again appeared. No experiments, however, were considered good unless the sun remained uniformly bright during the whole series of experiments. After each set of experiments light was admitted at H, and the position of the fixed lines D and F of the spectrum was read off on the scale at AB. It was found that after the instrument had been some time in use these positions were invariable, shewing that the eye-hole, the prisms, and the scale might be considered as rigidly connected. ON THE THEORY OF COMPOUND COLOURS. 423 § VI. Method of determining the Wave-length corresponding to any point of the Spectrum on the Scale AB. Two plane surfaces of glass were kept apart by two parallel strips of gold- beaters' leaf, so as to enclose a stratum of air of nearly uniform thickness. Light reflected from this stratum of air was admitted at E, and the spectrun formed by it was examined at AB by means of a lens. Thjs spectrum consists of a large number of bright bands, separated by dark spaces at nearly uniform intervals, these intervals, however, being considerably larger as we approach the violet end of the spectrum. The reason of these alternations of brightness is easily explained. By the theory of Newton's rings, the light reflected from a stratum of air consists of two parts, one of which has traversed a path longer than that of the other, by an interval depending on the thickness of the stratum and the angle of incidence. Whenever the interval of retardation is an exact multiple of a wave-length, these two portions of Kght destroy each other by interference ; and when the interval is an odd number of half wave-lengths, the resultant light is a maximum. In the ordinary case of Newton's rings, these alternations depend upon the varying thickness of the stratum ; while in this case a pencil of rays of different wave-lengths, but all experiencing the same retardation, is analysed into a spectrum, in which the rays are arranged in order of their respective wave-lengths. Every ray whose wave-length is an exact submultiple of the retardation will be destroyed by interference, and its place will appear dark in the spectrum; and there will be as many dark bands seen as there are rays whose wave-lengths fulfil this condition. If, then, we observe the positions of the dark bands on the scale AB, the wave-lengths corresponding to these positions will be a series of submultiples of the retardation. Let us caU the first dark band visible on the red side of the spectrum zero, and let us number them in order 1, 2, 3, &c. towards the violet end. Let N be the number of undulations corresponding to the band zero which are con- tained in the retardation R; then if *z. be the number of any other band, N+n wiU be the number of the corresponding wave-lengths in the retardation, or in symbols, R = {N+n)\ (6). 424 ON THE THEORY OP COMPOUND COLOUBS. Now observe the position of two of Fraunhofer's fixed lines with respect to the dark bands, and let n^, n, be their positions expressed in the number of bands, whole or fractional, reckoning from zero. Let \, \ be the wave-lengths of these fixed lines as determined by Fraunhofer, then R = {N-+n,)\ = {N+n,)K, (7) 5 whence js^=^>^^ Jj^X^-n, (8), and R = ^^'\X (9). Aj — Aj Having thus found N and E, we may find the wave-length corresponding to the dark band n from the formula \ = ^^ (10). N+n ^ ' In my experiments the line D corresponded with the seventh dark band, and F was between the 15th and 16th, so that n^=15-7. Here then for D, J » ET -.rrr \ ■lerr.^r "^ -C raunhoior s measure (11), and for F, n^=15-7, >^=1794:) ^ " whence we find iV=34, ^ = 89175 (12). There were 22 bands visible, corresponding to 22 different positions on the scale AB, as determined 4th August, 1859. Table I. Band. Scale. Band. Scale. Band. Scale. w= 1 17 n= 9 36 n= 16 57 2 19 10 39 17 61 3 21i 11 42 18 65 4 23J 12 45 19 69 5 26 13 48 20 73 6 28| 14 51 21 77 7 31 15 54 22 82 8 33| Sixteen equidistant points on the scale were chosen for standard colours in the experiments to be described. The following Table gives the reading on the scale AB, the value of N+n, and the calculated wave-length for each of these : — ON THE THEORY OF COMPOUND COLOURS. 425 Table II. Soale. (N+n). Wave-length. Colour. 20 36-4 2450 Red. 24 38-3 2328 Scarlet 28 39-8 2240 Orange. 32 41-4 2154 Yellow. 36 42-9 2078 Yellow-Green, 40 44-3 2013 Green. 44 45-7 1951 Green. 48 47-0 1879 Blniah green. 52 48-3 1846 Blue-green. 56 49-6 1797 Greenish blue. 60 50-8 1755 Blue. 64 51-8 1721 Blue. 68 52-8 1688 Blue. 72 53-7 1660 Indigo. 76 54-7 1630 Indigo. 80 55-6 1604 Indigo. Having thus selected sixteen distinct points of the spectrum on whicli to operate, and determined their wave-lengths and apparent colours, I proceeded to ascertain the mathematical relations between these colours in order to lay them down on Newton's diagram. For this purpose I selected three of these as points of reference, namely, those at 24, 44, and 68 of the scale. I chose these poiats because they are well separated from each other on the scale, and because the colour of the spectrum at these points does not appear to the eye to vary very rapidly, either in hue or brightness, in passing from one point to another. Hence a small error of position will not make so serious an alteration of colour at these points, as if we had taken them at places of rapid variation ; and we may regard the amount of the illumination produced by the light entering through the slits in these positions as sensibly proportional to the breadth of the slits. (24) corresponds to a bright scarlet about one-third of the distance from C to D; (44) is a green very near the line E ; and (68) is a blue about one- third of the distance from F to G. VOL. I. 54 426 ON THE THEOEY OF COMPOUND COLOURS. § VII. Method of Observation. The instrument is turned witli the end AB towards a board, covered with white paper, and lUuniinated by sunlight. The operator sits at the end AB, to move the sliders, and adjust the sHts ; and the observer sits at the end E, which is shaded from any bright light. The operator then places the shts so that their centres correspond to the three standard colours, and adjusts their breadths till the observer sees the prism illuminated with pure white light of the same intensity with that reflected by the mirror M. In order to do this, the observer must tell the operator what difference he observes in the two halves of the illuminated field, and the operator must alter the breadth of the slits accordingly, always keeping the centre of each slit at the proper point of the scale. The observer may call for more or less red, blue or green ; and then the operator must increase or diminish the width of the slits X, Y, and Z respectively. If the variable field is darker or lighter than the constant field, the operator must widen or narrow all the slits in the same proportion. When the variable part of the field is nearly adjusted, it often happens that the constant white light from the mirror appears tinged with the complementary colour. This is an indication of what is required to make the resemblance of the two parts of the field of view perfect. When no difference can be detected between the two parts of the field, either in colour or in brightness, the observer must look away for some time, to relieve the strain on the eye, and then look again. If the eye thus refreshed still judges the two parts of the field to be equal, the observation may be considered complete, and the operator must measure the breadth of each slit by means of the wedge, as before described, and write down the result as a colour-equation, thus — Oct. 18, J. 18-5 (24) + 27 (44) + 37 (68) = W^" (13). This equation means that on the 18th of October the observer J. (myself) made an observation in which the breadth of the slit X was 18*5, as measured by the wedge, while its centre was at the division (24) of the scale ; that the breadths of Y and Z were 27 and 37, and their positions (44) and (68) ; and that the illumination produced by these slits was exactly equal, in my estimation as an observer, to the constant white W. ON THE THEORY OF COMPOUND COLOURS. 427 The position of the slit X was then shifted from (24) to (28), and when the proper adjustments were made, I found a second colour-equation of this form — Oct. 18, J. 16(28) + 21(44) + 37(68) = W (14). Subtracting one equation from the other and remembering that the figures in brackets are merely symbols of position, not of magnitude, we find 16(28) = 18-5 (24) + 6(44) (15), shewing that (28) can be made up of (24) and (44), in the proportion of 18'5 to 6. In this way, by combining each colour with two standard colours, we may produce a white equal to the constant white. The red and yellow colours from (20) to (32) must be combined with green and blue, the greens from (36) to (52) with red and blue, and the blues from (56) to (80) with red and green. The following is a specimen of an actual series of observations made in this way by another observer (K.) : — Table III. Oct. 13, 1859. Observer (K.) (X) (Y) (Z) 18|(24) + 32|(44) + 32 (68) = W* 17^(24) + 32|(44) + 63 (80) =w. 18 (24) + 32^(44) + 35 (72) =w. 19 (24) + 32 (44) + 31|(68) =w*. 19 (24) + 30|(44) + 35 (64) =w. 20 (24) + 23 (44) + 39 (60) =w. 21 (24) + 14 (44) + 58 (56) = w. 22 (24) + 62 (52) + 11 (68) =w. 22 (24) + 42 (48) + 29^68) =w. 19 (24) + 31^(44) + 33 (68) =w* 16 (24) + 28 (40) + 32|(68) =w. 6 (24) + 27 (36) + 321(68) =w. 23 (32)+ll|(44) + 32J(68) = w. 17 (28) + 26 (44) + 32|(68 =w. 20 (24)+33|(44) + 32J(68 =w* 46 (20) + 33 (44) + 30 (68 = w. The equations marked with an asterisk (*) are those which involve the three standard colours, and since every other equation must be compared with them, they must be often repeated. 54—2 428 ON THE THEORY OP COMPOUND COLOURS. The following Table contains the means of four sets of observations by the same observer (K.) : — Table IV. (K.) 44-3 (20) + 31 16-1 (28) + 25 22-0 (32) + 12 6-4 (24) + 25 15-3 (24) + 26 19-8 (24) + 35 21 -2 (24) + 41 22 -0(24) + 62 21 -7 (24) + 10 20-5 (24) + 23 19-7 (24) + 30 18-0 (24) + 31 17-5 (24) + 30 18-3 (24) + 33 •0(44) + 27-7 (68; •6 (44) + 30-6 (68 1(44) + 30-6 (68 •2 (36) + 31 '3 (68' •0(40) + 30-7 (68 •0(46) + 30^2 (68 •4 (48) + 27^0 (68 •0(52) + 13-0 (68 4 (44) + 61 •? (56 7 (44) + 40^5 (60; •3 (44) + 33-7 (64 •2 (44) + 32 •S (72 •7 (44) + 44-0 (76' •2 (44) + 63 ^7 (80' = W. = W. = w. =w. =w. =w. =w. =w. =w. =w. =w. =w. =w. =w. § VIII. Determination of the Average Error in Observations of different kinds. In order to estimate the degree of accuracy of these observations, I have taken the differences between the values of the three standard colours as originally observed, and their means as given by the above Table. The sum of all the errors of the red (24) from the means, was 31'1, and the number of observations was 42, which gives the average error "74. The sum of errors in green (44) was 48 "0, and the number of observa- tions 31, giving a mean error 1"55. The sum of the errors in blue (68) was 46'9, and the number of observa- tions 35, giving a mean error 1*1 6. It appears therefore that in the observations generally, the average error does not exceed 1'5 ; and therefore the results, if confirmed by several obser- vations, may safely be trusted to that degree of accuracy. The equation between the three standard colours was repeatedly observed, in order to detect any alteration in the character of the light, or any other change of condition which would prevent the observations from being comparable Avith one another ; and also because this equation is used in the reduction of ON THE THEORY OP COMPOUND COLOURS. 429 all the others, and therefore requires to be carefully observed. There are twenty observations of this equation, the mean of which gives 18-6(24) + 31-4(44) + 30-5(68) = W* (16) as the standard equation. We may use the twenty observations of this equation as a means of determining the relations between the errors in the different colours, and thus of estimating the accuracy of the observer in distinguishing colours. The following Table gives the result of these operations, where E stands for (24), G for (44), and B for (68):— Table V. — Mean Errors in the Standard Equation. (R)= -54 (G) = l-22 (B) = M5 (G-B) = -99 (B - R) = -85 (R-G)=-86 G + B) = 2-67 (G + B) = 2-31 (B + R) = l-59 (R+G) = l-57 n/G'' + B^ = 1-67 sJB' + R' = 1-2Q >/R' + G''-1-33 (R + ^/R=' + G = + B^ = l-76 The first column gives the mean difference between the observed value of each of the colours and the mean of all the observations. The second column shews the average error of the observed differences between the values of the standards, from the mean value of those differences. The third column shews the average error of the sums of two standards, from the mean of such sums. The fourth column gives the square root of the sum of the squares of the quantities in the first column. I have also given the average error of the sum of R, G and B, from its mean value, and the value of JW+GF+W. It appears from the first column that the red is more accurately observed than the green and blue. § IX. Relative Accuracy in Observations of Colour and of Brightness. If the errors in the different colours occurred perfectly independent of each other, then the probable mean error in the sum or difference of any two colours would be the square root of the sum of their squares, as given in the fourth column. It will be seen, however, that the number in the second column is always less, and that in the third always greater, than that in the fourth ; shewing that the errors are not independent of each other, but that positive errors in any colour coincide more often with positive than with negative errors 430 ON THE THEORY OF COMPOUND COLOURS. in another colour. Now the hue of the resultant depends on the ratios of the components, while its brightness depends on their sum. Since, therefore, the difference of two colours is always more accurately observed than their sum, variations of colour are more easily detected than variations in brightness, and the eye appears to be a more accurate judge of the identity of colour of the two parts of the field than of their equal illumination. The same conclusion may be drawn from the value of the mean error of the sum of the three standards, which is 2-67, while the square root of the sum of the squares of the errors is 176. § X. Reduction of the Observations. By eliminating W from the equations of page 428 by means of the standard equation, we obtain equations involving each of the fourteen selected colours of the spectrum, along with the three standard colours ; and by transposing the selected colour to one side of the equation, we obtain its value in terms of the three standards. If any of the terms of these equations are negative, the equation has no physical interpretation as it stands, but by transposing the negative term to the other side it becomes positive, and then the equation may be verified. The following Table contains the values of the fourteen selected tints in terms of the standards. To avoid repetition, the symbols of the standard colours are placed at the head of each column. Table VI. Observer (K.). (24.) (44.) (68.) 44-3(20) = 18-6 + 0-4 + 2-8 16-1(28) = 18-6 + 5-8 - 0-1 22-0(32) = 18'6 + 19-3 - 0-1 25-2(36) = 12-2 + 31-4 - 0-8 26-0(40) = 3-3 + 31-4 - 0-2 35-0(46) = - 1-2 + 31-4 + 0-3 41-4(48) = - 2-6 + 31-4 + 3-5 62-0(52) = - 3-4 + 31-4 + 17-5 61-7(56) = - 3-1 + 21-0 + 30-6 40-5(60) = - 1-9 + 7-7 + 30-5 33-7(64) = - 1-1 + 1-1 + 30-5 32-3(72) = + 0-6 + 0-2 + 30-5 44-0(76) = + M + 0-7 + 30-5 63-7(80) = + 0-3 - 1-8 + 30-5 ON THE THEORY OP COMPOUND COLOURS. 431 From these equations we may lay down a chart of the spectrum on Newton's diagram by the following method :— Take any three points, A, B, C, and let A represent the standard colour (24), B (44), and G (68). Then, to find the position of any other colour, say (20), divide AC in P so that (18-6) ^P= (2-8) PC, and then divide BP in Q so that (l8-6 + 2-8)P^ = (0-4) ^P. At the point Q the colour corresponding to (20) must be placed. In this way the diagram of fig. 4, Plate VI., p. 444, has been constructed from the observations of all the colours. § XI. The Spectrum as laid down on Newton's Diagram. The curve on which these points lie has this striking feature, that two portions of it are nearly, if not quite, straight lines. One of these portions extends from (24) to (46), and the other from (48) to (64). The colour (20) and those beyond (64), are not far from the line joining (24) and (68). The spectrum, therefore, as exhibited in Newton's diagram, forms two sides of a triangle, with doubtful fragments of the third side. Now if three colours in Newton's diagram lie in a straight line, the middle one is a compound of the two others. Hence all the colours of the spectrum may be compounded of those which lie at the angles of this triangle. These correspond to the following colours : — Table VII. Scale. Wave-length. Index in water. Wave-length in water. E, Scarlet . 24 2328 1-332 1-747 G Green . . . 46| 19U 1-334 1-43.5 B Blue . . . 64| 1717 1-339 1-282 AU the other colours of the spectrum may be produced by combinations of these; and since all natural colours are compounded of the colours of the spec- trum, they may be compounded of these three primary colours. I have strong reason to believe that these are the three primary colours corresponding to three modes of sensation in the organ of vision, on which the whole system of colour, as seen by the normal eye, depends. § XII. Results found by a second Observer. We may now consider the results of three series of observations made by myself (J.) as observer, in order to determine the relation of one observer to 432 ON THE THEORY OF COMPOUND COLOURS. another in the perception of colour. The standard colours are connected by the following equation, as determined by six observations : — 18-l(24) + 27-5(44) + 37(68) = W* (17). The average errors in these observations were — E + G + B, -95 shewing that in this case, also, the power of distinguishing colour is more to be depended on than that of distinguishing degrees of illumination. The average error in the other observations from the means was "64 for red, "76 for green, and 1*02 for blue. Table VIII. R, -28 G + B, -83 G-B, -83 G, -83 B + R, -42 B-R, -28 B, -16 R+G, -95 R - G, -72 Table IX. Observations by J., October 1859. (24.) (44.) (68.) 44-3(20)= 18-1 - 2-5 + 2-3 16-0(28)= 18-1 + 6-2 - 0-7 21-5(32)= 18-1 + 25-2 - 0-7 19-3(36)= 8-1 + 27-5 - 0-3 20-7(40)= 2-1 + 27-5 - 0-5 52-3(48) = - 1-4 + 27-5 + 10-7 95-0(52) = - 2-4 + 27-5 + 37-0 51-7(56) = - 2-2 + 4-8 + 37-0 37-2(60) = - 1-2 + 0-8 + 37-0 36-7(64) = - 0-2 + 0-8 + 37-0 35-0(72) = + 0-6 - 0-2 + 37-0 40-0(76) = + 0-9 + 0-5 + 37-0 51-0(80) = + 1-1. + 0-5 + 37-0 § XIII. Comparison of Results hy Newton's Diagram. The relations of the colours, as given by these observations, are laid down in fig. 5, Plate VI., p. 444. It appears from this diagram, that the positions of the colours lie nearly in a straight line from (24) to (44), and from (48) to (60). The colours beyond (60) are crowded together, as in the other diagram, and the observations are not yet sufficiently accurate to distinguish their relative positions accurately. The colour (20) at the red end of the spectrum is further ON THE THEORY OF COMPOUND COLOTJBS. 433 from tlie line joining (24) and (68) than in the other diagram, but I have not obtained satisfactory observations of these extreme colours. It will be observed that (32), (36), and (40) are placed further to the right in fig. 5 than in fig. 4, shewing that the second observer (J.) sees more green in these colours than the first (K.), also that (48), (52), (56), and (60) are much further up in fig. 5, shewing that to the second observer they appear more blue and less green. These difi:erences were well seen in making: an observation. When the instru- ment was adjusted to suit the first observer (K.), then, if the selected colour were (32), (36), or (40), the second (J.), on looking into the instrument, saw it too green ; but if (48), (52), (56), or (60) were the selected colour, then, if right to the first observer, it appeared too blue to the second. If the instrument were adjusted to suit the second observer, then, in the first case, the other saw red, and in the second green ; shewing that there was a real difference in the eyes of these two individuals, producing constant and measurable differences in the apparent colour of objects. § XIV. Comparison by Curves of Intensity of the Primaries. Figs. 6 and 7, Plate VI. p. 444, are intended to indicate the intensities of the three standard colours at difierent points of the spectrum. The curve marked (R) indicates the intensity of the red or (24), (G) that of green or (44), and (B) that of blue or (68). The curve marked (S) has its ordinates equal to the sum of the ordinates of the other three curves. The intensities are found by dividing every colour-equation by the coefiGicient of the colour on the left-hand side. Fig. 6 represents the results of observations by K., and fig. 7 represents those of J. It will be observed that the ordinates in fig. 7 are smaller between (48) and (56) than in fig. 6. This indicates the feeble intensity of certain kinds of light as seen by the eyes of J., which made it impossible to get observations of the colour (52) at all without making the slit so wide as to include all between (48) and (56). This blindness of my eyes to the parts of the spectrum between the fixed lines E and F appears to be confined to the region surrounding the axis of vision, as the field of view, when adjusted for my eyes looking directly at the colour, is decidedly out of adjustment when I view it by indirect vision, turning the axis of my eye towards some other point. The prism then appears greener VOL. I. 55 434 ON THE THEORY OF COMPOUND COLOUKS. and brighter than the mirror, shewing that the parts of my eye at a distance from the axis are more sensitive to this blue-green light than the parts close to the axis. It is to be noticed that this insensibility is not to all light of a green or blue colour, but to light of a definite refrangibility. If I had a species of colour-blindness rendering me totally or partially insensible to that element of colour which most nearly corresponds with the light in question, then the light from the mirror, as well as that from the prism, would appear to me deficient in that colour, and I should still consider them chromatically identical ; or if there were any difierence, it would be the same for all colours nearly the same in appearance, such as those just beyond the line F, which appear to me quite bright. We must also observe that the peculiarity is confined to a certain portion of the retina, which is known to be of a yellow colour, and which is the seat of several ocular phenomena observed by Purkinje and Wheatstone, and of the sheaf or brushes seen by Haidinger in polarized light ; and also that though, of the two observers whose results are given here, one is much more affected with this peculiarity than the other, both are less sensible to the light between E and F than to that on either side ; and other observers, whose results are not here given, confirm this. § XY. Explanation of the Differences between the two Observers. I think, therefore, that the yellow spot at the foramen centrale of Soemmering wiU be found to be the cause of this phenomenon, and that it absorbs the rays between E and F, and would, if placed in the path of the incident light, produce a corresponding dark band in the spectrum formed by a prism. The reason why white light does not appear yellow in consequence, is that this absorbing action is constant, and we reckon as white the mean of all the colours we are accustomed to see. This may be proved by wearing spectacles of any strong colour for some time, when we shall find that we judge white objects to be white, in spite of the rays which enter the eye being coloured. Now ordinary white light is a mixture of all kinds of light, including that between E and F, which is partially absorbed. If, therefore, we compound an artificial white containing the absorbed ray as one of its three components, it ON THE THEORY OF COMPOUND COLOURS. 435 will be mucli more altered by the absorption than the ordinary light, which contains many rays of nearly the same colour, which are not absorbed. On the other hand, if the artificial light do not contain the absorbed ray, it will be less altered than the ordinary light which contains it. Hence the greater the absorption the less green will those colours appear which are near the absorbed part, such as (48), (52), (56), and the more greea will the colours appear which are not near it, such as (32), (36), (40). And these are the chief differences between fig. 4 and fig. 5. I first observed this peculiarity of my eyes when observing the spectrum formed by a very long vertical slit. I saw an elongated dark spot running up and down in the blue, as if confined in a groove, and following the motion of the eye as it moved up or down the spectrum, but refusing to pass out of the blue into other colours. By increasing the breadth of the spectrum, the dark portion was found to correspond to the foramen centrale, and to be visible only when the eye is turned towards the blue-green between E and F. The spot may be well seen by first looking at a yellow paper, and then at a blue one, when the spot will be distinctly seen for a short time, but it soon dis- appears when the eye gets accustomed to the blue*. I have been the more careful in stating this peculiarity of my eyes, as I have reason to believe that it affects most persons, especially those who can see Haidinger's brushes easily. Such persons, in comparing their vision with that of others, may be led to think themselves affected with partial colour-blindness, whereas their colour-vision may be of the ordinary kind, but the rays which reach their sense of sight may be more or less altered in their proportions by passing through the media of the eye. The existence of real, though partial colour-blindness will make itself apparent, in a series of observations, by the discrepancy between the observed values and the means being greater in certain colours than in others. § XVI. General Conclusions. Neither of the observers whose results are given here shew any indications of colour-blindness, and when the differences arising from the absorption of the rays between E and F are put out of account, they agree in proving that there are three colours in the spectrum, red, green, and blue, by the mixtures of * See the Report of the British Association for 1856, p. 12. 55—2 436 ON THE THEORY OF COMPOUND COLOURS. which colours chromatically identical with the other colours of the spectrum may be produced. The exact position of the red and blue is not yet ascer- tained ; that of the green is ^ from E towards F. The orange and yellow of the spectrum are chromatically equivalent to mixtures of red and green. They are neither richer nor paler than the corre- sponding mixtures, and the only difference is that the mixture may be resolved by a prism, whereas the colour in the spectrum cannot be so resolved. This result seems to put an end to the pretension of yellow to be considered a primary element of colour. In the same way the colours from the primary green to blue are chro- matically identical with mixtures of these ; and the extreme ends of the specti'um are probably equivalent to mixtures of red and blue, but they are so feeble in illumination that experiments on the same plan with the rest can give no result, but they must be examined by some special method. When observations have been obtained from a greater number of individuals, including those whose vision is dichromatic, the chart of the spectrum may be laid down independently of accidental differences, and a more complete discussion of the laws of' the sensation of colour attempted. POSTSCKIPT. [Received May 8, — Read May 24, I860.] Since sending the above paper to the Royal Society, I have obtained some observations of the colour of the spectrum by persons whose vision is " dichromic," and who are therefore said to be " colour-blind." The instrument used in making these observations was similar in principle to that formerly described, except that, in order to render it portable, the rays are reflected back through the prisms, nearly in their original direction ; thus rendering one of the limbs of the instrument unnecessary, and allowing the other to be shortened considerably on account of the greater angular dispersion. The principle of reflecting light, so as to pass twice through the same prism, was employed by me in an instrument for combining colours made in 1856 and a reflecting instrument for observing the spectrum has been constructed independently by M. Porro. ON THE THEORY OP COMPOUND COLOURS. 437 Light from a sheet of paper illuminated by sunlight is admitted at the slits X, Y, Z (fig. 8, Plate YII. p. 444), falls on the prisms P and P' (angles = 45"), then on a concave silvered glass, S, radius -34 inches. The light, after reflexion, passes again through the prisms P' and P, and is reflected by a small mirror, e, to the slit E, where the eye is placed to receive the light compounded of the colours corresponding to the positions and breadths of the slits X, Y, and Z. At the same time, another portion of the light from the illuminated paper enters the instrument at EC, is reflected at the mirror M, passes through the lens L, is reflected at the mirror M', passes close to the edge of the prism P, and is reflected along with the coloured light at e, to the eye-slit at E. In this way the compound colour is compared with a constant white light in optical juxtaposition with it. The mirror M is made of silvered glass, that at M' is made of glass roughened and blackened at the back, to reduce the intensity of the constant light to a convenient value for the experiments. This instrument gives a spectrum in which the lines are very distinct, and the length of the spectrum from A to jH" is 3 '6 inches. The outside measure of the box is 3 feet 6 inches, by 11 inches by 4 inches, and it can be carried about, and set up in any position, without readjustment. It was made by Messrs Smith and Ramage of Aberdeen. In obtaining observations from colour-blind persons, two slits only are required to produce a mixture chromatically equivalent to white ; and at one point of the spectrum the colour of the pure rays appears identical with white. This point is near the line F, a little on the less refrangible side. From this point to the more refrangible end of the spectrum appears to them "blue." The colours on the less refrangible side appear to them all of the same quahty, but of different degrees of brightness ; and when any of them are made sufficiently bright, they are called "yellow." It is convenient to use the term "yeUow" in speaking of the colours from red to green inclusive, since it will be found that a dichromic person in speaking of red, green, orange, and brown, refers to different degrees of brightness or purity of a single colour, and not to different colours perceived by him. This colour we may agree to call "yellow," though it is not probable that the sensation of it is like that of yellow as perceived by us. Of the three standard colours which I formerly assumed, the red appears to them "yellow," but so feeble that there is not enough in the whole red division of the spectrum to form an equivalent to make up the standard white. 438 ON THE THEORY OE COMPOUND COLOURS. The green at E appears a good "yellow," and the blue at f from F towards G appears a good "blue." I have therefore taken these as standard colours for reducing dichromic observations. The three standard colours will be referred to as (104), (88), and (68), these being the positions of the red, green, and blue on the scale of the new instrument. Mr James Simpson, formerly student of Natural Philosophy in my class, has furnished me with thirty-three observations taken in good sunlight. Ten of these were between the two standard colours, and give the following result : — 3.3-7 (88) + 33-1 (68) = W (1). The mean errors of these observations were as follows : — ■ Error of (88) = 2-5; of (68) = 2-3; of (88) + (68) = 4-8 ; of (88) -(68) = I'S. The fact that the mean error of the sum was so much greater than the mean error of the difference indicates that in this case, as in all others that I have examined, observations of equality of tint can be depended on much more than observations of equality of illumination or brightness. From six observations of my own, made at the same time, I have deduced the "trichromic" equation 22-6 (104) + 26 (88) + 37-4 (68) = W (2). If we suppose that the light which reached the organ of vision was the same in both cases, we may combine these equations by subtraction, and so find 22-6 (104) -7-7 (88) + 4-3 (68) = Z) (3), where D is that colour, the absence of the sensation of which constitutes the defect of the dichromic eye. The sensation which I have in addition to those of the dichromic eye is therefore similar to the full red (104), but different from it, in that the red (104) has 77 of green (88) in it which must be removed, and 4-3 of blue (68) substituted. This agrees pretty well with the colour which Mr Pole* describes as neutral to him, though crimson to others. It must be remembered, however, that different persons of ordinary vision require different proportions of the standard colours, probably owing to differences in the absorptive powers of the media of the eye, and that the above equation (2), if observed by K., would have been 23(104) + 32(88) + 31(68) = W (4). * Philosophical Transactions, 1859, Part I. p. 329. ON THE THEORY OF COMPOUND COLOURS. 439 and the value of D, as deduced from these observers, would have been 23 (104) -17 (88) -ri (68) -D (5), in which the defective sensation is much nearer to the red of the spectrum. It is probably a colour to which the extreme red of the spectrum tends, and which differs from the extreme red only in not containing that small proportion of "yellow" light which renders it visible to the colour-blind. From other observations by Mr Simpson the following results have been deduced : — Table a. (88.) (68.) (99-2 + ) = 33-7 1-9 31-3(96) = 33-7 2-1 28 (92) = 33-7 1-4 33-7(88) = 33-7 54-7(84) = 33-7 6-1 71 (82) = 33-7 15-1 99 (80) = 33-7 331 70 (78) = 15-7 33-1 56 (76) = 5-7 33-1 36 (72) = - 0-3 33-1 33-1(68) = 33-1 40 (64) = 0-2 33-1 55-5(60) = 1-7 33-1 (57-) = - 0-3 33-1 (88.) (68.) 100(96) = 108 7 100(92) = 120 5 100(88) = 100 100(84) = 61 11 100(82) = 47 21 100(80) = 34 33 100 (78) = 22 47 100(76) = 10 59 100(72)= ■ - 1 92 100(68) = 100 100(64) = 83 100 (60) = 3 60 In the Table on the left side (99"2-|-) means the whole of the spectrum beyond (99"2) on the scale, and (57 — ) means the whole beyond (57) on the scale. The position of the fixed lines with reference to the scale was as follows : — A, 116; a, 112; B, 110; C, 106; D, 98-3; E, 88; F, 79; G, 61; H, 44. The values of the standard colours in different parts of the spectrum are given on the right side of the above Table, and are represented by the curves of fig. 9, Plate VII. p. 444, where the left-hand curve represents the intensity of the " yellow " element, and the right-hand curve that of the " blue " element of colour as it appears to the colour-blind. The appearance of the spectrum to the colour-blind is as follows: — From ^ to ^ the colour is pure " yellow " very faint up to D, and reaching a maximum between D and E. From E to one-third beyond F towards 440 ON THE THEORY OE COMPOUND COLOURS. G the colour is mixed, varying from "yellow" to "blue," and becoming neutral or "white" at a point near F. In this part of the spectrum, the total inten- sity, as given by the dotted line, is decidedly less than on either side of it, and near the line F, the retina close to the "yellow spot" is less sensible to light than the parts further from the axis of the eye. This peculiarity of the light near F is even more marked in the colour-bhnd than in the ordinary eye. Beyond F the " blue " element comes to a maximum between F and G, and then diminishes towards H ; the spectrum from this maximum to the end being pure " blue." In fig. 10, Plate VII. p. 444, these results are represented in a different manner. The point D, corresponding to the sensation wanting in the colour-blind, is taken as the origin of coordinates, the "yellow" element of colour is represented by distances measured horizontally to the right from D, and the "blu.e" element by distances measured vertically from the horizontal line through D. The numerals indicate the different colours of the spectrum according to the scale shewn in fig. 9, and the coordinates of each point indicate the composition of the corresponding colour. The triangle of colours is reduced, in the case of dichromic vision, to a straight line "B" "Y," and the proportions of "blue" and "yellow" in each colour are indicated by the ratios in which this line is cut by the line from D passing through the position of that colour. The results given above were all obtained with the light of white paper, placed in clear sunshine. I have obtained similar results, when the sun was hidden, by using the light of uniformly illuminated clouds, but I do not consider these observations sufficiently free from disturbing circumstances to be employed in calculation. It is easy, however, by means of such observations, to verify the most remarkable phenomena of colour-blindness, as for instance, that the colours from red to green appear to differ only in brightness, and that the brightness may be made identical by changing the width of the slit ; that the colour near F is a neutral tint, and that the eye in viewing it sees a dark spot in the direction of the axis of vision ; that the colours beyond are aU blue of different intensities, and that any "blue" may be combined with any "yellow" in such proportions as to form "white." These results I have verified by the observations of another colour-blind gentleman, who did not obtain sunlight for his observations ; and as I have now the means of carrying the requisite apparatus easily, I hope to meet with other colour-blind observers, and to obtain their observations under more favourable circumstances. ON THE THEORY OP COMPOUND COLOURS. 441 0)1 the Comparison of Colour-blind with ordinary Vision by means of Observations with Coloured Papers. In March 1859 I obtained a set of observations by Mr Simpson, of the relations between six coloured papers as seen by him. The experiments were made with the colour-top in the manner described in my paper in the Trans- actions of the Royal Society of Edinburgh, Vol. xxi. pt. 2, p. 286 ; and the colour-equations were arranged so as to be equated to zero, as in those given in the Philosophical Magazine, July, 1857. The colours were — Vermilion (V), ultramarine (U), emerald-green (G), ivory-black (B), snow-white (W), and pale chrome-yellow (Y). These six colours afford fifteen colour-bhnd equations, since four colours enter into each equation. Fourteen of these were observed by Mr Simpson, and from these I deduced three equations, giving the relation of the three standards (V), (U), (G) to the other colours, according to his kind of vision. From these three equations I then deduced fifteen equations, admitting of comparison with the observed equations, and necessarily consistent in themselves. The comparison of these equations furnishes a test of the truth of the theory that the colour-blind see by means of two colour-sensations, and that therefore every colour may be expressed in terms of two given colours, just as in ordinary vision it may be expressed in terms of three given colours. The one set of equations are each the result of a single observation; the other set are deduced from three equations in accordance with this theory, and the two sets agree to within an average error = 2*1. 2. Table b. V. V. G. B. W. Y. Observed . . -100 + 45 + 22 + 33 = 0. Calculated . -100 + 37-5 + 26-5 + 36 =0. Observed . . + 58 -69 -31 -42 =0. Calculated . + 58-3 -67-3 -32-7 + 41-7 = 0. Observed . . + 32 -100 + 12 + 56 =0. Calculated . + 32-3 -100 + 8-3 + 59-4 = 0. Observed . . + 38 - 89 -11 + 62 =0. Calculated . . + 40 - 85 -15 + 60 =0. Observed . . + 32 + 68 -60 -40 =0. Calculated . . + 34 + 66 - 63-5 -36'5 =0. VOL. I. 56 442 ON THE THEORY OP COMPOUND COLOURS. Table b (continued). 6. 7. 10. 11. 12. 13. 14. 15. V. U. G. B. W. Y. Observed . . .-100 + 82 + 5 + 13 =0. Calculated . .-100 + 83-9 + 4-5 + 11-6 = 0. Observed . . .+ 47 - 100 + 22 + 31 =0. Calculated . .+ 44-7 - 100 + 24-5 + 30-8 = 0. Observed . . .-100 + 20 + 77 + 3 =0. Calculated . .-100 + 17 + 77-5 + 5-5=0. Not Observed. Calculated . .+ 96 - 31 -69 + 4 0=0. Observed . . .- 70 + 53 -30 + 47 =0. Calculated . .- 73-5 + 53 -26-5 + 47 =0. Observed . . .-100 + 8 + 71 + 21 =0. Calculated . .-100 + 8 + 74-5 + 17-5=0. Observed . . .+ 85 + 15 -88 -12 =0. Calculated . .+ 86 + 14 -88-5 -11-5 =0. Observed . . .- 20 + 39 - 80 + 61 =0. Calculated . .- 19 + 40 - 81 + 60 =0. Observed . . .- 66 + 30 + 70 -34 =0. Calculated . .- 70 + 27 + 73 -30 =0. Observed . . . + 100 - 2 - 27 -71 =0. Calculated . .+ 96 + 4 - 24 -76 =0. But, according to our theory, colour-blind vision is not only dichromic, but the two elements of colour are identical with two of the three elements of colour as seen by the ordinary eye ; so that it differs from ordinary vision only in not perceiving a particular colour, the relation of which to known colours may be numerically defined. This colour may be expressed under the form aV + hV + cG^J) (16), where V, U, and G are the standard colours used in the experiments, and D is the colour which is visible to the ordinary eye, but invisible to the colour- blind. If we know the value of D, we may always change an ordinary colour- equation into a colour-blind equation by subtracting from it nD (n being chosen so that one of the standard colours is eliminated), and adding n of black. In September 1856 I deduced, from thirty-six observations of my own, the chromatic relations of the same set of six coloured papers. These observations with a comparison of them with the trichromic theory of vision, are to be found in the Philosophical Magazine for July 1857. The relations of the ON THE THEOEY OF COMPOUND COLOURS. 443 six colours may be deduced from two equations, of which the most convenient form is V. U. G. B. "W. Y. + 39-7 +26-6 +33-7 -227 -77-3 =0 (17). -62-4 +18-6 -37-6 +457 +357 = (18). The value of D, as deduced from .a comparison of these equations with the colour-blind equations, is 1-198 V + 0-078U-0-276G = D (19). By making D the same thing as black (B), and eliminating W and Y respectively from the two ordinary colour-equations by means of D, we obtain three colour-blind equations, calculated from the ordinary equations and con- sistent with them, supposing that the colour (D) is black to the colour-blind. The following Table is a comparison of the colour-bhnd equations deduced from Mr Simpson's observations alone, with those deduced from my observations and the value of D. (15) Calculated By (19) . . . (14) Calculated By (17) and (19) (13) Calculated By (18) and (19) Table c. V. V. G. B. W. Y. + 96 +4 -24 -76 + 93-9 + 6-1 -21-7 -78-3 -70 +27 +73 0-30 -70 +27-2 -72-8 0-30 -19 +40 -81 +60 -13-6 +38-5 -86-4 +61-5 The average error here is 1-9, smaller than the average error of the indi- vidual colour-blind observations, shewing that the theory of colour-bhndness being the want of a certain colour-sensation which is one of the three ordinary colour- sensations, agrees with observation to within the limits of error. In fig. 11, Plate VII. p. 444, I have laid down the chromatic relations of these colours according to Newton's method. V (vermilion), U (ultramarine), and G (emerald-green) are assumed as standard colours, and placed at the angles of an equilateral triangle. The position of W (white) and Y (pale chrome-yellow) with respect to these are laid down from equations (17) and (18), deduced from my own observations. The positions of the defective colour, of white, and of yellow, as deduced from Mr Simpson's equations alone, are given at " d," "w," and "y." The positions of these points, as deduced from a combination 56—2 444 ON THE THEOEY OF COMPOUND COLOUES. of these equations with my own, are given at "D," "W," and "Y." The difference of these positions from those of " d" " w" and "y" shews the amount of discrepancy between observation and theory. It will be observed that D is situated near V (vermUion), but that a line from D to W cuts UV at C near to V. D is therefore a red colour, not scarlet, but further from yellow. It may be called crimson, and may be imitated by a mixture of 86 vermilion and 14 ultramarine. This compound colour will be of the same hue as D ; but since C lies between D and W, C must be regarded as D diluted with a certain amount of white ; and therefore D must be imagined to be Hke C in hue, but without the intermixture of white which is unavoidable in actual pigments, and which reduces the purity of the tint. Lines drawn from D through "W" and "Y," the colour-blind positions of white and yellow, pass through W and Y, their positions in ordinary vision. The reason why they do not coincide with W and Y, is that the white and yellow papers are much brighter than the colours corresponding to the points W and Y of the triangle V; U, G; and therefore lines from D, which represent them in intensity as weU as in quality, must be longer than DW and DY in the proportion of their brightness. VOL. I. PLATE VI. (I) Ch' Compound/ (blears. Pig. 1. We Eg. 4 . jz6S..^ 1) •Gr ^ Et\ .■■'G- '--A & :Aa- '"■■■^ — i,.. i : Us W M 28 SZ ~36'^ 44'18~S&-S6-eO-Gf''68 12 16 80 K "2/ 78 32 36 W M 48"Sl SS'60 G4- 68 72 16 80 J ^o /ace 'pagu 444 Cambridge University Preu VOL. I. PLATE VIL (ii) ^ig.8. ^^^^^^^s^^^^^^ss^^^^^^^^^^^^^^^^^^^^^^^^^ S ^''' If the vortices are not circular, and if the angular velocity and the density are not uniform, but vary according to the same law for all the vortices, Pi-p^=Cp'if, ON PHYSICAL LINES OF FORCE. 457 where p is the mean density, and C is a numerical quantity depending on the distribution of angular velocity and density in the vortex. In future we shall write -7- instead of Co, so that An '^ Pi-P^ = ^l^'"" (1)> where /i is a quantity bearing a constant ratio to the density, and v is the linear velocity at the circumference of each vortex. A medium of this kind, filled with molecular vortices having their axes parallel, differs from an ordinary fluid in having different pressures in different directions. If not prevented by properly arranged pressures, it would tend to expand laterally. In so doing, it would allow the diameter of each vortex to expand and its velocity to diminish in the same proportion. In order that a medium having these inequalities of pressure in different directions should be in equilibrium, certain conditions must be fulfilled, which we must investigate. Prop. II. — If the direction-cosines of the axes of the vortices with respect to the axes of x, y, and 2 be I, m, and n, to find the normal and tangential stresses on the co-ordinate planes. The actual stress may be resolved into a simple hydrostatic pressure p^ acting in all directions, and a simple tension Pi—p^, or j- fiv', acting along the axis of stress. Hence if p^, Pyy, and p„ be the normal stresses parallel to the three axes, considered positive when they tend to increase those axes ; and if Py^, p^^, and p^ be the tangential stresses in the three co-ordinate planes, considered positive when they tend to increase simultaneously the symbols subscribed, then by the resolution of stresses'''', Pyy = l^l^'"^-Piy Pzz = J^F'V''"^-P^> * Kankine's Applied Mechanics, Art. 106. VOL. I. 58 458 ON PHYSICAL LINES OF FORCE. 1 P=^ = ^H'V'i^- If we write then Pyy = 4^ /^^' -Pi' -P^« = 4^ /^r''^ (2). Prop. III. — To find the resultant force on an element of the medium, arising from the variation of internal stress. We have in general, for the force in the direction of x per unit of volume by the law of equihbrium of stresses''''', ■XT- ^ Oj Qj f ^ ^^d^P^ + dyP-' + dzP- (^)- In this case the expression may be written X— ^ j^il^°') I ^" 1 dpi .(^{H'^) , pda d (fjiy) da\ , , ~ 4:TT \ dx ^ dx dx dy ^ dy dz "^ dz\ ^ '' Remembering that a t- + /8 -i^ + y ^ = - -^ (a' + ^' + /), this becomes z=<.ii^(H+|w)4(^r)} + ^^s(«-+^+/) 47r\c?x ■Z.^- 1 Id^ da\ 1 /da d/y\ dp^ Ait \dx dy) '^ Av \dz dx) dx (5). The expressions for the forces parallel to the axes of y and 2 may be written down from analogy. * Eankine's Api)lied Mechanics, Art. 116. ON PHYSICAL LINES OF FORCE, 459 We have now to interpret the meaning of each term of this expression. We suppose a, /3, y to be the components of the force which would act upon that end of a unit magnetic bar which points to the north. ft represents the magnetic inductive capacity of the medium at any point referred to air as a standard, /xa, fji/3, /xy represent the quantity of magnetic induction through unit of area perpendicular to the three axes of x, y z respectively. The total amount of magnetic induction through a closed surface surrounding the pole of a magnet, depends entirely on the strength of that pole; so that if dxdydz be an element, then I -7- jita + -y- ju.;8 + -T- [ly] dx dy dz = iirm dxdydz (6), which represents the total amount of magnetic induction outwards through the surface of the element dxdydz, represents the amount of "imaginary magnetic matter" within the element, of the kind which points north. The Jlrst term of the value of X, therefore, 1 / d d ^ d \ ,„. "4^U^" + d^'^^ + dz^>'j (^)' may be written am (8), where a is the intensity of the magnetic force, and m is the amount of mag- netic matter pointing north in unit of volume. The physical interpretation of this term is, that the force urging a north pole in the positive direction of x is the product of the intensity of the magnetic force resolved in that direction, and the strength of the north pole of the magnet. Let the parallel lines from left to right in fig. 1 represent a field of mag- netic force such as that of the earth, sn being the direction from south to north. The vortices, according to our hypothesis, will be in the direction shewn by the arrows in fig. 3, that is, in a plane perpendicular to the lines of force, and revolving in the direction of the hands of a watch when observed from s looking towards n. The parts of the vortices above the plane of the paper will be moving towards e, and the parts below that plane towards w. 58—2 460 ON PHYSICAL LINES OF FORCE. Mg. 1. Kg. 2. -^ T ^m^ -7^ V- ^ "We shall always mark by an arrow-head the direction in which we must look in order to see the vortices rotating in the direction of the hands of a watch. The arrow-head will then indicate the northward direction in the magnetic field, that is, the direction in which that end of a magnet which points to the north would set itself in the field. Now let A be the end of a magnet which points north. Since it repels the north ends of other magnets, the fines of force will be directed from A outwards in all directions. On the north side the line AD will be in the same direction with the lines of the magnetic field, and the velocity of the vortices will be increased. On the south side the line AC will be in the opposite direction, and the velocity of the vortices will be diminished, so that the lines of force are more powerful on the north side of A than on the south side. We have seen that the mechanical efiect of the vortices is to produce a tension along their axes, so that the resultant effect on A will be to pull it more powerfully towards D than towards C; that is, A will tend to move to the north. Let B in fig. 2 represent a south pole. The lines of force belonging to B will tend towards B, and we shall find that the lines of force are rendered stronger towards E than towards F, so that the effect in this case is to urge B towards the south. It appears therefore that, on the hypothesis of molecular vortices, our first term gives a mechanical explanation of the force acting on a north or south pole in the magnetic field. We now proceed to examine the second term, Here a' + yS' + y" is the square of the intensity at any part of the field, and jji is the magnetic inductive capacity at the same place. Any body therefore ON PHYSICAL LINES OF FORCE, 461 placed in the field wiU be urged towards places of stronger magnetic intensity with a force depending partly on its own capacity for magnetic induction, and partly on the rate at which the square of the intensity increases. If the body be placed in a fluid medium, then the medium, as weU as the body, will be urged towards places of greater intensity, so that its hydrostatic pressure wiU be increased in that direction. The resultant effect on a body placed m the medium will be the difference of the actions on the body and on the portion of the medium which it displaces, so that the body will tend to or from places of greatest magnetic intensity, according as it has a greater or less capacity for magnetic induction than the surrounding medium. In fig. 4 the Hnes of force are represented as converging and becoming more powerful towards the right, so that the magnetic tension at B is stronger than at A, and the body AB wiU be urged to the right. If the capacity for magnetic induction is greater in the body than in the surrounding medium, it will move to the right, but if less it wUl move to the left. Fig. 4. We may suppose in this case that the lines of force are converging to a magnetic pole, either north or south, on the right hand. In fig. 5 the lines of force are represented as vertical, and becoming more numerous towards the right. It may be shewn that if the force increases towards the right, the lines of force will be curved towards the right. The efiect of the magnetic tensions will then be to draw any body towards the right with a force depending on the excess of its inductive capacity over that of the surrounding medium. We may suppose that in this figure the lines of force are those surrounding an electric current perpendicular to the plane of the paper and on the right hand of the figure. These two illustrations will shew the mechanical effect on a paramagnetic or diamagnetic body placed in a field of varying magnetic force, whether the increase of force takes place along the lines or transverse to them. The form 462 ON PHYSICAL LINES 01" FOECE. of the second term of our equation indicates the general law, which is quite independent of the direction of the lines of force, and depends solely on the manner in which the force varies from one part of the field to another. We come now to the third term of the value of X, 1 [d^ da\ At: \dx dy, Here /i/3 is, as before, the quantity of magnetic induction through unit of area perpendicular to the axis of y, and -^ — -f- is a quantity which would disap- pear if adx + /3dy + ydz were a complete differential, that is, if the force acting on a unit north pole were subject to the condition that no work can be done upon the pole in passing round any closed curve. The quantity represents the work done on a north pole in travelling round unit of area in the direction from +x to +y parallel to the plane of xy. Now if an electric current whose strength is r is traversing the axis of z, which, we may suppose, points vertically upwards, then, if the axis of x is east and that of y north, a unit north pole will be urged round the axis of z in the direction from x to y, so that in one revolution the work done will be =4irr. Hence — (-/^ — ^] repre- 47r \dx ay] ^ sents the strength of an electric current parallel to z through unit of area ; and if we write _l_(dy_d§\^ ±/da_dy\ J^m da\_ iTr\dy dz) P' iTr\dz dx)~^' 4iT[dx dy)~^ ^^^' then p, q, r will be the quantity of electric current per unit of area perpen- dicular to the axes of x, y, and z respectively. The physical interpretation of the third term of X, —[x/Sr, is that if fifi is the quantity of magnetic induction parallel to y, and r the quantity of electricity flowing in the direction of z, the element will be urged in the direction of —x, transversely to the direction of the current and of the lines of force; that is, an ascending current in a field of force magnetized towards the north would tend to move west. To illustrate the action of the molecular vortices, let sn be the direction of magnetic force in the field, and let C be the section of an ascending mag- netic current perpendicular to the paper. The lines of force due to this current ON PHYSICAL LINES OP POECE. 463 will be circles drawn in the opposite direction from that of the hands of a watch ; that is, in the direction nwse. At e the lines of force Fig. 6. will be the sum of those of the field and of the current, and at w they will be the difference of the two sets of lines ; so that the vortices on the east side of the current will be more powerful than those on the west side. Both sets of vortices have their equatorial parts turned towards C, so that they tend to expand towards C, but those on the east side have the greatest effect, so that the resultant effect on the current is to urge it towards the west. The fourth term, 1 /da dy\ . ^, + '^^i^U-5SJ'''^+^^^ (^^)' may be interpreted in the same way, and indicates that a current q in the direction of y, that is, to the north, placed in a magnetic field in which the lines are vertically upwards in the direction of z, will be urged towards the east. The fifth term, -£' (")> merely implies that the element will be urged in the direction in which the hydrostatic pressure j?i diminishes. "We may now write down the expressions for the components of the resultant force on an element of the medium per unit of volume, thus : ^='^^'^l^i^^')-^^^^^y'i-% (12)' Y=^-'-^i^Ty^^')-^yp^^^''~'^ (1^)' ^=r^ + ^/*^(^')-/^«?+/^^i'-^' (14 The first term of each expression refers to the force acting on magnetic poles. The second term to the action on bodies capable of magnetism by induction. The third and fourth terms to the force acting on electric currents. And the fifth to the effect of simple pressure. 464 On physical lines of foece. Before going further in the general investigation, we shall consider equations (12, 13, 14), in particular cases, corresponding to those simplified cases of the actual phenomena which we seek to obtain in order to determine their laws by experiment. We have found that the quantities p, q, and r represent the resolved parts of an electric current in the three co-ordinate directions. Let us suppose in the first instance that there is no electric current, or that p, q, and r vanish. We have then by (9), dy dz ' dz dx ' dx dy ' ^ '' whence we learn that adx + ^dy + ydz = d d%\ represents the amount of imaginary magnetic matter in unit of volume. That there may be no resultant force on that unit of volume arising from the action represented by the first term of equations (12, 13, 14), we must have m = 0, or dx''^df^ dz' ~ " (19)- Now it may be shewn that equation (19), if true within a given space, implies that the forces acting within that space are such as would result from a distribution of centres of force beyond that space, attracting or repelling inversely as the square of the distance. Hence the lines of force in a part of space where /t is uniform, and where there are no electric currents, must be such as would result from the theory of "imaginary matter" acting at a distance. The assumptions of that theory are unlike those of ours, but the results are identical. m-- ON PHYSICAL LINES OF FORCE. 465 Let us first take the case of a single magnetic pole, that is, one end of a long magnet, so long that its other end is too far off to have a perceptible influence on the part of the field we are considering. The conditions then are, that equation (18) must be fulfilled at the magnetic pole, and (19) everywhere else. The only solution under these conditions is ^--,-r (^«). where r is the distance from the pole, and m the strength of the pole. The repulsion at any point on a unit pole of the same kind is d(^ _m 1 dr fj. 7^ (21). In the standard medium ju. = 1 ; so that the repulsion is simply — in that medium, as has been shewn by Coulomb. In a medium having a greater value of fi (such as oxygen, solutions of salts of iron, &c.) the attraction, on our theory, ought to be less than in air, and in diamagnetic media (such as water, melted bismuth, &c.) the attraction between the same magnetic poles ought to be greater than in air. The experiments necessary to demonstrate the difference of attraction of two magnets according to the magnetic or diamagnetic character of the medium in which they are placed, would require great precision, on account of the limited range of magnetic capacity in the fluid media known to us, and the small amount of the difference sought for as compared with the whole attraction. Let us next take the case of an electric current whose quantity is C, flowing through a cylindrical conductor whose radius is M, and whose length is infinite as compared with the size of the field of force considered. Let the axis of the cylinder be that of z, and the direction of the current positive, then within the conductor the quantity of current per unit of area is C _ 1 (d^ da\ , . so that within the conductor a=-2-^2/, ^ = 2^x, 7 = (23). VOL. I. 59 466 ON PHYSICAL LINES OF FORCE. Beyond the conductor, in the space round it, d, = 2Ctan-^^ (24), T IT, X # ^p y /D_#_2C-^- v = ^ = (25). If p = JW+f is the perpendicular distance of any point from the axis of 2(7 the conductor, a unit north pole will experience a forces — , tending to move it round the conductor in the direction of the hands of a watch, if the observer view it in the direction of the current. Let us now consider a current running parallel to the axis of z in the plane of xz at a distance p. Let the quantity of the current be c, and let the length of the part considered be I, and its section s, so that - is its strength per unit of section. Putting this quantity for p in equations (12, 13, 14), we find c •^ s per unit of volume ; and multiplying by Is, the volume of the conductor con- sidered, we find Pc'l = -2/.— (26), shewing that the second conductor wiU be attracted towards the first with a force inversely as the distance. We find in this case also that the amount of attraction depends on the value of p., but that it varies directly instead of inversely as /t ; so that the attraction between two conducting wires will be greater in oxygen than in air, and greater in air than in water. We shall next consider the nature of electric currents and electromotive forces in connexion with the theory of molecular vortices. ON PHYSICAL LINES OF FORCE. 467 PART II. The Theory op Molecular Vortices applied to Electric Currents. We have already shewn that aU the forces acting between magnets, sub- stances capable of magnetic induction, and electric currents, may be mechanically accounted for on the supposition that the surroxmding medium is put into such a state that at every point the pressures are different in different directions, the direction of least pressure being that of the observed lines of force, and the difference of greatest and least pressures being proportional to the square of the intensity of the force at that point. Such a state of stress, if assumed to exist in the medium, and to be arranged according to the known laws regulating lines of force, will act upon the magnets, currents, &c. in the field with precisely the same resultant forces as those calculated on the ordinary hypothesis of direct action at a distance. This is true independently of any particular theory as to the cause of this state of stress, or the mode in which it can be sustained in the medium. We have therefore a satisfactory answer to the question, " Is there any mechanical hypothesis as to the condition of the medium indicated by lines of force, by which the observed resultant forces may be accounted fori" The answer is, the lines of force indicate the direction of minimum pressure at every point of the medium. The second question must be, " What is the mechanical cause of this difference of pressure in different directions ? " We have supposed, in the first part of this paper, that this difference of pressures is caused by molecular vortices, having their axes parallel to the lines of force. We also assumed, perfectly arbitrarily, that the direction of these vortices is such that, on looking along a line of force from south to north, we should see the vortices revolving in the direction of the hands of a watch. We found that the velocity of the circumference of each vortex must be proportional to the intensity of the magnetic force, and that the density of the substance of the vortex must be proportional to the capacity of the medium for magnetic induction. We have as yet given no answers to the questions, "How are these vortices set in rotation?" and "Why are they arranged according to the known laws 59—2 468 ON PHYSICAL LINES OF FOECE. of lines of force about magnets and currents?" These questions are certainly of a higher order of difficulty than either of the former ; and I wish to separate the suggestions I may offer by way of provisional answer to them, from the mechanical deductions which resolved the first question, and the hypothesis of vortices which gave a probable answer to the second. We have, in fact, now come to inquire into the physical connexion of these vortices with electric currents, whUe we are still in doubt as to the nature of electricity, whether it is one substance, two substances, or not a substance at all, or in what way it differs from matter, and how it is connected with it. We know that the lines of force are affected by electric currents, and we know the distribution of those lines about a current ; so that from the force we can determine the amount of the current. Assuming that our explanation of the lines of force by molecular vortices is correct, why does a particular distribution of vortices indicate an electric current ? A satisfactory answer to this question would lead us a long way towards that of a very important one, " What is an electric current ? " I have found great difficulty in conceiving of the existence of vortices in a medium, side by side, revolving in the same direction about parallel axes. The contiguous portions of consecutive vortices must be moving in opposite directions ; and it is difficult to understand how the motion of one part of the medium can coexist with, and even produce, an opposite motion of a part in contact with it. The only conception which has at all aided me in conceiving of this kind of motion is that of the vortices being separated by a layer of particles, revolving each on its own axis in the opposite direction to that of the vortices, so that the contiguous surfaces of the particles and of the vortices have the same motion. In mechanism, when two wheels are intended to revolve in the same direc- tion, a wheel is placed between them so as to be in gear with both, and this wheel is called an "idle wheel." The hypothesis about the vortices which I have to suggest is that a layer of particles, acting as idle wheels, is interposed between each vortex and the next, so that each vortex has a tendency to make the neighbouring vortices revolve in the same direction with itself In mechanism, the idle wheel ia generally made to rotate about a fixed axle ; but in epicyclic trains and other contrivances, as, for instance, in Siemens's ON PHYSICAL LINES OP FORCE. 469 governor for steam-engines*, we find idle wheels whose centres are capable of motion. In all these cases the motion of the centre is the half sum of the motions of the circumferences of the wheels between which it is placed. Let us examine the relations which must subsist between the motions of our vortices and those of the layer of particles interposed as idle wheels between them. Prop. IV. — To determine the motion of a layer of particles separating two vortices. Let the circumferential velocity of a vortex, multiplied by the three direc- tion-cosines of its axis respectively, be a, ^, y, as in Prop. II. Let I, m, n be the direction-cosines of the normal to any part of the surface of this vortex, the outside of the surface being regarded positive. Then the components of the velocity of the particles of the vortex at this part of its surface will be nfi — my parallel to x, ly — na parallel to y^ ma — lfi parallel to z. If this portion of the sxirface be in contact with another vortex whose velocities are a, /S', 7', then a layer of very small particles placed between them will have a velocity which will be the mean of the superficial velocities of the vortices which they separate, so that if u is the velocity of the particles in the direction of x, u = im{y'-y)-in{^-^) (27), since the normal to the second vortex is in the opposite direction to that of the first. Prop. V. — To determine the whole amount of particles transferred across unit of area in the direction of x in unit of time. Let a?!, 2/i, z^ be the co-ordinates of the centre of the first vortex, x^, y^, z., those of the second, and so on. Let Fj, V^, &c. be the volumes of the first, second, &c. vortices, and V the sum of their volumes. Let dS be an element of the surface separating the first and second vortices, and x, 3/, z its co-ordinates. Let p be the quantity of particles on every unit of surface. Then if j9 be the whole quantity of particles transferred across unit of area in unit of time in * See Goodeve's Elements of Mechanism, p. 118. 470 ON PHYSICAL LINES OF FOECE. the direction of x, the whole momentum parallel to x of the particles within the space whose volume is V will be Vp, and we shall have Vp = tupdS (28), the summation being extended to every surface separating any two vortices^ within the volume V. Let us consider the surface separating the first and second vortices. Let an element of this surface be dS, and let its direction-cosines be l^, m„ n^ with respect to the first vortex, and \, m^, n„ with respect to the second; then we know that ^1 + ^2 = 0, m, + m, = 0, n^ + n^ = (29). The values of a, ^, y vary with the position of the centre of the vortex; so that we may write a, = a, + ^(^,-xO + |(y,-2/0 + |(^.-^0 (30), with similar equations for fi and y. The value of u may be written : — u = i^ {m^ (x — iCi) + mi{x — x^)} + i^H(2/-2/i) + ™2(2/-y2)} + i^K(z-2:) + m,(3-z,)} - i- ^f K (^-^i) + ^» ('»-^^)}-i ^ K (3/-2/1) + «= (2/-^»)} -if K(^-20 + «.(^-«.)} (31)- In effecting the summation of 'tupdS, we must remember that round any closed surface XldS and all similar terms vanish; also that terms of the form %lydS, where I and y are measured in different directions, also vanish ; but that terms of the form "tlxdS, where I and x refer to the same axis of co-ordinates, do not vanish, but are equal to the volume enclosed by the surface. The result is f5'=ip(|-f)('^.+ '^.+*-) (32)^ ON PHYSICAL LINES OF FORCE. 471 or dividing by V=Vi+ V^ + kc, ^=iK|-f) (^^)- If we make p = — (34^)j then equation (33) will be identical with the first of equations (9), which give the relation between the quantity of an electric current and the intensity of the lines of force surrounding it. It appears therefore that, according to our hypothesis, an electric current is represented by the transference of the moveable particles interposed between the neighbouring vortices. We may conceive that these particles are very small compared with the size of a vortex, and that the mass of all the particles together is inappreciable compared with that of the vortices, and that a great many vortices, with their surrounding particles, are contained in a single complete molecule of the medium. The particles must be conceived to roll without sliding between the vortices which they separate, and not to touch each other, so that, as long as they remain within the same complete molecule, there is no loss of energy by resistance. When, however, there is a general transference of par- ticles in one direction, they must pass from one molecule to another, and in doing so, may experience resistance, so as to waste electrical energy and generate heat. Now let us suppose the vortices arranged in a medium in any arbitrary manner. The quantities rr^ — -f^ > <^c. will then in general have values, so that there wiU at first be electrical currents in the medium. These wUl be opposed by the electrical resistance of the medium ; so that, unless they are kept up by a continuous supply of force, they will quickly disappear, and we shall then have -r- — ;p = 0> ^^ > ^^^^ is, adx + ^dy + ydz wiU be a complete diflferential (see equations (15) and (16)) ; so that our hypothesis accounts for the distri- bution of the lines of force. In Plate VIII. p. 488, fig. 1, let the vertical circle EE represent an electric current flowing from copper C to zinc Z through the conductor EE', as shewn by the arrows. 472 ON PHYSICAL LINES OF FORCE. Let the horizontal circle MM' represent a line of magnetic force embracing the electric circuit, the north and south directions being indicated by the lines SN and NS. Let the vertical circles V and V represent the molecular vortices of which the line of magnetic force is the axis. V revolves as the hands of a watch, and V the opposite way. It wiU appear from this diagram, that if V and F' were contiguous vortices, particles placed between them would move downwards ; and that if the particles were forced downwards by any cause, they would make the vortices revolve as in the figure. We have thus obtained a point of view from which we may regard the relation of an electric current to its lines of force as analogous to the relation of a toothed wheel or rack to wheels which it drives. In the first part of the paper we investigated the relations of the statical forces of the system. "We have now considered the connexion of the motions of the parts considered as a system of mechanism. It remains that we should investigate the dynamics of the system, and determine the forces necessary to produce given changes in the motions of the different parts. Prop. YI. — To determine the actual energy of a portion of a medium due to the motion of the vortices within it. Let a, /3, y be the components of the circumferential velocity, as in Prop, II., then the actual energy of the vortices in unit of volume will be proportional to the density and to the square of the velocity. As we do not know the distribution of density and velocity in each vortex, we cannot determine the numerical value of the energy directly ; but since /x also bears a constant though unknown ratio to the mean density, let us assume that the energy in unit of volume is i:=Cii,{a' + ^' + '/), where (7 is a constant to be determined. Let us take the case in which dx' '^ dy- ^ dz ^^^)- Let = , + ({>, (36), , , , /x /d', , d%, , d'(f>,\ , u. /d'6, d'ib, d%,\ , , ON PHYSICAL LINES OF PORCE. 473 then ^1 is tlie potential at any point due to the magnetic system m^ and ^^ that dup to the distribution of magnetism represented by m^. The actual energy of all the vortices is E = 'ZCiJi{a' + fi' + Y)dV.... (38), the integration being performed over all space. This may be shewn by integration by parts (see Green's ' Essay on Elec- tricity,' p. 10) to be equal to E= -4:TrCt{({)r'>n^-^(f>im^ + ^m^ + (j)^m^)dV (39). Or since it has been proved (Green's 'Essay,' p. 10) that S^iWiaC? F= ^(p^mjd V, E= -ATrC{jn^ + (j>^m, + 2^ depends on mj only, it will remain as before, so that ^^m^ will be constant ; and since (j)^ depends on m^ only, the distribution of ^j about m^ wUl remain the same, so that if>^m^ will be the same as before the change. The only part of E that will be altered is that depending on 2(j)^m^, because ^^ becomes <^i + -j-i Sx on account of the displacement. The variation of actual energy due to the displacement is therefore SE=-i'!rCt(2^m}jdnx (41). But by equation (12) the work done by the mechanical forces on nir, during the motion is BW=-zl^m,dv]Sx .....(42); and since bur hypothesis is a purely mechanical one, we must have by the conservation of force, SE + SW^O (43); that is, the loss of energy of the vortices must be made up by work done iu moving magnets, so that - 4:iTCt U ^ m,d v)Sx + 'Z (^ m,d v\ Sx = 0, C'=^..-. (44); VOL. L 60 474 ON PHYSICAL LINES OP FOBCB. SO that the energy of the vortices in unit of volume is 1 ^^lx{a^ + ^ + y) (45); and that of a vortex whose volume is V is ±f,{a^ + ^ + y^)V. (46). In order to produce or destroy this energy, work must be expended on, or received from, the vortex, either by the tangential action of the layer of particles in contact with it, or by change of form in the vortex. We shall first investigate the tangential action between the vortices and the layer of particles in contact with them. Prop. VII. — To find the energy spent upon a vortex in unit of time by the layer of particles which surrounds it. Let P, Q, R be the forces acting on unity of the particles in the three co-ordinate directions, these quantities being functions of x, y, and z. Since each particle touches two vortices at the extremities of a diameter, the reaction of the particle on the vortices wUl be equally divided, and will be -iP, -IQ, -iR on each vortex for unity of the particles; but since the superficial density of the particles is — - (see equation (34) ), the forces on unit of surface of a vortex will be -T-^' -T-Q' -7-^- ^TT Air Att Now let dS be an element of the surface of a vortex. Let the direction-cosines of the normal be I, m, n. Let the co-ordinates of the element be x, y, z. Let the component velocities of the surface be u, v, w. Then the work expended on that element of surface wUl be j-p 1 '!±=-±^{Pu+Qv + Rw)dS (47). Let us begin with the first term, PudS. P may be written „ dP dP dP ^» + ^'^+%2/ + -^^ (48), and ti = w^ — my. ON PHYSICAL LINKS OF FORCE. 475 Remembering that the surface of the vortex is a closed one, so that tnxdS = tmxdS = tmydS = tmzdS = 0, and %mydS = tnzdS= V, we find ip„ds^('^p-!^y)r. dz (49). and the whole work done on the vortex in tmit of time will be d^ dt dW 1 = -~t (Pu+Qv + Rw) dS 4:TT Prop. VIII. — To find the relations between the alterations of motion of the vortices, and the forces P, Q, R which they exert on the layer of particles between them. Let V be the volume of a vortex, then by (46) its energy is ^=8i/^(«' + )S' + y) V (51), and dE dt ' 1 ^r( da /tK I a 477- dt ^^t^yf) .(52). Comparing this value with that given in equation (50), we find a dQ dR dz dR da\ rt /( -d^-f^dij-^^i. dR_dP_ d§ dx dz ^ dt ,dP dQ dx dy -^1.-^,1 = 0. .(53). This equation being true for all values of a, ^, and y, first let ^ and y vanish, and divide by a. We find dQ dR _ da, dz c?v ~ dt Similarly, and dy dR_dP^ dS dx dz dt dP^_dQ^ dry dx ■^* (54). dy dx '" dt From these equations we may determine the relation between the alterations of motion -rr, &c. and the forces exerted on the layers of particles between 60—2 (55), 476 ON PHYSICAL LINES OF FORCE. the vortices, or, in the language of our hypothesis, the relation between changes in the state of the magnetic field and the electromotive forces thereby brought into play. In a memoir "On the Dynamical Theory of Diffraction" {Cambridge Philo- sophical Transactions, Vol. ix. Part 1, section 6), Professor Stokes has given a method by which we may solve equations (54), and find P, Q, and R in terms of the quantities on the right hand of those equations. I have pointed out* the appHcation of this method to questions in electricity and magnetism. Let us then find three quantities F, G, H from the equations dG _dH^ dz dy ^ dH_dF^ dx dz '^^ d^_dG^ dy dx with the conditions -7- [j~ 1^"- + "J' H-^ + ^ H-y) — ""^ "= ^ i^^)> dF dG dH ^ ,^^. dx dy dz Differentiating (55) with respect to t, and comparing with (54), we find ^ dF ^ dG „ dH ,,„, P = 1I' ^ = -di' ^ = W (^«)- We have thus determined three quantities, F, G, H, from which we can find P, Q, and P by considering these latter quantities as the rates at which the former ones vary. In the paper already referred to, I have given reasons for considering the quantities F, G, H as the resolved parts of that which Faraday has conjectured to exist, and has called the electrotonic state. In that paper I have stated the mathematical relations between this electrotonic state and the lines of magnetic force as expressed in equations (55), and also between the electrotonic state and electromotive force as expressed in equations (58). We must now endeavour to interpret them from a mechanical point of view in connexion with our hypothesis. * Gamhridge PhUosopMcal Transactions, Vol. X. Part i. Art. 3. " On Faraday's Lines of Force," pp. 205—209 of tMs vol. ON PHYSICAL LINES OF TOECE. 477 We shall in the first place examine the process by which the lines of force ^re produced by an electric current. Let AB, Plate VIII., p. 488, fig. 2, represent a current of electricity in the direction from A to B. Let the large spaces above and below AB represent the vortices, and let the small circles separating the vortices represent the layers of particles placed between them, which in our hypothesis represent electricity. Now let an electric current from left to right commence in AB. The row of vortices gh above AB will be set in motion in the opposite direction to that of a watch. (We shall call this direction +, and that of a watch — .) We shall suppose the row of vortices kl still at rest, then the layer of particles between these rows will be acted on by the row gh on their lower sides, and will be at rest above. If they are free to move, they wiU rotate in the negative direction, and will at the same time move from right to left, or in the opposite direction from the current, and so form an induced electric current. If this current is checked by the electrical resistance of the medium, the rotating particles wiU act upon the row of vortices hi, and make them revolve in the positive direction till they arrive at such a velocity that the motion of the particles is reduced to that of rotation, and the induced current disappears. If, now, the primary current AB be stopped, the vortices in the row gh will be checked, while those of the row hi still continue in rapid motion. The momentum of the vortices beyond the layer of particles pq will tend to move them from left to right, that is, in the direction of the primary current ; but if this motion is resisted by the medium, the motion of the vortices beyond pq will be gradually destroyed. It appears therefore that the phenomena of induced currents are part of the process of communicating the rotatory velocity of the vortices from one part of the field to another. As an example of the action of the vortices in producing induced currents, let us take the following case :— Let B, Plate YIII., p. 488, fig. 3, be a circular ring, of uniform section, lapped uniformly with covered wire. It may be shewn that if an electric current is passed through this wire, a magnet placed within the coil of wire will be strongly afiected, but no magnetic effect will be produced on any external point. The effect will be that of a magnet bent round till its two poles are in contact. If the coil is properly made, no effect on a magnet placed outside it can 478 ON PHYSICAL LINES OF FORCE. be discovered, whether the current is kept constant or made to vary in strength ; but if a conducting wire C be made to embrace the ring any number of times, an electromotive force will act on that wire whenever the current in the coU is made to vary ; and if the circuit be closed, there will be an actual current in the wire C. This experiment shews that, in order to produce the electromotive force, it is not necessary that the conducting wire should be placed in a field of magnetic force, or that lines of magnetic force should pass through the substance of the wire or near it. All that is required is that hues of force should pass through the circuit of the conductor, and that these lines of force should vary in quantity during the experiment. In this case the vortices, of which we suppose the lines of magnetic force to consist, are all withia the hollow of the ring, and outside the ring all is at rest. If there is no conducting circuit embracing the ring, then, when the primary current is made or broken, there is no action outside the ring, except an instantaneous pressure between the particles and the vortices which they separate. If there is a continuous conducting circuit embracing the ring, then, when the primary current is made, there will be a current in the opposite direction through C; and when it is broken, there will be a current through C in the same direction as the primary current. We may now perceive that induced currents are produced when the elec- tricity yields to the electromotive force, — this force, however, still existing when the formation of a sensible current is prevented by the resistance of the circuit. The electromotive force, of which the components are P, Q, R, arises from the action between the vortices and the interposed particles, when the velocity of rotation is altered in any part of the field. It corresponds to the pressure on the axle of a wheel in a machine when the velocity of the driving wheel is increased or diminished. The electrotonic state, whose components are F, G, H, is what the electromotive force would be if the currents, &c. to which the lines of force are due, instead of arriving at their actual state by degrees, had started instantaneously from rest with their actual values. It corresponds to the impulse which would act on the axle of a wheel in a machine if the actual velocity were suddenly given to the driving wheel, the machine being previously at rest. ON PHYSICAL LINES OF FORCE. 479 If the machine were suddenly stopped by stopping the driving wheel, each wheel would receive an impulse equal and opposite to that which it received when the machine was set in motion. This impulse may be calculated for any part of a system of mechanism, and may be called the reduced momentum of the machine for that point. In the varied motion of the machine, the actual force on any part arising from the variation of motion may be fovmd by differentiating the reduced momentum with respect to the time, just as we have found that the electromotive force may be deduced from the electrotonic state by the same process. Having found the relation between the velocities of the vortices and the electromotive forces when the centres of the vortices are at rest, we must extend our theory to the case of a fluid medium containing vortices, and subject to aU the varieties of fluid motion. If we fix our attention on any one elementary portion of a fluid, we shall find that it not only travels from one place to another, but also changes its form and position, so as to be elon- gated in certain directions and compressed in others, and at the same time (in the most general case) turned round by a displacement of rotation. These changes of form and position produce changes in the velocity of the molecular vortices, which we must now examine. The alteration of form and position may always be reduced to three simple extensions or compressions in the direction of three rectangular axes, together with three angular rotations about any set of three axes. We shall first con- sider the efiect of three simple extensions or compressions. Prop. IX. — To find the variations of a, /3, y in the parallelepiped x, y, z when X becomes a; -I- 8a;; y, y + By; and z, z + Bz; the volume of the figure remaining the same. By Prop. II. we find for the work done by the vortices against pressure, SW=p,B{xyz)-^{a'yzBx+^zxBy+'/xyBz) (59); and by Prop. VI. we find for the variation of energy, SE=-^(aSa + ^Sl3 + yBy)xyz (60), 47r 480 OIJ PHYSICAL LINES OF TOECE. The sum 8W+SE must be zero by the conservation of energy, and S{xyz) = 0, since xyz is constant ; so that Scc\ a (Sa — a X :)+/3(8/3-^|)+y(8y-r^) = (61). In order that this should be true independently of any relations between a, /S, and y, we must have Ba = J-l, 8/3 = /3|. 8y = y| (62). Peop. X. — To find the variations of a, /S, y due to a rotation ^i about the axis of X from y to z, a rotation 6^ about the axis of y from 2 to », and a rotation 6^ about the axis of z from x to y. The axis of /8 will move away from the axis of x by an angle 0^ ; so that )S resolved in the direction of x changes from to —18^3. The axis of y approaches that of a; by an angle d^i so that the resolved part of 7 in direction x changes from to yO^. The resolved part of a in the direction of x changes by a quantity depending on the second power of the rotations, which may be neglected. The variations of a, fi, y from this cause are therefore Sa = y9,-/30„ B^^ae,-y0„ 8y = /30,-a0, (63). The most general expressions for the distortiop. of an element produced by the displacement of its different parts depend on the nine quantities -^ OX, -J- ox, -^ox; -y- by, -j-oy, -7- ow ; -7- oz, -y- 62, -7- Sz : ax ay az ax ^ d-y dz ^ ax ay dz and these may always be expressed in terms of nine other quantities, namely, three simple extensions or compressions, Zx' By' Bz' along three axes properly chosen, x', y', 2', the nine direction-cosines of these axes with their six connecting equations, which are equivalent to three inde- pendent quantities, and the three rotations 0^, 0^, 0^ about the axes of x, y, z.- Let the direction-cosines of x' with respect to x, y, 2 be l^, m^, %, those of y, l„, mj, Wj, and those of z', l^, m^, n^ ; then we find ON PHYSICAL LINES OF FORCE. 481 dx d d -^ , Bx' , 8y' , dz' . (64), with similar equations for quantities involving Sy and Sz. Let a, fi', y be the values of a, /8, y referred to the axes x' , y, z ; then a = l^a + Trijfi + n^y " P' = l,a + mS + n,y ■ (65). y=l.,a + m^^ + n^y\ We shall then have ha = \ha +l^h^ + \hy' + yd^- ^9, =i,a:^+i^'^-l+i,y'^+ye.^-pe, Jb y z .(66), (67). By substituting the values of a', yS', y', and comparing with equations (64), we find Sa = a ^ Sx + /8 -T- Sx + y -y- Sec , . dx dz (68) as the variation of a due to the change of form and position of the element. The variations of ;S and y have similar expressions. Prop. XI. — To find the electromotive forces in a moving body. The variation of the velocity of the vortices in a moving element is due to two causes — the action of the electromotive forces, and the change of form and position of the element. The whole variation of a is therefore Sa^lf^-'^^St + a-Bx + IS^Bx + y^Sx (69). jLi\az dyj dx ' dy ' dz ^ ' But since a is a function of x, y, z and t, the variation of a may be also written (70). _ da ^ da ^ \ da ^ da ^, UjJb \AjU \Ajfy tvO Equating the two values of Sa and dividing by St, and remembering that in the motion of an incompressible medium d dx d dy d dz dx dt dy dt dz dt VOL. I. (71), 61 482 ON PHYSICAL LINES OF FOECE. and that in the absence of free magnetism da dl3 dy _ dx dy dz ■(72). we find l/dQ li.\dz Putting dy) d dx dz dt d dz dz dt +r-r.-i^-'<^-jzzj:-^:j7.:i7+^:j7.u^ d dy ' dydt d dx dy dt dy dx da dz ^da dy dfi dx _ da _ ,„^-. c[z~di~ dz dt dy dt dy dt dt ^ ' aiid da 'dt' ,_\ldG _dH\ lj,\dz dy I " dm\ dydt) (74), .(75), Ifd'G yu \dz dt where F, G, and H are the values of the electrotonic components for a fixed point of space, our equation becomes dy p dx dH\ di~^'^di~lU d I ^ dx dz dG\ d I -r, ^^ a^-^ — . ^ ,(76). The expressions for the variations of ^ and y give us two other equations which may be written down from symmetry. The complete solution of the three equations is ^^'^~dt~^^ dt'^ dt dx dG d^ dy „ dz dx dt dt dt R^ixl3 dx dy ^ dH ~dt -'"'dt^ dt d^ dz (77). The first and second terms of each equation indicate the effect of the motion of any body in the magnetic field, the third term refers to changes in the electrotonic state produced by alterations of position or intensity of magnets or currents in the field, and '^ is a function of x, y, z, and t, which is inde- terminate as far as regards the solution of the original equations, but which may always be determined in any given case from the circumstances of the problem. The physical interpretation of ^ is, that it is the electric tension at each point of space. ON PHYSICAL LINES OF FORCE. 483 The physical meaning of the terms in the expression for the electromotive force depending on the motion of the body, may be made simpler by supposing the field of magnetic force uniformly magnetized with intensity a in the direction of the axis of x. Then if I, m, n be the direction-cosines of any portion of a linear conductor, and S its length, the electromotive force resolved in the direction of the conductor will be e = S{Pl + Qm + Rn) (78), ^ = '^'^'^(^^'-^1) (^^)' that is, the product of iia, the quantity of magnetic induction over unit of area multiplied by S im -.- — n ~-\ , the area swept out by the conductor S in unit of time, resolved perpendicular to the direction of the magnetic force. The electromotive force in any part of a conductor due to its motion is therefore measured by the number of lines of magnetic force which it crosses in unit of time ; and the total electromotive force in a closed conductor is measured by the change of the number of lines of force which pass through it ; and this is true whether the change be produced by the motion of the con- ductor or by any external cause. In order to understand the mechanism by which the motion of a conductor across lines of magnetic force generates an electromotive force in that conductor, we must remember that in Prop. X. we have proved that the change of form of a portion of the medium containing vortices produces a change of the velocity of those vortices ; and in particular that an extension of the medium in the direction of the axes of the vortices, combined with a contraction in aU direc- tions perpendicular to this, produces an increase of velocity of the vortices ; while a shortening of the axis and bulging of the sides produces a diminution of the velocity of the vortices. This change of the velocity of the vortices arises from the internal effects of change of form, and is independent of that produced by external electro- motive forces. If, therefore, the change of velocity be prevented or checked, electromotive forces will arise, because each vortex wiU press on the surrounding particles in the direction in which it tends to alter its motion. Let A, fig. 4, p. 488, represent the section of a vertical wire moving in the direction of the arrow from west to east, across a system of lines of magnetic force 61—2 484 ON PHYSICAL LINES OF FORCE. running north and south. The curved Hnes in fig. 4 represent the lines of fluid motion about the wire, the wire being regarded as stationary, and the fluid as having a motion relative to it. It is evident that, from this figure, we can trace the variations of form of an element of the fluid, as the form of the element depends, not on the absolute motion of the whole system, but on the relative motion of its parts. In front of the wire, that is, on its east side, it will be seen that as the wire approaches each portion of the medium, that portion is more and more compressed in the direction from east to west, and extended in the direction from north to south ; and since the axes of the vortices lie in the north and south direction, their velocity will continually tend to increase by Prop. X., unless prevented or checked by electromotive forces acting on the circumference of each vortex. We shall consider an electromotive force as positive when the vortices tend to move the interjacent particles upwards perpendicularly to the plane of the paper. The vortices appear to revolve as the hands of a watch when we look at them from south to north ; so that each vortex moves upwards on its west side, and downwards on its east side. In front of the wire, therefore, where each vortex is striving to increase its velocity, the electromotive force upwards must be greater on its west than on its east side. There will therefore be a con- tinual increase of upward electromotive force from the remote east, where it is zero, to the front of the moving wire, where the upward force will be strongest. Behind the wire a different action takes place. As the wire moves away from each successive portion of the medium, that portion is extended from east to west, and compressed from north to south, so as to tend to diminish the velocity of the vortices, and therefore to make the upward electromotive force greater on the east than on the west side of each vortex. The upward electro- motive force will therefore increase continually from the remote west, where it is zero, to the back of the moving wire, where it wiU be strono-est. It appears, therefore, that a vertical wire moving eastwards will experience an electromotive force tending to produce in it an upward current. If there is no conducting circuit in connexion with the ends of the wire, no current will be formed, and the magnetic forces will not be altered; but if such a circuit exists, there wUl be a current, and the lines of magnetic force and the velocity ON PHYSICAL LINES OP FORCE. 485 of the vortices will be altered from their state previous to the motion of the wire. The change in the hnes of force is shewn in fig, 5. The vortices in front of the wire, instead of merely producing pressures, actually increase in velocity, while those behind have their velocity diminished, and those at the sides of the wire have the direction of their axes altered; so that the final effect is to produce a force acting on the wire as a resistance to its motion. We may now recapitulate the assumptions we have made, and the results we have obtained. (1) Magneto-electric phenomena are due to the existence of matter under certain conditions of motion or of pressure in every part of the magnetic field, and not to direct action at a distance between the magnets or currents. The substance producing these effects may be a certain part of ordinary matter, or it may be an aether associated with matter. Its density is greatest in iron, and least in diamagnetic substances ; but it must be in all cases, except that of iron, very rare, since no other substance has a large ratio of magnetic capacity to what we call a vacuum. (2) The condition of any part of the field, through which lines of magnetic force pass, is one of unequal pressure in different directions, the direction of the lines of force being that of least pressure, so that the lines of force may be considered lines of tension, (3) This inequality of pressure is produced by the existence in the medium of vortices or eddies, having their axes in the direction of the lines of force, and having their direction of rotation determined by that of the lines of force. "We have supposed that the direction was that of a watch to a spectator looking from south to north. We might with equal propriety have chosen the reverse direction, as far as known facts are concerned, by supposing resinous elec- tricity instead of vitreous to be positive. The effect of these vortices depends on their density, and on their velocity at the circumference, and is independent of their diameter. The density must be proportional to the capacity of the substance for magnetic induction, that of the vortices in air being 1. The velocity must be very great, in order to produce so powerful effects in so rare a medium. The size of the vortices is indeterminate, but is probably very small as compared with that of a complete molecule of ordinary matter''-. * The angular momentum of the system of vortices depends on their average diameter ; so that if the diameter were sensible, we migtt expect that a magnet would behave as if it contained a revolving body 486 ON PHYSICAL LINES OF FORCE. (4) The vortices are separated from each other by a single layer of round particles, so that a system of cells is formed, the partitions being these layers of particles, and the substance of each cell being capable of rotating as a vortex. (5) The particles forming the layer are in rolling contact with both the vortices which they separate, but do not rub against each other. They are perfectly free to roll between the vortices and so to change their place, provided they keep within one complete molecule of the substance ; but in passing from one molecule to another they experience resistance, and generate irregular motions, which constitute heat. These particles, in our theory, play the part of electricity. Their motion of translation constitutes an electric current, their rotation serves to transmit the motion of the vortices from one part of the field to another, and the tangential pressures thus called into play constitute electromotive force. The conception of a particle having its motion connected with that of a vortex by perfect rolling contact may appear somewhat awkward. I do not bring it forward as a mode of connexion existing in nature, or even as that which I would willingly assent to as an electrical hypothesis. It is, however, a mode of connexion which is mechanically conceivable, and easily investigated, and it serves to bring out the actual mechanical connexions between the known electro-magnetic phenomena; so that I venture to say that any one who understands the provisional and temporary character of this hypothesis, will find himself rather helped than hindered by it in his search after the true interpretation of the phenomena. The action between the vortices and the layers of particles is in part tangential; so that if there were any slipping or differential motion between the parts in contact, there would be a loss of the energy belonging to the lines of force, and a gradual transformation of that energy into heat. Now we know that the lines of force about a magnet are maintained for an indefinite time without any expenditure of energy; so that we must conclude that wherever there is tangential action between different parts of the medium, there is no motion of slipping between those parts. We must therefore conceive that the vortices and particles roll together without slipping; and that the interior strata of each vortex receive their proper velocities from the exterior stratum without slipping, that is, the angular velocity must be the same throughout each vortex. within it, and that the existence of this rotation might be detected by experiments on the free rotation of a magnet. I have made experiments to investigate this question, but have not yet fully tried the apparatus. ON PHYSICAL LINES OP FOECE, 487 The only process in wHch electro-magnetic energy is lost and transformed LQto heat, is in the passage of electricity from one molecule to another. In all other cases the energy of the vortices can only be diminished when an equivalent quantity of mechanical work is done by magnetic action. (6) The effect of an electric current upon the surrounding medium is to make the vortices in contact with the current revolve so that the parts next to the current move in the same direction as the current. The parts furthest from the current will move in the opposite direction; and if the medium is a conductor of electricity, so that the particles are free to move in any direction, the particles touching the outside of these vortices will be moved in a direction contrary to that of the current, so that there will be an induced current in the opposite direction to the primary one. If there were no resistance to the motion of the particles, the induced current would be equal and opposite to the primary one, and would continue as long as the primary current lasted, so that it would prevent all action of the primary current at a distance. If there is a resistance to the induced current, its particles act upon the vortices beyond them, and transmit the motion of rotation to them, till at last all the vortices in the medium are set in motion with such velocities of rotation that the particles between them have no motion except that of rotation, and do not produce currents. In the transmission of the motion from one vortex to another, there arises a force between the particles and the vortices, by which the particles are pressed in one direction and the vortices in the opposite direction. We call the force acting on the particles the electromotive force. The reaction on the vortices is equal and opposite, so that the electromotive force cannot move any part of the medium as a whole, it can only produce currents. When the primary current is stopped, the electromotive forces all act in the opposite direction. (7) When an electric current or a magnet is moved in presence of a conductor, the velocity of rotation of the vortices in any part of the field is altered by that motion. The force by which the proper amount of rotation is transmitted to each vortex, constitutes in this case also an electromotive force, and, if permitted, wUl produce currents. (8) When a conductor is moved in a field of magnetic force, the vortices in it and in its neighbourhood are moved out of their places, and are changed in form. The force arising from these changes constitutes the electromotive 488 ON PHYSICAL LINES OF FORCE. fprce on a moving conductor, and is found by calculation to correspond with that determined by experiment. We have now shewn in what way electro-magnetic phenomena may be imitated by an imaginary system of molecular vortices. Those who have been already inclined to adopt an hypothesis of this kind, will find here the con- ditions which must be fulfilled in order to give it mathematical coherence, and a comparison, so far satisfactory, between its necessary results and known facts. Those who look in a different direction for the explanation of the facts, may be able to compare- this theory with that of the existence of currents flowing freely through bodies, and with that which supposes electricity to act at a distance with a force depending on its velocity, and therefore not subject to the law of conservation of energy. The facts of electro-magnetism are so complicated and various, that the explanation of any number of them by several different hypotheses must bfr interesting, not only to physicists, but, to all who desire to understand how much evidence the explanation of phenomena lends to the credibility of a theory, or how far we ought to regard a coincidence in the mathematical expression of two sets of phenomena as an indication that these phenomena are of the same kind. We know that partial coincidences of this kind have been discovered ; and the fact that they are only partial is proved by the divergence of the laws of the two sets of phenomena in other respects. We may chance to find, in the higher parts of physics, instances of more complete coincidence, which may require much investigation to detect their loltimate divergence. NOTE. Since the firsc part of this paper was written, I have seen in Crelle's Journal for 1859 a paper by Prof. Helmholtz on Fluid Motion, in which he has pointed out that the lines of fluid motion are arranged according to the same laws as the lines of magnetic force, the path of an electric current corresponding to a line of axes of those particles of the fluid which are in a state of rotation. This is an additional instance of a physical analogy, the investigation of which may illustrate both electro-magnetism and hydrodynamics. ON PHYSICAL LINES OF FORCE. 489 [From the Philosophical Magazine for January and February, 1862.] PART III. THE THEORY OF MOLECULAR VORTICES APPLIED TO STATICAL ELECTRICITY. In the first part of this paper ^' I have shewn how the forces acting between magnets, electric currents, and matter capable of magnetic induction may be accounted for on the hypothesis of the magnetic field being occupied with innumerable vortices of revolving matter, their axes coinciding with the direction of the magnetic force at every point of the field. The centrifugal force of these vortices produces pressures distributed in such a way that the final effect is a force identical in direction and magnitude with that which we observe. In the second partf I described the mechanism by which these rotations may be made to coexist, and to be distributed according to the known laws of magnetic lines of force. I conceived the rotating matter to be the substance of certain cells, divided from each other by cell-walls composed of particles which are very small com- pared with the cells, and that it is by the motions of these particles, and their tangential action on the substance in the cells, that the rotation is communi- cated from one cell to another. I have not attempted to explain this tangential action, but it is necessary to suppose, in order to account for the transmission of rotation from the exterior to the interior parts of each cell, that the substance in the cells possesses elasticity of figure, similar in kind, though difierent in degree, to that observed in solid bodies. The undulatory theory of light requires us to admit this kind of elasticity in the luminiferous medium, in order to account for transverse vibrations. We need not then be surprised if the magneto-electric medium possesses the same property. * Phil Mag. March, 1861 [pp. 4.51—466 of this vol.]. t Phil. Mag. April and May, 1861 [pp. 467—488 of this vol.]. VOL. I. 62 490 ON PHYSICAL LINES OF FORCE. According to our theory, the particles which form the partitions between the cells constitute the matter of electricity. The motion of these particles constitutes an electric current; the tangential force with which the particles are pressed by the matter of the cells is electromotive force, and the pressure of the particles on each other corresponds to the tension or potential of the electricity. If we can now explain the condition of a body M'ith respect to the surrounding medium when it is said to be "charged" with electricity, and account for the forces acting between electrified bodies, we shall have established a connexion between all the principal phenomena of electrical science. We know by experiment that electric tension is the same thing, whether observed in statical or in current electricity; so that an electromotive force produced by magnetism may be made to charge a Ley den jar, as is done by the coil machine. When a difference of tension exists in different parts of any body, the electricity passes, or tends to pass, from places of greater to places of smaller tension. If the body is a conductor, an actual passage of electricity takes place; and if the difference of tensions is kept up, the current continues to flow with a velocity proportional inversely to the resistance, or directly to the conductivity of the body. The electric resistance has a very wide range of values, that of the metals being the smallest, and that of glass being so great that a charge of electricity has been preserved '"■ in a glass vessel for years without penetrating the thick- ness of the glass. Bodies which do not permit a current of electricity to flow through them are called insulators. But though electricity does not flow through them, the electrical effects are propagated through them, and the amount of these effects differs according to the nature of the body; so that equally good insu- lators may act differently as dielectricsf. Here then we have two independent qualities of bodies, one by which they allow of the passage of electricity through them, and the other by which they allow of electrical action being transmitted through them without any electri- city being allowed to pass. A conducting body may be compared to a porous membrane which opposes more or less resistance to the passage of a fluid, * By Professor W. Thomson. t Faraday, Experimental Researches, Series xi. ON PHYSICAL LINES OF FORCE. 491 while a dielectric is like an elastic membrane whicli may be impervious to the fluid, but transmits the pressure of the fluid on one side to that on the other. As long as electromotive force acts on a conductor, it produces a current which, as it meets with resistance, occasions a continual transformation of electrical energy into heat, which is incapable of being restored again as electri- cal energy by any reversion of the process. Electromotive force acting on a dielectric produces a state of polarization of its parts similar in distribution to the polarity of the particles .of iron under the influence of a magnet*, and, like the magnetic polarization, capable of being described as a state in which every particle has its poles in opposite conditions. In a dielectric under induction, we may conceive that the electricity in each molecule is so displaced that one side is rendered positively, and the other negatively electrical, but that the electricity remains entirely connected with the molecule, and does not pass from one molecule to another. The effect of this action on the whole dielectric mass is to produce a general displacement of the electricity in a certain direction. This displace- ment does not amount to a current, because when it has attained a certain value it remains constant, but it is the commencement of a current, and its variations constitute currents in the positive or negative direction, according as the displacement is increasing or diminishing. The amount of the displacement depends on the nature of the body, and on the electromotive force; so that if h is the displacement, R the electromotive force, and E a coefiicient depending on the nature of the dielectric, R= -iTrE'h; and if r is the value of the electric current due to displacement, _dh ^^^ df These relations are independent of any theory about the internal mechanism of dielectrics; but when we find electromotive force producing electric displace- ment in a dielectric, and when we find the dielectric recovering from its state of electric displacement with an equal electromotive force, we cannot help * See Prof. Mossotti, "Discussione Analitica," Memorie della Soc. Italiana (Modena), Vol. xxiv. Part 2, p. 49. 62—2 492 ON PHYSICAL LINES OF FORCE. regarding the phenomena as those of an elastic body, yielding to a pressure, and recovering its form when the pressure is removed. According to our hypothesis, the magnetic medium is divided into cells, separated by partitions formed of a stratum of particles which play the part of electricity. When the electric particles are urged in any direction, they will, by their tangential action on the elastic substance of the cells, distort each cell, and call into play an equal and opposite force arising from the elasticity of the cells. When the force is removed, the cells will recover their form, and the electricity will return to its former position. In the following investigation I have considered the relation between the displacement and the force producing it, on the supposition that the cells are spherical. The actual form of the cells probably does not differ from that of a sphere suiEciently to make much difference in the numerical result. I have deduced from this result the relation between the statical and dynamical measures of electricity, and have shewn, by a comparison of the electro-magnetic experiments of MM. Kohlrausch and Weber with the velocity of light as found by M. Fizeau, that the elasticity of the magnetic medium in air is the same as that of the luminiferous medium, if these two coex- istent, coextensive, and equally elastic media are not rather one medium. It appears also from Prop. XY. that the attraction between two electrified bodies depends on the value of E'^, and that therefore it would be less in turpentine than in air, if the quantity of electricity in each body remains the same. If, however, the potentials of the two bodies were given, the attraction between them would vary inversely as E'^, and would be greater in turpentine than in air. Prop. XII. To find the conditions of equilibrium of an elastic sphere whose surface is exposed to normal and tangential forces, the tangential forces being proportional to the sine of the distance from a given point on the sphere. Let the axis of z be the axis of spherical' co-ordinates. Let f, rj, t, be the displacements of any particle of the sphere in the direC'^ tions of X, y, and z. Let p^a;, pyy, ^ ^^ bc tho stresses normal to planes perpendicular to the three axes, and let p^„ p,^, p^y be the stresses of distortion in the planes yz, zx, and xy. ON PHYSICAL LINES OF FORCE, 493 Let fi be the coefficient of cubic elasticity, so that if Pxx Pyy Pzz P> \dx dy dzj ^ ' Let m be the coefficient of rigidity, so that (d^ dr,\ ,_ ^g^^_ Then we have the following equations of elasticity in an isotropic medium, d^ j,.. = (^-»(| + | + |)+m-^^^ with similar equations in y and z, and also ^-^I'lS+D'^'^ In the case of the sphere, let us assume the radius = a, and ^=exz, 7] = ezy, l=f{x' + y'')+gz' + d Then ]p^ = 2{ii, — ^m){e + g)z-\-mez=Py Pa = 2 (ju, — -Jm) {e + g)z + 2mgz .(82); (83). .(84). m 2^^x=^{e + 2f)z (85). (86), Pxy = The equation of internal equilibrium with respect to z is d d d ^P.. + ^i>,. + ^i>. = which is satisfied in this case if m{e + 2f+1g) + 2{iL-^m){e+g) = Q (87). The tangential stress on the surface of the sphere, whose radius is a at an angular distance 6 from the axis in plane xz, T={p^^-p,,)smeG03e+p^,{Gos'9-am'd) (88) ma = 2m {e+f-g) aainOcos'O — — (e + 2/)sin^ .(89). 494 ON PHYSICAL LINES Off FORCE. In order that T may be proportional to sin ^, the first term must vanish, and therefore 9 = e+f (90), r=-!^'(e + 2/)sin^ (91). The normal stress on the surface at any point is N—p^^ sin° 6 + pyy 003^ + 2p^2 sin 6 cos 9 = 2 (/x-i??z)(e + ^)acos^ + 2macos6'{(e+/)sin'6' + g'cos'(9} (92); or by (87) and (90), N^ -ma {e + 2f) cos (93). The tangential displacement of any point is i = fcos^-^sin^= - (cr/+cZ)sin^ (94). The normal displacement is n = ^ sin + 1 cos = {a' {e +f) + d} cos e (95), If we make d'{e+f) + d = (96), there will be no normal displacement, and the displacement will be entirely tangential, and we shall have « = a'esln6' (97). The whole work done by the superficial forces is U=^X{Tt)dS, the summation being extended over the surface of the sphere. The energy of elasticity in the substance of the sphere is the summation being extended to the whole contents of the sphere. We find, as we ought, that these quantities have the same value, namely U= - ^Tra'me (e + 2/) (98). We may now suppose that the tangential action on the surface arises from a layer of particles in contact with it, the particles being acted on by their own mutual pressure, and acting on the surfaces of the two cells with which they are in contact. ON PHYSICAL LINES OF FOKCE. 495 We assume the axis of z to be in the direction of maximum variation of the pressure among the particles, and we have to determine the relation between an electromotive force R acting on the particles in that direction, and the electric displacement h which accompanies it. Prop. XIII. — To find the relation between electromotive force and electric displacement when a uniform electromotive force R acts parallel to the axis of z. Take any element S>S of the surface, covered with a stratum whose density is p, and having its normal inclined Q to the axis of z ; then the tangential force upon it will be pRSS sine-- 2 TSS (99), T being, as before, the tangential force on each side of the surface. Putting p — :r- as in equation (34)*, we find R= -2'!Tma{e+2f) (100). The displacement of electricity due to the distortion of the sphere is XSS^pt sin 6 taken over the whole surface (101)^ and if h is the electric displacement per unit of volume, we shall have f7ra% = |a^e (102), or h=--ae (103); s6 that R = i7r'm^-^h (104), or we may write R= — -IvE'-h (105), provided we assume E-= — vm — (lOG). Finding e and / from (87) and (90), we get ^^=—4^ (^«^)- ^ + 3^ The ratio of m to jjl varies in different substances; but in a medium whose elasticity depends entirely upon forces acting between pairs of particles, this ratio is that of 6 to 5, and in this case £= = 7rm ....(108). * Fhil. Mag. April, 1861 [p. 471 of this vol.]. 496 ON PHYSICAL LINES OF FORCE. When the resistance to compression is infinitely greater than the resistance to distortion, as in a liquid rendered slightly elastic by gum or jelly, E' = ZTrm (109). The value of E" must lie between these limits. It is probable that the substance of our cells is of the former kind, and that we must use the first value of E-, which is that belonging to a hypothetically "perfect" solid'*, in which 5m = 6/A (110), so that we must use equation (108). Peop. XIV. — To correct the equations (9)t of electric currents for the efiect due to the elasticity of the medium. We have seen that electromotive force and electric displacement are connected by equation (105). Differentiating this equation with respect to t, we find f=-^-^§ ('"). shewing that when the electromotive force varies, the electric displacement also varies. But a variation of displacement is equivalent to a current, and this current must be taken into account in equations (9) and added to r. The three equations then become ^~ I^ [cly dz E' dt J _ J_ /da_dy _V_ dQ\ ^~Att \dy dx E' dtj ,_^(d§ _da. _l_ dR\ '' ~lTT\dx dy W 'dtj x\'here p, q, r are the electric currents in the directions of x, y, and z; a, fi, y are the components of magnetic intensity; and P, Q, R are the electromotive forces. Now if e be the quantity of free electricity in unit of volume, then the equation of continuity will be dp do dr de ^ , , dx + Ty+dz + dt = ^ (113). * See Rankine "On Elasticity," Camh. and Dub. Math. Journ. 1851. t Phil. Mag. March, 1861 [p. 462 of this vol.]. (112), ON PHYSICAL LINES OF FOECE. 497 Differentiating (112) with respect to x, y, and z respectively, and substituting, we find de^_}_d_(dP dQ dR\ dt AirE'dt \dx '^ dy'^ dz) ^^^^>' 1 1 (dP dQ dR\ , , the constant being omitted, because e = when there are no electromotive forces. Peop. XV. — To find the force acting between two electrified bodies. The energy in the medium arising from the electric displacements is U=-ti{Pf+Qg + Eh)SV. (116), where P, Q, R are the forces, and f, g, h the displacements. Now when there is no motion of the bodies or alteration of forces, it appears from equations (77)* that ^ d-^ ^ d'^ -r, d^ , , P=-irx' ^=-%' ^=-^ (^18); and we know by (105) that P=-iTTE% Q=-4:TtP:% E=-iTTl?h (119); whence ^=8^^^"^^ /d^^ d^' d^W^Y /,oA\ \dx dy dz ) ^ ■'" Integrating by parts throughout all space, and remembering that '^ vanishes at an infinite distance, 1 ^, fd^ d"^ d^\^^^ ,_,, or by (115), U=it{^e)SV (122). Now let there be two electrified bodies, and let gj be the distribution of electricity in the first, and '^i the electric tension due to it, and let 1 fd^ d^ d^,\ ^'-I^\d^'^ df ^~d^) ^^''^^• Let e^ be the distribution of electricity in the second body, and '^^ the tension due to it; then the whole tension at any point wHl be "^i + '^j, and the expansion for U will become U=i%{%e, + %e, + %e, + %e,)SV. (124). * Phil. Mag. May, 1861 [p. 482 of this vol.]. VOL. I. 63 498 ON PHYSICAL LINES OP FORCE. Let the body whose electricity is e^ be moved in any way, the electricity moving along with the body, then since the distribution of tension '^j moves with the body, the value of ^i^i remains the same. ^262 also remains the same; and Green has shewn (Essay on Electricity, p. 10) that ^163 = "5^361, so that the work done by moving the body against electric forces Tr=SC7=S2(^A)SF (125). And if 61 is confined to a small body, or Fdr = e, ^dr (126), where F is the resistance and dr the motion. If the body e^ be small, then if r is the distance from e^, equation (123) gives r whence F= -E'%' (127); or the force is a repulsion varying inversely as the square of the distance. Now let 7^1 and tj, be the same quantities of electricity measured stati- cally, then we know by definition of electrical quantity F^-'f (128); and this will be satisfied provided r)^ = Ee^ and rf^ = Ee., (129); so that the quantity E previously determined in Prop. XIII. is the number by which the electrodynamic measure of any quantity of electricity must be multiplied to obtain its electrostatic measure. That electric current which, circulating round a ring whose area is unity, produces the same effect on a distant magnet as a magnet would produce whose strength is unity and length unity placed perpendicularly to the plane of the ring, is a unit current; and E units of electricity, measured statically, ON PHYSICAL LINES OP FOBCE. 499 traverse the section of this current in one second, — these units being such that any two of them, placed at unit of distance, repel each other with unit of force. We may suppose either that E units of positive electricity move in the positive direction through the wire, or that E units of negative electricity move in the negative direction, or, thirdly, that \E units of positive electricity move in the positive direction, while ^E units of negative electricity move in the negative direction at the same time. The last is the supposition on which MM. Weber and Kohlrausch* proceed, who have found ^^=155,370,000,000 (130), the unit of length being the millimetre, and that of time being one second, whence ^ = 310,740,000,000 (131). Prop. XVI. — To find the rate of propagation of transverse vibrations through the elastic medium of which the cells are composed, on the suppo- sition that its elasticity is due entirely to forces acting between pairs of particles. By the ordinary method of investigation we know that y-sM (132). where m is the coefficient of transverse elasticity, and p is the density. By referring to the equations of Part I., it will be seen that if p is the density of the matter of the vortices, and ju. is the " coefficient of magnetic induction," ti = 'np (133); whence TTm=V'ii (134); and by (108), E= Vj]^ (135). In air or vacuum [i=l, and therefore V=E = 310,740,000,000 millimetres per second ■ (136). = 193,088 mUes per second * Abhandlungm der Kmig. Sdchsischen Gesellschaft, Vol. in. (1857), p. 260. 63—2 500 ON PHYSICAL LINES OF FOKCE. The velocity of light in air, as determined by M. Fizeau"*', is 70,843 leagues per second (25 leagues to a degree) which gives 7=314,858,000,000 millimetres = 195,647 miles per second (137). The velocity of transverse undulations in our hypothetical medium, calculated from the electro-magnetic experiments of MM. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena. Pbop. XVII. — To find the electric capacity of a Leyden jar composed of any given dielectric placed between two conducting surfaces. Let the electric tensions or potentials of the two surfaces be ^^ and '^j. Let S be the area of each surface, and 6 the distance between them, and let e and — e be the quantities of electricity on each surface ; then the capacity C=^4^ (138). Within the dielectric we have the variation of ^ perpendicular to the surface ~ e ■ Beyond either surface this variation is zero. Hence by (115) applied at the surface, the electricity on unit of area is 4.TrE'e (139); and we deduce the whole capacity of the apparatus. so that the quantity of electricity required to bring the one surface to a * Gomptes Rmdus, Vol. xxix. (1849), p. 90. In Galbraith and Haughton's Manual of Astronomy, M. Fizeau's result is stated at 169,944 geographical miles of 1000 fathoms, which cives 193 118 statute miles; the value deduced from aberration is 192,000 miles. ON PHYSICAL LINES OF FORCE. 501 given tension varies directly as the surface, inversely as the thickness, and inversely as the square of E. Now the coefficient of induction of dielectrics is deduced from the capacity of induction-apparatus formed of them ; so that if D is that coefficient, D varies inversely as E^, and is unity for air. Hence ^ = W (141), where V and V^ are the velocities of light in air and in the medium. Now if i is the index of refraction, -pr = i, and D = ~ (142); so that the inductive power of a dielectric varies directly as the square of the index of refraction, and inversely as the magnetic inductive power. In dense media, however, the optical, electric, and magnetic phenomena may be modified in different degrees by the particles of gross matter; and their mode of arrangement may influence these phenomena differently in different directions. The axes of optical, electric, and magnetic properties will probably coincide ; but on account of the unknown and probably complicated nature of the reactions of the heavy particles on the setherial medium, it may be im- possible to discover any general numerical relations between the optical, electric, and magnetic ratios of these axes. It seems probable, however, that the value of E, for any given axis, depends upon the velocity of light whose vibrations are parallel to that axis, or whose plane of polarization is perpendicular to that axis. In a uniaxal crystal, the axial value of E will depend on the velocity of the extraordinary ray, and the equatorial value will depend on that of the ordinary ray. In " positive " crystals, the axial value of E will be the least and in negative the greatest. The value of D^, which varies inversely as E\ will, ccBteris paribus, be greatest for the axial direction in positive crystals, and for the equatorial direction in negative crystals, such as Iceland spar. If a spherical portion of a crystal, radius =a, be suspended in a field of electric force which would act on unit of 502 ON PHYSICAL LINES OF FOECE. electricity with force = 1, and if A and A be the coefficients, of dielectric induction along the two axes in the plane of rotation, then if 6 be the incli- nation of the axis to the electric force, the moment tending to turn the sphere will be and the axis of greatest dielectric induction (Dj) will tend to become parallel to the lines of electric force. PART lY. THE THEORY OF MOLECULAR VORTICES APPLIED TO THE ACTION OF MAGNETISM ON POLARIZED LIGHT. The connexion between the distribution of lines of magnetic force and that of electric currents may be completely expressed by saying that the work done on a unit of imaginary magnetic matter, when carried round any closed curve, is proportional to the quantity of electricity which passes through the closed curve. The mathematical form of this law may be expressed as in equations (9)*, which I here repeat, where a, /8, y are the rectangular components of magnetic intensity, and p, q, r are the rectangular components of steady electric currents, ^ Air \dy dz _ 1 /da dy " A,TT\dz dx^ _ 1 fdjB dd Air \dx dy (9). The same mathematical connexion is found between other sets of phenomena in physical science. (1) If a, ^, 7 represent displacements, velocities, or forces, then p, q, r will be rotatory displacements, velocities of rotation, or moments of couples pro- ducing rotation, in the elementary portions of the mass. * Fhil. Mag. March, 1861 [p. 462 of this vol.]. ON PHYSICAL LINES OP POECE. 503 (2) If a, ^, y represent rotatory displacements in a uniform and continuous substance, tlien p, q, r represent the relative linear displacement of a particle with respect to those in its immediate neighbourhood. See a paper by Prof. W. Thomson "On a Mechanical Representation of Electric, Magnetic, and Galvanic Forces," Camh. and Dublin Math. Journal, Jan. 1847. (3) If a, ^, y represent the rotatory velocities of vortices whose centres are fixed, then p, q, r represent the velocities with which loose particles placed between them would be carried along. See the second part of this paper {Phil. Mag. April, 1861) [p. 469]. It appears from all these instances that the connexion between magnetism and electricity has the same mathematical form as that between certain pairs of phenomena, of which one has a linear and the other a rotatory character. Professor Challis* conceives magnetism to consist in currents of a fluid whose dhection corresponds with that of the lines of magnetic force ; and electric currents, on this theory, are accompanied by, if not dependent on, a rotatory motion of the fluid about the axis of the current. Professor Helmholtzf has investigated the motion of an incompressible fluid, and has conceived lines drawn so as to correspond at every point with the instantaneous axis of rotation of the fluid there. He has pointed out that the lines of fluid motion are arranged according to the same laws with respect to the lines of rotation, as those by which the lines of magnetic force are arranged with respect to electric currents. On the other hand, in this paper I have regarded magnetism as a phenomenon of rotation, and electric currents as consisting of the actual translation of particles, thus assuming the inverse of the relation between the two sets of phenomena. Now it seems natural to suppose that all the direct efiects of any cause which is itself of a longitudinal character, must be themselves longitudinal, and that the direct effects of a rotatory cause must be themselves rotatory. A motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw, which connects a motion in a given direction along the axis with a rotation in a given dhection round it ; -and a motion of rotation, though it may produce tension along the axis, cannot of itself produce a current in one direction along the axis rather than the other. * Fhil Mag. December, 1860, January and February, 1861. t Crelle, Journal, Vol lv. (1858), p. 25. 504 ON PHYSICAL LINES OF FORCE. Electric currents are known to produce effects of transference in the direc- tion of the current. They transfer the electrical state from one body to another, and they transfer the elements of electrolytes in opposite directions, but they do not* cause the plane of polarization of light to rotate when the light tra- verses the axis of the current. On the other hand, the magnetic state is not characterized by any strictly longitudinal phenomenon. The north and south poles differ only in their names, and these names might be exchanged without altering the statement of any magnetic phenomenon; whereas the positive and negative poles of a battery are completely distinguished by the different elements of water which are evolved there. The magnetic state, however, is characterized by a well-marked rotatory phenomenon discovered by Faraday f — the rotation of the plane of polarized light when transmitted along the lines of magnetic force. When a transparent diamagnetic substance has a ray of plane-polarized light passed through it, and if lines of magnetic force are then produced in the substance by the action of a magnet or of an electric current, the plane of polarization of the transmitted light is found to be changed, and to be turned through an angle depending on the intensity of the magnetizing force within the substance. The direction of this rotation in diamagnetic substances is the same as that in which positive electricity must circulate round the substance in order to produce the actual magnetizing force within it ; or if we suppose the horizontal part of terrestrial magnetism to be the magnetizing force acting on the sub- stance, the plane of polarization would be turned in the direction of the earth's true rotation, that is, from west upwards to east. In paramagnetic substances, M. YerdetJ has found that the plane of polari- zation is turned in the opposite direction, that is, in the direction in which negative electricity would flow if the magnetization were effected by a helix surrounding the substance. In both cases the absolute direction of the rotation is the same, whether the light passes from north to south or from south to north, — a fact which dis- tinguishes this phenomenon from the rotation produced by quartz, turpentine, &c., * Faraday, Experimental Researches, 951 — 954, and 2216—2220. t Ibid., Series xix. ' X Comptes Rendus, Vol. XLiii. p. 529 ; Vol. xliv. p. 1209. ON PHYSICAL LINES OF FORCE. 505 in which the absolute direction of rotation ia reversed when that of the light is reversed. The rotation in the latter case, whether related to an axis, as in quartz, or not so related, as in fluids, indicates a relation between the direction of the ray and the direction of rotation, which is similar in its formal expression to that between the longitudinal and rotatory motions of a right-handed or a left-handed screw; and it indicates some property of the substance the mathe- matical form of which exhibits right-handed or left-handed relations, such as are known to appear in the external forms of crystals having these properties. In the magnetic rotation no such relation appears, but the direction of rotation is directly connected with that of the magnetic lines, in a way which seems to indicate that magnetism is really a phenomenon of rotation. The transference of electrolytes in fixed directions by the electric current, and the rotation of polarized light in fixed directions by magnetic force, are the facts the consideration of which has induced me to regard magnetism as a phenomenon of rotation, and electric currents as phenomena of translation, instead of following out the analogy pointed out by Hehnholtz, or adopting the theory propounded by Professor Challis. The theory that electric currents are linear, and magnetic forces rotatory phenomena, agrees so far with that of Ampere and Weber ; and the hypothesis that the raagnetic rotations exist wherever magnetic force extends, that the centrifugal force of these rotations accounts for magnetic attractions, and that the inertia of the vortices accounts for induced currents, is supported by the opinion of Professor W. Thomson*. In fact the whole theory of molecular vor- tices developed in this paper has been suggested to me by observing the direction in which those investigators who study the action of media are looking for the explanation of electro-magnetic phenomena. Professor Thomson has pointed out that the cause of the magnetic action on light must be a real rotation going on in the magnetic field. A right-handed circularly polarized ray of light is found to travel with a different velocity according as it passes from north to south, or from south to north, along a line of magnetic force. Now, whatever theory we adopt about the direction of vibrations in plane-polarized Hght, the geometrical arrangement of the parts of the medium during the passage of a right-handed circularly polarized ray is exactly the same whether the ray is moving north or south. The only difference * See Nichol's Gyclopmdia, art. "Magnetism, Dynamical Eolations of," edition 1860; Proceedings of Royal Society, June 1856 and June 1861 ; and Phil. Mag. 1857. VOL. I. 64 506 ON PHYSICAL LINES OP FORCE. is, that the particles describe their circles in opposite directions. Since, therefore, the configuration is the same in the two cases, the forces acting between par- ticles must be the same in both, and the motions due to these forces must be equal in velocity if the medium was originally at rest; but if the medium be in a state of rotation, either as a whole or in molecular vortices, the circular vibrations of light may differ in velocity according as their direction is similar or contrary to that of the vortices. We have now to investigate whether the hypothesis developed in this paper — that magnetic force is due to the centrifugal force of smaU vortices, and that these vortices consist of the same matter the vibrations of which constitute light — leads to any conclusions as to the effect of magnetism on polarized light. We suppose transverse vibrations to be transmitted through a magnetized medimn. How will the propagation of these vibrations be affected by the circumstance that portions of that medium are in a state of rotation ? In the following investigation, I have found that the only effect which the rotation of the vortices will have on the light will be to make the plane of polarization rotate in the same direction as the vortices, through an angle proportional — [A) to the thickness of the substance, [B) to the resolved part of the magnetic force parallel to the ray, (C) to the index of refraction of the ray, (D) inversely to the square of the wave-length in air, {E) to the mean radius of the vortices, {F) to the capacity for magnetic induction. A and B have been fully investigated by M. Verdet*, who has shewn that the rotation is strictly proportional to the thickness and to the magnetizing force, and that, when the ray is inclined to the magnetizing force, the rotation is as the cosine of that inclination. D has been supposed to give the true relation between the rotation of different rays; but it is probable that C must be taken into account in an accurate statement of the phenomena. The rotation varies, not exactly inversely as the square of the wave length, but a little faster; so that for the highly refrangible rays the rotation is greater than that given by this law, but more nearly as the index of refraction divided by the square of the wave-length. * Annales de Chimie et de Physique, sgr. 3, Vol. XLi. p. 370; Vol. xliii. 'p. 37. ON PHYSICAL LINES OP FORCE. 507 The relation (^) between the amount of rotation and the size of the vortices shews that different substances may differ in rotating power inde- pendently of any observable difference in other respects. We know nothing of the absolute size of the vortices ; and on our hypothesis the optical phenomena are probably the only data for determining their relative size in different sub- stances. On our theory, the direction of the rotation of the plane of polarization depends on that of the mean moment of momenta, or angular momentum, of the molecular vortices ; and since M. Verdet has discovered that magnetic substances have an effect on light opposite to that of diamagnetic substances, it follows that the molecular rotation must be opposite in the two classes of substances. We can no longer, therefore, consider diamagnetic bodies as being those whose coefficient of magnetic induction is less than that of space empty of gross matter. We must admit the diamagnetic state to be the opposite of the paramagnetic ; and that the vortices, or at least the influential majority of them, in diamagnetic substances, revolve in the direction in which positive electricity revolves in the magnetizing bobbin, while in paramagnetic substances they revolve in the opposite direction. This result agrees so far with that part of the theory of M. Weber* which refers to the paramagnetic and diamagnetic conditions. M. Weber sup- poses the electricity in paramagnetic bodies to revolve the same way as the surrounding helix, while in diamagnetic bodies it revolves the opposite way. Now if we regard negative or resinous electricity as a substance the absence of which constitutes positive or vitreous electricity, the results will be those actually observed. This will be true independently of any other hypothesis than that of M. Weber about magnetism and diamagnetism, and does not require us to admit either M. Weber's theory of the mutual action of electric particles in motion, or our theory of cells and cell-walls. I am inclined to believe that iron differs from other substances in the manner of its action as well as in the intensity of its magnetism ; and I think its behaviour may be explained on our hypothesis of molecular vortices, by supposing that the particles of the iron itself are set in rotation by the tan- gential action of the vortices, in an opposite direction to their own. These large heavy particles would thus be revolving exactly as we have supposed the * Taylor's Scientific Memoirs, Vol. v. p. 477. 64—2 508 ON PHYSICAL LINES OF FOECE. infinitely small particles constituting electricity to revolve, but without being free like them to change their place and form currents. The whole energy of rotation of the magnetized field would thus be greatly increased, as we know it to be ; but the angular momentum of the iron particles would be opposite to that of the sethereal cells and immensely greater, so that the total angular momentum of the substance wiU be in the direction of rotation of the iron, or the reverse of that of the vortices. Since, however, the angular momentum depends on the absolute size of the revolving portions of the substance, it may depend on the state of aggregation or chemical arrangement of the elements, as well as on the ultimate nature of the com- ponents of the substance. Other phenomena in nature seem to lead to the conclusion that all substances are made up of a number of parts, finite in size, the particles composing these parts being themselves capable of internal motion. Prop. XVIII. — To find the angular momentum of a vortex. The angular momentum of any material system about an axis is the sum of the products of the mass, dm, of each particle multipHed by twice the area it describes about that axis in unit of time ; or if ^ is the angular momentum about the axis of x, dz dy\ A = "tdm ( y 2j dt dt As we do not kiiow the distribution of density within the vortex, we shall determine the relation between the angular momentum and the energy of the vortex which was found in Prop. VI. Since the time of revolution is the same throughout the vortex, the mean angular velocity w wUl be uniform and =-, where a is the velocity at the circumference, and r the radius. Then A = tdmr^o), and the energy E = ^tdmrW = ^Aw, = — ju,a^F by Prop. VI.* whence A = ~ix,raV (144) * Fhil Mag. April 1861 [p. 472 of this vol.]. X ON PHYSICAL LINES OF FOBCB. 509 for the axis of x, with similar expressions for the other axes, V being the volume, and r the radius of the vortex. Prop. XIX. — To determine the conditions of undulatory motion in a medium containing vortices, the vibrations being perpendicular to the direction of pro- pagation. Let the waves be plane-waves propagated in the direction of z, and let the axis of x and y be taken in the directions of greatest and least elasticity in the plane xy. Let x and y represent the displacement parallel to these axes, which will be the same throughout the same wave-surface, and therefore we shall have x and y functions of z and t only. Let X be the tangential stress on unit of area parallel to xy, tending to move the part next the origin in the direction of x. Let Y be the corresponding tangential stress in the ^-^ direction of y. ^ - Let ^1 and k^ be the coefficients of elasticity with respect ^^/" to these two kinds of tangential stress ; then, if the medium y is at rest, ^'"'dz' ^-"^'dz- Now let us suppose vortices in the medium, whose velocities are represented as usual by the symbols a, fi, y, and let us suppose that the value of a is increasing at the rate -j- , on account of the action of the tangential stresses alone, there being no electromotive force in the field. The angular momentum in the stratum whose area is unity, and thickness dz, is therefore increasing at the rate -— jLtr -7- dz; and if the part of the force Y which produces this effect 477 dt is F', then the moment of Y' is - Y'dz, so that Y' = - -- [ir -r- . The complete value of Y when the vortices are in a state of varied motion is (145). ~tT_T dy 1 da ^^"'di'I^'^^'dt Similarly, ^ ^='^'^ + 4^^^^^ 510 ON PHYSICAL LINES OF FORCE. The whole force acting upon a stratum whose thickness is dz and area unity, is -^ dz in the direction of x, and -i- dz in direction of y. The mass of the stratum is pdz, so that we have as the equations of motion, d''x_dX_,dh^ ^ J^ dS Pdf~~d^~'dz''^dz Att'^^' dt dY dz d^y d J_ dz 4ir' da ~di ] (146). Now the changes of velocity -j- and -^ are produced by the motion of the medium containing the vortices, which distorts and twists every element of its mass; so that we must refer to Prop. X.* to determine these quantities in terms of the motion. We find there at equation (68), da = a^r- Sx + /3 -^^ Bx + y -j- Sa; (68). ' dx dy ' dz Since hx and Sy are functions of z and t only, we may write this equation da dt' d^x ^ dzdt and in like manner. d^ _ d'y dt '^ dzdt . .(147), so that if we now put \ — a?p, \ = ¥p, and — — y = c', we may write the 1 jar equations of motion 47r p d^x „ d^x df d'i dz^dt d^x df dz' dz'dtj These equations may be satisfied by the values x = A cos (nt — mz + a) y = B sin {nt — mz + a) P^*^^^ ^ {n'-mJ'af)A=m''rufB and (n^ - w?}f) B = m'nc'A (148). .(149), .(150). Phil Mag. May 1861 [p. 481 of this vol.]. ON PHYSICAL LINES OF FOBCE. 511 Multiplying the last two equations together, we find (?i'-mV)(n=-m'&0=m%V (151) an equation quadratic with respect to m", the solution of which is m'= , ^"^^ (152). These values of m^ being put in the equations (150) wiU each give a ratio of A and B, A _ o?-lf + J {a' - hj + 4wV B ~ 2n& which being substituted in equations (149), will satisfy the original equations (148). The most general undulation of such a medium is therefore compounded of two elliptic undulations of difierent eccentricities travelHng with different velocities and rotating in opposite directions. The results may be more easily explained in the case in which a — h; then n m' = - a +710 and A^+B (153). Let us suppose that the value of A is unity for both vibrations, then we shall have m \ I . nz 05 = cos \nt — y sja^ — nc nz + cos nt sja? + nc- nz • / ^2 \ , • / * nz \ = -sm {nt--p==]+sv[i.[nt- ,-— A .(154). The first terms of x and y represent a circular vibration in the negative direction, and the second term a circular vibration in the positive direction, the positive having the greatest velocity of propagation. Combining the terms, we may write a; = 2 cos (jz« - j9z) cos gzl (155), y = 2 cos{nt-pz)&va.qz} where and n n n n ^'^.Ja'-nc' 2^/a' + wc^ .(156). 512 ON PHYSICAL LINES OF TOECB. These are the equations of an undulation consisting of a plane vibration whose periodic time is — , and wave-length — =\, propagated in the direction of % with a velocity - — "v, while the plane of the vibration revolves about the 2_. axis of z in the positive direction so as to complete a revolution when z = — . Now let us suppose & small, then we may write P = a^''^ ^^2^^ ^^^^)' 1 T and remembering that c^ = t — I^J, we find g = f^?^ (158). Here r is the radius of the vortices, an unknown quantity, p is the density of the luminiferous medium in the body, which is also unknown ; but if we adopt the theoiy of Fresnel, and make s the density in space devoid of gross matter, then p = si' (159), where i is the index of refraction. On the theory of MacCullagh and Neumann, p = s (160) in all bodies. /x is the coefficient of magnetic induction, which is unity in empty space or in air. y is the velocity of the vortices at their circumference estimated in the ordinary units. Its value is unknown, but it is proportional to the intensity of the magnetic force. Let Z be the magnetic intensity of the field, measured as in the case of terrestrial magnetism, then the intrinsic energy in air per unit of volume is where s is the density of the magnetic medium in air, which we have reason to believe the same as that of the luminiferous medium. We therefore put 7 = -]=-^ (161), ON PHYSICAL LINES OF FORCE. 513 \ is the wave-length of the undulation in the substance. Now if A be the wave-length for the same ray in air, and i the index of refraction of that ray in the body, ^ = J (162). Also V, the velocity of light in the substance, is related to V, the velocity of light in air, by the equation ^ = T (163). Hence if z be the thickness of the substance through which the ray passes, the angle through which the plane of polarization wiH be turned will be in degrees, /I 180 , , e = qz (164); or, by what we have now calculated, '-^^h-yi^ <'«=)■ In this expression all the quantities are known by experiment except r, the radius of the vortices in the body, and s, the density of the luminiferous medium in air. The experiments of M. Verdet* supply all that is wanted except the deter- mination of Z in absolute measure ; and this would also be known for aU his experiments, if the value of the galvanometer deflection for a semi-rotation of the testing bobbin in a known magnetic field, such as that due to terrestrial magnetism at Paris, were once for all determined. * Annales de Ghimie et de Physique, ser. 3, Vol. xli. p. 370. VOL. I. 65 [From the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. Vol. XXVII. Fourth Series.] XXIV. On Reciprocal Figures and Diagrams of Forces. Eeciprocal figures are such that the properties of the first relative to the second are the same as those of the second relative to the first. Thus inverse figures and polar reciprocals are instances of two different kinds of reciprocity. The kind of reciprocity vrhich we have here to do with has reference to figures consisting of straight lines joining a system of points, and forming closed rectilinear figures ; and it consists in the directions of all lines in the one figure having a constant relation to those of the lines in the other figure which correspond to them. In plane figures, corresponding lines may be either parallel, perpendicular, or at any constant angle. Lines meeting in a point in one figure form a closed polygon in the other* In figures in space, the lines in one figure are perpendicular to planes in the other, and the planes corresponding to lines which meet in a point form a closed polyhedron. The conditions of reciprocity may be considered from a purely geometrical point of view; but their chief importance arises from the fact that either of the figures being considered as a system of points acted on by forces along the lines of connexion, the other figure is a diagram of forces, in which these forces are represented in plane figures by lines, and in solid figures by the areas of planes. The properties of the "triangle" and "polygon" of forces have been long known, and the "diagram" of forces has been used in the case of the funicular polygon; but I am not aware of any more general statement of the method ON RECIPROCAL FIGURES AND DIAGRAMS OP FORCES. 515 of drawing diagrams of forces before Professor Kankine applied it to frames, roofs, &c. in Hs Applied Mechanics, p. 137, &c. The "polyhedron of forces," or the equilibrium of forces perpendicular and proportional to the areas of the faces of a polyhedron, has, I believe, been enunciated independently at various times; but the apphcation to a "frame" is given by Professor Rankine in the Philosophical Magazine, February, 1864. I propose to treat the question geometrically, as reciprocal figures are subject to certain conditions besides those belonging to diagrams of forces. On Reciprocal Plane Figures. Definition. — Two plane figures are reciprocal when they consist of an equal number of lines, so that corresponding lines in the two figures are parallel, and corresponding hues which converge to a point in one figure form a closed polygon in the other. Note. — If corresponding lines in the two figures, instead of being parallel are at right angles or any other angle, they may be made parallel by turning one of the figures round in its own plane. Since every polygon in one figure has three or more sides, every point in the other figure must have three or more lines converging to it ; and since every line in the one figure has two and only two extremities to which lines converge, every line in the other figure must belong to two, and only two closed polygons. The simplest plane figure fulfilling these conditions is that formed by the six lines which join four points in pairs. The reciprocal figure consists of six lines parallel respectively to these, the points in the one figure corresponding to triangles in the other. General Relation between the Numbers of Points, Lines, and Polygons in Reciprocal Figures. The effect of drawing a line, one of whose extremities is a point connected with the system of lines already drawn, is either to introduce one new point into the system, or to complete one new polygon, or to divide a polygon into two parts, according as it is drawn to an isolated point, or a point already connected with the system. Hence the sum of points and polygons in the 65—2 516 ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. system is increased by one for every new line. But the simplest figure consists of four points, four polygons, and six lines. Hence the sum of the points and polygons must always exceed the number of lines by two. Note. — This is the same relation which connects the numbers of summits, faces, and edges of polyhedra. Conditions of indeterminateness and impossibility in drawing reciprocal Diagrams. Taking any line parallel to one of the lines of the figure for a base, every new point is to be determined by the intersection of two new lines. Calling s the number of points or summits, e the number of lines or edges, and f the number of polygons or faces, the assumption of the first line deter- mines two points, and the remaining 5 — 2 points are determined by 2 (s — 2) lines. Hence if e = 2s-3, every point may be determined. If e be less, the form of the figure will be in some respects indeterminate ; and if e be greater, the construction of the figure will be impossible, unless certain conditions among the directions of the lines are fulfilled. These are the conditions of drawing any diagram in which the directions of the lines are arbitrarily given; but when one diagram is already drawn in which e is greater than 2s - 3, the directions of the lines will not be altogether arbitrary, but will be subject to e — (2s — 3) conditions. Now if e, s', /' be the values of e, s, and / in the reciprocal diagram e = e', s=f', f=s', e = s+f-2, e' = s'+/'-2. Hence if s =/, e = 2^ — 2 ; and there will be one condition connecting the directions of the lines of the original diagram, and this condition will ensure the possibility of constructing the reciprocal diagram. If s>f, e>2s-2, and e'<2s'-2; so that the construction of the reciprocal diagram will be possible, but inde- terminate to the extent of s — / variables. If s Eeciprocal figures are mechanically reciprocal; that is, either may be taken as representing a system of points, and the other as representing the magnitudes of the forces acting between them. In figures like 1, 2 and II., 3 and III., in which the equation e = 2s - 2 is true, the forces are determinate in their ratios ; so that one being given, the rest may be found. When e>2s-2, as in figs. 4 and 5, the forces are indeterminate, so that more than one must be known to determine the rest, or else certain relations among them must be given, such as those arising from the elasticity of the parts of a frame. When e<2s-2, the determination of the forces is impossible except under certain conditions. Unless these be fulfilled, as in figs. IV. and V., no forces along the lines of the figure can keep its points in equilibrium, and the figure, considered as a frame, may be said to be loose. When the conditions are fulfilled, the pieces of the frame can support forces, but in such a way that a small disfigurement of the frame may produce in- ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 523 finitely great forces in some of the pieces, or may throw the frame into a loose condition at once. The conditions, however, of the possibility of determining the ratios of the forces in a frame are not coextensive with those of finding a figure perfectly reciprocal to the frame. The condition of determinate forces is e = 2s-2; the condition of reciprocal figures is that every line belongs to two polygons only, and e = s+f-2. In fig. 7 we have six points connected by ten lines in such a way that the forces are all determinate ; but since the line Z is a side of three triangles, we cannot draw a reciprocal figure, for we should have to draw a straight line I with three ends. If we attempt to draw the reciprocal figure as in fig. VII., we shall find that, in order to represent the reciprocals of all the lines of fig. 7 and fix their relations, we must repeat two of them, as h and e by h' and e, so as to form a parallelogram. Fig. YII. is then a complete representation of the rela- tions of the force which would produce equilibrium in fig. 7 ; but it is redundant by the repetition of h and e, and the two figures are not reciprocal. Fig. VII, Fig. 7. On Reciprocal Figures in three dimensions. Definition. — Figures in three dimensions are reciprocal when they can be so placed that every line in the one figure is perpendicular to a plane face of the other, and every point of concourse of lines in the one figure is represented by a closed polyhedron with plane faces. 66—2 524 ON EECIPROCAL FIGURES AND DIAGRAMS OF FORCES. The simplest case is that of five points in space with their ten connecting fines, forming ten triangular faces enclosing five tetrahedrons. By joining the five points which are the centres of the spheres circumscribing these five tetrahedrons, we have a reciprocal figure of the kind described by Professor Eankine in the Philosophical Magazine, February 1864; and forces proportional to the areas of the triangles of one figure, if applied along the corresponding lines of connexion of the other figure, will keep its points in equihbrium. In order to have perfect reciprocity between two figures, each figure must be made up of a number of closed polyhedra having plane faces of separation, and such that each face belongs to two and only two polyhedra, corresponding to the extremities of the reciprocal line in the other figure. Every line in the figure is the intersection of three or more plane faces, because the plane face in the reciprocal figure is bounded by three or more straight lines. Let s be the number of points or summits, e the number of lines or edges, f the number of faces, and c the number of polyhedra or cells. Then if about one of the summits in which polyhedra meet, and a edges and rj faces, we describe a polyhedral cell, it wUl have (^ faces and tr summits and 77 edges, and we shall have ,; = ^ + o--2; s, the number of summits, will be decreased by one and increased by cr ; c, the number of cells, will be increased by one ; /, the number of faces, will be increased by <^ ; 6, the number of edges, will be increased by 77 ; so that e + c — {s+f) will be increased by 17 + 1 — (cr + ^ — 1), which is zero, or this quantity is constant. Now in the figure of five points already discussed, e = 10, c = 5, s = 5, f= 1 ; so that generally e + c = s+f, in figures made up of cells in the way described. The condition of a reciprocal figure being indeterminate, determinate, or im- possible except in particular cases, is e = 3s- 5. < This condition is sufiicient to determine the possibility of finding a system of forces along the edges which will keep the summits in equilibrium; but it is ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 525 manifest that the mechanical problem may be solved, though the reciprocal figure cannot be constructed owing to the condition of all the sides of a face lying in a plane not being fulfilled, or owing to a face belonging to more than two cells. Hence the mechanical interest of reciprocal figures in space rapidly diminishes with their complexity. Diagrams of forces in which the forces are represented by lines may be always constructed in space as well as in a plane, but in general some of the lines must be repeated. Thus in the figure of five points, each point is the meeting place of four lines. The forces in these lines may be represented by five gauche quadrilaterals (that is, quadrilaterals not in one plane) ; and one of these being chosen, the other four may be applied to its sides and to each other so as to form five sides of a gauche hexahedron. The sixth side, that opposite the original quad- rilateral, wUl be a parallelogram, the opposite sides of which are repetitions of the same line. We have thus a complete but redundant diagram of forces consisting of eight points joined by twelve lines, two pairs of the lines being repetitions. This is a more convenient though less elegant construction of a diagram of forces, and it never becomes geometrically impossible as long as the problem is mechanically possible, however complicated the original figure may be. [From the Royal Society Transactions, Vol. CLV.J XXV. A Dynamical Theory of the Electromagnetic Field. (Received October 27,— Read December 8, 1864.) PAET I. INTRODUCTORY. (l) The most obvious mechanical phenomenon in electrical and magnetical experiments is the mutual action by which bodies in certain states set each other in motion while still at a sensible distance from each other. The first step, therefore, in reducing these phenomena into scientific form, is to ascertain the magnitude and direction of the force acting between the bodies, and when it is found that this force depends in a certain way upon the relative position of the bodies and on their electric or magnetic condition, it seems at first sight natural to explain the facts by assuming the existence of something either at rest or in motion in each body, constituting its electric or magnetic state, and capable of acting at a distance according to mathematical laws. In this way mathematical theories of statical electricity, of magnetism, of the mechanical action between conductors carrying currents, and of the induction of currents have been formed. In these theories the force acting between the two bodies is treated with reference only to the condition of the bodies and their relative position, and without any express consideration of the surrounding medium. These theories assume, more or less explicitly, the existence of substances the particles of which have the property of acting on one another at a distance by attraction or repulsion. The most complete development of a theory of this A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 527 kind is that of M. W. Weber*, who has made the same theory include elec- trostatic and electromagnetic phenomena. In doing so, however, he has found it necessary to assume that the force between two electric particles depends on their relative velocity, as well as on their distance. This theory, as developed by MM. W. Weber and C. Neumannt, is ex- ceedingly ingenious, and wonderfully comprehensive in its application to the phenomena of statical electricity, electromagnetic attractions, induction of currents and diamagnetic phenomena ; and it comes to us with the more authority, as it has served to guide the speculations of one who has made so great an advance in the practical part of electric science, both by introducing a consistent system of units in electrical measurement, and by actually determining electrical quantities with an accuracy hitherto unknown. (2) The mechanical difficulties, however, which are involved in the assump- tion of particles acting at a distance with forces which depend on their velocities are such as to prevent me from considering this theory as an ultimate one, though it may have been, and may yet be useful in leading to the coordina- tion of phenomena. I have therefore preferred to seek an explanation of the fact in another direction, by supposing them to be produced by actions which go on in the surrounding medium as well as in the excited bodies, and endeavouring to explain the action between distant bodies without assuming the existence of forces capable of acting directly at sensible distances. ^, -- (3) The theory I propose may therefore be called a theory of the Electro- magnetic Field, because it has to do with the space in the neighbourhood of the electric or magnetic bodies, and it may be called a Dynamical Theory,, because it assumes that in that space there is matter in motion, by which the observed electromagnetic phenomena are produced. (4) The electromagnetic field is that part of space which contains and surrounds bodies in electric or magnetic conditions. ♦ " Electrodynamische Maassbestimmungen." Leipzic Trans. Vol. i. 1849, and Taylor's Scientific Mernmrs, Vol. v. art. xiv. t Explicare tentatur quomodo fiat ut lucis planum polarizationis per vires electricas vel magneticas declinetur. — Halis Saxonum, 1858. 528 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. It may be filled with any kind of matter, or we may endeavour to render it empty of all gross matter, as in the case of Geissler's tubes and other so- caUed vacua. There is always, however, enough of matter left to receive and transmit the undulations of light and heat, and it is because the transmission of these radiations is not greatly altered when transparent bodies of measurable density are substituted for the so-called vacuum, that we are obliged to admit that the undulations are those of an sethereal substance, and not of the gross matter, the presence of which merely modifies in some way the motion of the sether. We have therefore some reason to believe, from the phenomena of light and heat, that there is an sethereal medium filling space and permeating bodies, capable of being set in motion and of transmitting that motion from one part to another, and of communicating that motion to gross matter so as to heat it and affect it in various ways. (5) Now the energy communicated to the body in heating it must have formerly existed in the moving medium, for the undulations had left the source of heat some time before they reached the body, and during that time the energy must have been half in the form of motion of the medium and half in the form of elastic resilience. From these considerations Professor W. Thomson has argued*, that the medium must have a density capable of comparison with that of gross matter, and has even assigned an inferior limit to that density. (6) We may therefore receive, as a datum derived from a branch of science independent of that with which we have to deal, the existence of a pervading medium, of small but real density, capable of being set in motion, and of trans- mitting motion from one part to another with great, but not infinite, velocity. Hence the parts of this medium must be so connected that the motion of one part depends in some way on the motion of the rest; and at the same time these connexions must be capable of a certain kind of elastic yielding, since the communication of motion is not instantaneous, but occupies time. The medium is therefore capable of receiving and storing up two kinds of energy, namely, the "actual" energy depending on the motions of its parts, and "potential" energy, consisting of the work which the medium will do in recover- ing from displacement in virtue of its elasticity. * "On the Possible Density of the Luminiferous Medium, and on the Mechanical Value of a Cubic Mile of Sunlight," Transactions of the Royal Society of Edinburgh (1854), p. 57. A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 529 The propagation of undulations consists in the continual transformation of one of these forms of energy into the other alternately, and at any instant the amount of energy in the whole medium is equally divided, so that half is energy of motion, and half is elastic resilience. (7) A medium having such a constitution may be capable of other kinds of motion and displacement than those which produce the phenomena of light and heat, and some of these may be of such a kind that they may be evidenced to our senses by the phenomena they produce. (8) Now we know that the luminiferous medium is in certain cases acted on by magnetism ; for Faraday* discovered that when a plane polarized ray traverses a transparent diamagnetic medium in the direction of the lines of magnetic force produced by magnets or currents in the neighbourhood, the plane of polarization is caused to rotate. This rotation is always in the direction in which positive electricity must be carried round the diamagnetic body in order to produce the actual mag- netization of the field. M. Verdetf has since discovered that if a paramagnetic body, such as solution of perchloride of iron in ether, be substituted for the diamagnetic body, the rotation is in the opposite direction. Now Professor W. Thomson J has pointed out that no distribution of forces acting between the parts of a medium whose only motion is that of the lumi- nous vibrations, is sufficient to account for the phenomena, but that we must admit the existence of a motion in the medium depending on the magnetization, in addition to the vibratory motion which constitutes light. It is true that the rotation by magnetism of the plane of polarization has been observed only in media of considerable density ; but the properties of the magnetic field are not so much altered by the substitution of one medium for another, or for a vacuum, as to allow us to suppose that the dense medium does anything more than merely modify the motion of the ether. We have therefore warrantable grounds for inquiring whether there may not be a motion of the ethereal medium going on wherever magnetic effects are observed, and * Experimented Researches, Series xix, t Oomptes Rendus (1856, second lialf year, p. 529, and 1857, first half year, p. 1209). J Proceedings of the Royal Society, June 1856 and June 1861. VOL. I. 67 530 A DYNAMICAL THEORY OF THE ELECTBOMAGNETIO FIELD. we have some reason to suppose that this motion is one of rotation, having the direction of the magnetic force as its axis. (9) We may now consider another phenomenon observed in the electro- magnetic field. When a body is moved across the lines of magnetic force it experiences what is called an electromotive force ; the two extremities of the body tend to become oppositely electrified, and an electric current tends to flow through the body. When the electromotive force is sufiiciently powerful, and is made to act on certain compound bodies, it decomposes them, and causes one of their components to pass towards one extremity of the body, and the other in the opposite direction. Here we have evidence of a force causing an electric current in spite of resistance ; electrifying the extremities of a body in opposite ways, a condition which is sustained only by the action of the electromotive force, and which, as soon as that force is removed, tends, with an equal and opposite force, to produce a counter current through the body and to restore the original electrical state of the body ; and finally, if strong enough, tearing to pieces chemical compounds and carrying their components in opposite directions, while their natural tendency is to combine, and to combine with a force which can generate an electromotive force in the reverse direction. This, then, is a force acting on a body caused by its motion through the electromagnetic field, or by changes occurring in that field itself; and the effect of the force is either to produce a current and heat the body, or to decompose the body, or, when it can do neither, to put the body in a state of electric polarization, — a state of constraint in which opposite extremities are oppositely electrified, and from which the body tends to relieve itself as soon as the disturbing force is removed. (10) According to the theory which I propose to explain, this "electro- motive force" is the force called into play during the communication of motion from one part of the medium to another, and it is by means of this force that the motion of one part causes motion in another part. When electromotive force acts on a conducting circuit, it produces a current, which, as it meets with resistance, occasions a continual transformation of electrical energy into heat, which is incapable of being restored again to the form of electrical energy by any reversal of the process. A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 531 (11) But when electromotive force acts on a dielectric it produces a state of polarization of its parts similar in distribution to tlie polarity of the parts of a mass of iron under the influence of a magnet, and like the magnetic polarization, capable of being described as a state in which every particle has its opposite poles in opposite conditions*". In a dielectric under the action of electromotive force, we may conceive that the electricity in each molecule is so displaced that one side is rendered positively and the other negatively electrical, but that the electricity remains entirely connected with the molecule, and does not pass from one molecule to another. The efiect of this action on the whole dielectric mass is to produce a general displacement of electricity in a certain direction. This displacement does not amount to a current, because when it has attained to a certain value it remains constant, but it is the commencement of a current, and its varia- tions constitute currents in the positive or the negative direction according as the displacement is increasing or decreasing. In the interior of the dielectric there is no indication of electrification, because the electrification of the surface of any molecule is neutralized by the opposite electrification of the surface of the molecules in contact with it; but at the bounding surface of the dielectric, where the electrification is not neutralized, we find the phenomena which indicate positive or negative electrification. The relation between the electromotive force and the amount of electric displacement it produces depends on the nature of the dielectric, the same electromotive force producing generally a greater electric displacement in solid dielectrics, such as glass or sulphur, than in air. (12) Here, then, we perceive another efiect of electromotive force, namely, electric displacement, which according to our theory is a kind of elastic yielding to the action of the force, similar to that which takes place in structures and machines owing to the want of perfect rigidity of the connexions. (13) The practical investigation of the inductive capacity of dielectrics is rendered difficult on account of two disturbing phenomena. The first is the conductivity of the dielectric, which, though in many cases exceedingly small, is not altogether insensible. The second is the phenomenon called electric absorp- '' Faraday, Experimental Researches, Series xi. ; Mossotti, Mem. delta Soo. Italiana (Modena), Vol. XXIV. Part 2, p. 49. 67—2 5:32 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. tion*, in virtue of which, when the dielectric is exposed to electromotive force, the electric displacement gradually increases, and when the electromotive force is removed, the dielectric does not instantly return to its primitive state, but only discharges a portion of its electrification, and when left to itself gradually acquires electrification on its surface, as the interior gradually becomes depolarized. Almost all solid dielectrics exhibit this phenomenon, which gives rise to the residual charge in the Leyden jar, and to several phenomena of electric cables described by Mr F. Jenkint. (14) We have here two other kinds of yielding besides the yielding of the perfect dielectric, which we have compared to a perfectly elastic body. The yielding due to conductivity may be compared to that of a viscous fluid (that is to say, a fluid having great internal friction), or a soft solid on which the smallest force produces a permanent alteration of figure increasing with the time during which the force acts. The yielding due to electric absorption may be compared to that of a cellular elastic body containing a thick fluid in its cavities. Such a body, when subjected to pressure, is compressed by degrees on account of the gradual yielding of the thick fluid ; and when the pressure is removed it does not at once recover its figure, because the elasticity of the substance of the body has gradually to overcome the tenacity of the fluid before it can regain complete equilibrium. Several solid bodies in which no such structure as we have supposed can be found, seem to possess a mechanical property of this kind J ; and it seems probable that the same substances, if dielectrics, may possess the analogous electrical property, and if magnetic, may have corresponding properties relating to the acquisition, retention, and loss of magnetic polarity. (15) It appears therefore that certain phenomena in electricity and mag- netism lead to the same conclusion as those of optics, namely, that there is an aithereal medium pervading all bodies, and modified only in degree by their presence ; that the parts of this medium are capable of being set in motion by electric currents and magnets; that this motion is communicated from one * Faraday, Experimental Researches, 1233 — 1250. t Reports of British Association, 1859, p. 218; and Report of Committee of Board of Trade on Submarine Cables, pp. 136 & 464. X As, for instance, the composition of glue, treacle, &c., of wMch small plastic figures are made, which after being distorted gradually recover their shape. A DYNAMICAL THEORY OP THE ELECTEOMAGNETIC FIELD. 533 part of the medium to another by forces arising from the connexions of those parts ; that under the action of these forces there is a certain yielding depending on the elasticity of these connexions ; and that therefore energy in two different forms may exist in the medium, the one form bemg the actual energy of motion of its parts, and the other being the potential energy stored up in the con- nexions, in virtue of their elasticity. (16) Thus, then, we are led to the conception of a complicated mechanism capable of a vast variety of motion, but at the same time so connected that the motion of one part depends, according to definite relations, on the motion of other parts, these motions being communicated by forces arising from the relative displacement of the connected parts, in virtue of their elasticity. Such a mechanism must be subject to the general laws of Dynamics, and we ought to be able to work out all the consequences of its motion, provided we know the form of the relation between the motions of the parts. (17) We know that when an electric current is established in a conducting circuit, the neighbouring part of the field is characterized by certain magnetic properties, and that if two circuits are in the field, the magnetic properties of the field due to the two currents are combined. Thus each part of the field is in connexion with both currents, and the two currents are put in connexion with each other in virtue of their connexion with the magnetization of the field. The first result of this connexion that I propose to examine, is the induction of one current by another, and by the motion of conductors in the field. The second result, which is deduced firom this, is the mechanical action between conductors carrying currents. The phenomenon of the induction of currents has been deduced from their mechanical action by Helmholtz * and Thomson!. I have followed the reverse order, and deduced the mechanical action from the laws of induction. I have then described experimental methods of determining the quantities L, M, IST, on which these phenomena depend. (18) I then apply the phenomena of induction and attraction of currents to the exploration of the electromagnetic field, and the laying down systems of lines of magnetic force which indicate its magnetic properties. By exploring '' "Conservation of Force," Physical Society of Berlin, 1847; and Taylor's Scientific Memoirs, 1853, p. 114. t Reports of the British Association, 1848; Philosophical Magazine, Dec. 1851. 534 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. the same field with a magnet, I shew the distribution of its equipotential magnetic surfaces, cutting the fines of force at right angles. In order to bring these results within the power of symbolical calculation, I then express them in the form of the General Equations of the Electro- magnetic Field. These equations express — (A) The relation between electric displacement, true conduction, and the total current, compounded of both. (B) The relation between the lines of magnetic force and the inductive coefficients of a circuit, as already deduced from the laws of induction. (C) The relation between the strength of a current and its magnetic effects, according to the electromagnetic system of measurement. (D) The value of the electromotive force in a body, as arising from the motion of the body in the field, the alteration of the field itself, and the variation of electric potential from one part of the field to another. (E) The relation between electric displacement, and the electromotive force which produces it. (F) The relation between an electric current, and the electromotive force which produces it. (G) The relation between the amount of free electricity at any point, and the electric displacements in the neighbourhood. (H) The relation between the increase or diminution of free electricity and the electric currents in the neighbourhood. There are twenty of these equations in all, involving twenty variable quantities. (19) I then express in terms of these quantities the intrinsic energy of the Electromagnetic Field as depending partly on its magnetic and partly on its electric polarization at every point. From this I determme the mechanical force acting, 1st, on a moveable con- ductor carrying an electric current; 2ndly, on a magnetic pole; 3rdly, on an electrified body. The last result, namely, the mechanical force acting on an electrified body, gives rise to an independent method of electrical measurement founded on its A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 535 electrostatic effects. The relation between tlie units employed in the two methods is shewn to depend on what I have called the "electric elasticity" of the medium, and to be a velocity, which has been experimentally determined by MM. Weber and Kohlrausch. I then shew how to calculate the electrostatic capacity of a condenser, and the specific inductive capacity of a dielectric. The case of a condenser composed of parallel layers of substances of different electric resistances and inductive capacities is next examined, and it is shewn that the phenomenon called electric absorption will generally occur, that is, the condenser, when suddenly discharged, will after a short time shew signs of a residual charge. (20) The general equations are next applied to the case of a magnetic disturbance propagated through a non-conducting field, and it is shewn that the only disturbances which can be so propagated are those which are transverse to the direction of propagation, and that the velocity of propagation is the velocity v, found from experiments such as those of Weber, which expresses the number of electrostatic units of electricity which are contained in one electro- magnetic unit. This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws. If so, the agree- ment between the elasticity of the medium as calculated from the rapid alterna- tions of luminous vibrations, and as found by the slow processes of electrical experiments, shews how perfect and regular the elastic properties of the medium must be when not encumbered with any matter denser than air. If the same character of the elasticity is retained in dense transparent bodies, it appears that the square of the index of refraction is equal to the product of the specific dielectric capacity and the specific magnetic capacity. Conducting media are shewn to absorb such radiations rapidly, and therefore to be generally opaque. The conception of the propagation of transverse magnetic disturbances to the exclusion of normal ones is distinctly set forth by Professor Faraday* in his " Thoughts on Eay Vibrations." The electromagnetic theory of light, as * Philosophical Magazine, May 1846, or Experimental Researches, ill. p. 447. 536 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. proposed by him, is the same in substance as that which I have begun to develope in this paper, except that in 1846 there were no data to calculate the velocity of propagation. (21) The general equations are then applied to the calculation of the coef- ficients of mutual induction of two circular currents and the coefficient of self- induction in a coil. The want of uniformity of the current in the different parts of the section of a wire at the commencement of the current is investi- gated, I believe for the first time, and the consequent correction of the coefficient of self-induction is found. These results are applied to the calculation of the self-induction of the coil used in the experiments of the Committee of the British Association on Standards of Electric Resistance, and the value compared with that deduced from the experiments. PART II. ON ELECTROMAGNETIC INDUCTION. Electromagnetic Momentum of a Current. (22) We may begin by considering the state of the field in the neigh- bourhood of an electric current. We know that magnetic forces are excited in the field, their direction and magnitude depending according to known laws upon the form of the conductor carrying the current. When the strength of the current is increased, all the magnetic effects are increased in the same pro- portion. Now, if the magnetic state of the field depends on motions of the medium, a certain force must be exerted, in order to increase or diminish these motions, and when the motions are excited they continue, so that the effect of the connexion between the current and the electromagnetic field surrounding it, is to endow the current with a kind of momentum, just as the connexion between the driving-point of a machine and a fly-wheel endows the driving-point with an additional momentum, which may be called the momentum of the fly- wheel reduced to the driving-point. The unbalanced force acting on the driving- point increases this momentum, and is measured by the rate of its increase. A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 537 In the case of electric currents, the resistance to sudden increase or dimi- nution of strength produces effects exactly like those of momentum, but the amount of this momentum depends on the shape of the conductor and the relative position of its different parts. Mutual Action of two Currents. (23) If there are two electric currents in the field, the magnetic force at any point is that compounded of the forces due to each current separately, and since the two currents are in connexion with every point of the field, they will be in connexion with each other, so that any increase or diminution of the one will produce a force acting with or contrary to the other. Dynamical Illustration of Reduced Momentum. (24) As a dynamical illustration, let us suppose a body C so connected with two independent driving-points A and B that its velocity is p times that of A together with q times that of B. Let u be the velocity of A, v that of B, and iv that of C, and let hx, hy, 8z be their simultaneous displacements, then by the general equation of dynamics*, C^hz^XZx+Yhy, where X and Y axe the forces acting at A and B. T, , dw du dv ^^* dt=Pdt^'^dt' and Sz = pStc + qhy. Substituting, and remembering that hx and hy are independent, X^'^^iCfu+Cpqv) (1)- dt We may call Cp'u+Cpqv the momentum of C referred to A, and Cpqu + Cq^v its momentum referred to i? ; . then we may say that the effect of the force X is to increase the momentum of C referred to A, and that of Y to increase its momentum referred to B. * Lagrange, Mec. Anal. ii. 2, § 5, VOL. L 68 538 A DYNAMICAL THEORY OF THE ELECTKOMAGNETIC FIELD. If there are many bodies connected with A and B in a similar way but with different values of p and q, we may treat the question in the same way by assuming L = t {Cp% M= t {Cpq), and N= t {Cq% where the summation is extended to all the bodies with their proper values of C, p, and q. Then the momentum of the system referred to A is Lu + Mv, and referred to B, Mu + Nv, d and we shall have ^~'Tf {Lu + Mv) dt Y=^(Mu + Nv) (2), dt where X and Y are the external forces acting on A and B. (25) To make the illustration more complete we have only to suppose that the motion of A is resisted by a force proportional to its velocity," which we may call Bu, and that of i? by a similar force, which we may call Sv, B and S being coefficients of resistance. Then if ^ and r] are the forces on A and B, ^^X + Bu=^Bu + j^ {Lit + Mv) 'q=Y + Sv= Sv +j^{Mu + Nv) (3). If the velocity of A be increased at the rate -j: , then in order to prevent B from moving a force, tj — -,- [Mu] must be applied to it. This effect on B, due to an increase of the velocity of A, corresponds to the electromotive force on one circuit arising from an increase in the strength of a neighbouring circuit. This dynamical illustration is to be considered merely as assisting the reader to understand what is meant in mechanics by Reduced Momentum. The facts of the induction of currents as depending on the variations of the quantity called Electromagnetic Momentum, or Electrotonic State, rest on the experiments of Faraday'", Felicif, &c. * Experimental Researches, Series i., ix. t Annales de Chimie, ser. 3, xxxiv. (1852), p. 64. A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 539 Coefficients of Induction for Two Circuits. (26) In the electromagnetic field the values of L, M, N depend on the distribution of the magnetic effects due to the two circuits, and this distri- bution depends only on the form and relative position of the circuits. Hence L, M, N are quantities depending on the form and relative position of the circuits, and are subject to variation with the motion of the conductors. It will be presently seen that L, M, N are geometrical quantities of the nature of lines, that is, of one dimension in space ; L depends on the form of the first conductor, which we shall call ^, iV on that of the second, which we shall call B, and M on the relative position of A and B. (27) Let ^ be the electromotive force acting on A, x the strength of the current, and R the resistance, then Rx will be the resisting force. In steady currents the electromotive force just balances the resisting force, but in variable currents the resultant force ^ — Rx is expended in increasing the "electro- magnetic momentum," using the word momentum merely to express that which is generated by a force acting during a time, that is, a velocity existing in a body. In the case of electric currents, the force in action is not ordinary mechanical force, at least we are not as yet able to measure it as common force, but we call it electromotive force, and the body moved is not merely the electricity in the conductor, but something outside the conductor, and capable of being afiected by other conductors in the neighbourhood carrying currents. In this it resembles rather the reduced momentum of a driving-point of a machine as influenced by its mechanical connexions, than that of a simple moving body like a cannon ball, or water in a tube. Electromagnetic Relations of two Conducting Circuits. (28) In the case of two conducting circuits, A and B, we shall assume that the electromagnetic momentum belonging to A is Lx + My, and that belonging to B, Mx + Ny, where L, M, N correspond to the same quantities in the dynamical illustration, except that they are supposed to be capable of variation when the conductors A or B are moved. 68—2 540 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. Then the equation of the current x in A will be i^Rx + ^^{Lx + My) (4), and that of y in B 7]:^'Sy+ -j-{Mx + Ny) (5), where ^ and rj are the electromotive forces, x and y the currents, and M and S the resistances in A and B respectively. Induction of one Current by another. (29) Case 1st. Let there be no electromotive force on B, except that which arises from the action of A, and let the current of A increase from to the value x, then Sy + ^^{Mx + Ny) = 0, ft M whence ^= ydt=—-^x, (6) that is, a quantity of electricity Y, being the total induced current, will flow through B when x rises from to x. This is induction by variation of the current in the primary conductor. When M is positive, the induced Current due to increase of the primary current is negative. Induction by Motion of Conductor. (30) Case 2nd; Let x remain constant, and let M change from M to M', then Y--' -^ -x; (7) so that if M is increased, which it will be by the primary and secondary circuits approaching each other, there will be a negative induced current, the total quantity of electricity passed through B being Y. This is induction by the relative motion of the primary and secondary con- ductors. A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 541 Equation of Work and Energy. (31) To form the equation between work done and energy produced, multiply (1) by x and (2) by y, and add ix + r)y = Eaf + Sy- + Xj^{Lx + My) + yj^{Mx + Ny) (8). Here ix is the work done in unit of time by the electromotive force f acting on the current x and maintaining it, and rjy is the work done by the elec- tromotive force 7). Hence the left-hand side of the equation represents the work done by the electromotive forces in unit of time. Heat produced by the Current. (32) On the other side of the equation we have, first, Ra? + Sf = H (9), which represents the work done in overcoming the resistance of the circuits in unit of time. This is converted into heat. The remaining terms represent work not converted into heat. They may be written 1 <^ / r 2 , o 71^ , AT 2N -i dL „ dM , dN , i ^ {Lx^ + 2Mxy + Nf) +^ ^ ^H_ ^^ + ^_ ^^ Intrinsic Energy of the Currents. (33) If L, M, N are constant, the whole work of the electromotive forces which is not spent against resistance will be devoted to the development of the currents. The whole intrinsic energy of the currents is therefore ^Lx' + M'xy + ^]Yy' = E (10). This energy exists in a form imperceptible to our senses, probably as actual motion, the seat of this motion being not merely the conducting circuits, but the space surrounding them. Mechanical Action between Conductors. (34) The remaining terms, , dL , dM , dN , „^ . , 542 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. represent the work done in unit of time arising from the variations of L, M, and N, or, what is the same thing, alterations in the form and position of the conducting circuits A and B. Now if work is done when a body is moved, it must arise from ordinary mechanical force acting on the body while it is moved. Hence this part of the expression shews that there is a mechanical force urging every part of the conductors themselves in that direction in which L, M, and N will be most increased. The existence of the electromagnetic force between conductors carrying currents is therefore a direct consequence of the joint and independent action of each current on the electromagnetic field. If A and B are allowed to approach a distance ds, so as to increase M from M to M' while the currents are x and y, then the work done will be {M'-M)xy, and the force in the direction of ds wUl be ^xv (12) and this will be an attraction if x and y are of the same sign, and if M is increased as A and B approach. It appears, therefore, that if we admit that the unresisted part of electro- motive force goes on as long as it acts, generating a self-persistent state of the current, which we may call (from mechanical analogy) its electromagnetic momentum, and that this momentum depends on circumstances external to the conductor, then both induction of currents and electromagnetic attractions may be proved by mechanical reasoning. What I have called electromagnetic momentum is the same quantity which is -called by Faraday^' the electrotonic state of the circuit, every change of which involves the action of an electromotive force, just as change of momentum involves the action of mechanical force. If, therefore, the phenomena described by Faraday in the Ninth Series of his Experimental Researches were the only known facts about electric currents, the laws of Ampere relating to the attraction of conductors carrying currents, * Experimental Researches, Series I. 60, (fcc. A DYNAMICAL THEORY OF THK ELECTROMAGNETIC FIELD. 543 as well as those of Faraday about the mutual induction of currents, might be deduced by mechanical reasoning. In order to bring these results within the range of experimental verifica- tion, I shall next investigate the case of a single current, of two currents, and of the six currents in the electric balance, so as to enable the experimenter to determine the values of L, M, N. Case of a single Circuit. (35) The equation of the current re in a circuit whose resistance is R, and whose coefficient of self-induction is L, acted on by an external electro- motive force $, is ^-Rx = j^Lx (13). When f is constant, the solution is of the form x = h + {a — h) e~L \ where a is the value of the current at the commencement, and b is its final value. The total quantity of electricity which passes in time t, where t is great, is I xdt = ht + {a-h)-p (14). R The value of the integral of of with respect to the time is [W« = &^« + (a-&)|(5^) (15). The actual current changes gradually from the initial value a to the final value h, but the values of the integrals of x and x' are the same as if a steady current of intensity ^{a + h) were to flow for a time 2 -p , and were then suc- ceeded by the steady current b. The time 2 -^ is generally so minute a fraction of a second, that the effects on the galvanometer and dynamometer may be calculated as if the impulse were instantaneous. If the circuit consists of a battery and a coil, then, when the circuit is first completed, the effects are the same as if the current had only half its 544 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. final strength during the time 2-^. This diminution of the current, due to induction, is sometimes called the counter-current. (36) If an additional resistance r is suddenly thrown into the circuit, as by breaking contact, so as to force the current to pass through a thin wire of resistance r, then the original current is a — -^, and the final current is R + r The current of induction is then ^^ „ /r» \ . and continues for a time ^ R{R + r) 2 -p , This current is greater than that which the battery can maintain in the two wires R and r, and may be sufficient to ignite the thin wire r. When contact is broken by separating the wires in air, this additional resistance is given by the interposed air, and since the electromotive force across the new resistance is very great, a spark will be forced across. If the electromotive force is of the form Esaipt, as in the case of a coil revolving in the magnetic field, then E . , . X — — sm {pt — a), p ' where p' — R^ + Dp", and tan a = -^ . Case of tivo Circuits. (37) Let R be the primary circuit and S the secondary circuit, then we have a case similar to that of the induction coil. The equations of currents are those marked A and B, and we may here assume L, M, N as constant because there is no motion of the conductors. The equations then become ^^+^5+^1 = ^ (13«). A DYNAMICAL THEORY OF THE ELECTROMAGNETIC EIELD. 545 To find tlie total quantity of electricity which passes, we have only to integrate these equations with respect to t ; then if a;„, y, be the strengths of the currents at time 0, and x^, y^ at time t, and if X, Y be the quantities of electricity passed through each circuit during time t, , {in X=-^{^t + L{x,- X,) + M{y, - y,)] Y^^^{M{x,-x,) + N{y,-y,)] When the circuit R is completed, then the total currents up to time t, when t is great, are found by makiag x, = Q, x, = ^, y, = 0, 2/1 = ; then X = xjt-^), Y^-^x, (15*). EJ' S The value of the total counter- current in i2 is therefore independent of the secondary circuit, and the induction current in the secondary circuit depends only on M, the coefficient of induction between the coils, S the resistance of the secondary coil, and x^ the final strength of the current in R. When the electromotive force ^ ceases to act, there is an extra current in the primary circuit, and a positive induced current in the secondary circuit, whose values are equal and opposite to those produced on making contact. (38) All questions relating to the total quantity of transient currents, as measured by the impulse given to the magnet of the galvanometer, may be solved in this way without the necessity of a complete solution of the equa- tions. The heating effect of the current, and the impulse it gives to the suspended coil of Weber's dynamometer, depend on the square of the current a,t every instant during the short time it lasts. Hence we must obtain the solution of the equations, and from the solution we may find the effects both on the galvanometer and dynamometer; and we may then make use of the method of Weber for estimating the intensity and duration of a current uniform while it lasts which would produce the same effects. VOL. L 69 546 A DYNAMICAL THEORY 0^ THE ELECTBOMAaNETIC FIELD. (39) Let m„ n, be the roots of the equation {LN-M')7f + {RN-+LS)n + RS=0 (16), and let the primary coil be acted on by a constant electromotive force Ec, so that c is the constant current it could maintain; then the complete solution of the equations for making contact is c jy^ r/>S \ „,_/^_^^\ „,^^n^.| (17), 2/ = ^-^^{e-*-e"»*} (18). From these we obtain for calculating the impulse on the dynamometer, K^i = c^|«-f^-i^^^^} (19), ^y'^' = '''^ S{RN+LS) ('°)- The effects of the current in the secondary coil on the galvanometer and dynamometer are the same as those of a uniform current MR ^ RN+LS . (L N' lor a tune ^ I "p "^ "^ (40) The equation between work and energy may be easily verified. The work done by the electromotive force is ^lxdt = c'{Rt-L). Work done in overcoming resistance and producing heat, R]afdt + Slfdt = c' {Rt - f X). Energy remaining in the system, =-Jc^i. (41) If the circuit R is suddenly and completely interrupted while carrying a current c, then the equation of the current in the secondary coil would be s M -kt N M This current begins with a value c-jj, and gradually disappears. A DYNAMICAL THEORY OP THE ELECTEOMAGNETIC FIELD. 547 M . M"" The total quantity of electricity is c -^ , and the value of ly'dt is c° oCtv" ' The effects on the galvanometer and djraamometer are equal to those of a uniform current -ic irp for a time 2 -^ . ^ N S The heating effect is therefore greater than that of the current on making contact. (42) If an electromotive force of the form ^=E cos pt acts on the circuit R, then if the circuit S is removed, the value of x will be JE where and 33 = -J sin {pt — a), tan a: Lp "R The effect of the presence of the circuit S in the neighbourhood is to alter the value of A and a, to that which they would be if ^ became R+p' MS and L became L—p S'+pi'N" MN S'+p'N'' Hence the effect of the presence of the circuit S is to increase the apparent resistance and duninish the apparent self-induction of the circuit R. On the Determination of Coefficients of Induction by the Electric Balance. (43) The electric balance consists of six con- ductors joining four points, A, C, D, E, two and two. One pair, AC, of these points is con- nected through the battery B. The opposite pair, DE, is connected through the galvanometer G. Then if the resistances of the four remaining conductors are represented by P, Q, R, S, and the currents in them by x, x-z, y, and y + z, 69—2 548 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. the current through G will be z. Let the potentials at the four points be ^, C, D, E. Then the conditions of steady currents may be found from the equations Px = A-D, Q{x-z)=D-C Ry^A-E, S{y + z)^E-C ■ (21). Gz=D-E, B{x + y)=-A + C+F . Solving these equations for z, we find .(22). In this expression F is the electromotive force of the battery, z the current through the galvanometer when it has become steady. P, Q, R, S the resistances in the four arms. B that of the battery and electrodes, and G that of the galvanometer. (44) If PS=QR, then z — 0, and there will be no steady current, but a transient current through the galvanometer may be produced on making or breaking circuit on account of induction, and the indications of the galvano- meter may be used to determine the coeflBcients of induction, provided we understand the actions which take place. We shall suppose PS=QR, so that the current z vanishes when sufficient time is allowed, and Let the induction coefficients between P, Q, R, S be given by the following Table, the coefficient of in- duction of P on itself being p, between P and Q, h, and . so on. Let g be the coefficient of induction of the gal- vanometer on itself, and let it be out of the reach of the inductive influence of P, Q, R, S (as it must be in order to avoid direct action of P, Q, R, S on the needle) Let A', Y, Z be the integrals of x, y, z with respect to t. At P Q R S P P h Jc I Q h 1 m n R h m r S I n s A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 549 making contact x, y, z are zero. After a time z disappears, and x and y reach constant values. The equations for each conductor will therefore be PX +{p + h)x + {h+l)y = lAdt-lDdf Q{X-Z) + {h + q)x + {m + n)y = lDdt-lCdt RY +{k+m)x + {r +o)y = jAdt-jmt [ (24). S{Y+Z) +{l +n)x + {o +s)y=jEdt-jCdt GZ=\Ddt-\Edt. Solving these equations for Z, we find PS\P Q R^ S^'^Kp'^^^'" Q. R~ p)^^\R'^Q, 1 -m^p + ^j+n(^^-^). + o 1 S R .(25). (45) Now let the deflection of the galvanometer by the instantaneous current whose intensity is Z be a. '"Let the permanent deflection produced by making the ratio of PS to QR, p instead of unity, be 0. Also let the time of vibration of the galvanometer needle from rest to rest be T. Then calling the quantity p P we I ~ :i "^ S "^ ^(p ~ ^) "^ ^'"s - i) +^ fi + iV^fi + ^ R P 1 P^ s. find +^l^-^)+^(s-i)="' Z_2sin|ar_ r z tan 6 IT 1 — p .(26), (27). * [In those circumstances the values of x and y found in Art. 44 require modification before being inserted in equation (24). This has been pointed out by Lord Eayleigh, who employed the method described in the text in his second determination of the British unit of resistance in absolute measure. See the Philosophical Transactions, 1882, Part ii. pp. 677, 678.] 550 A DYNAMICAL THEORY OF THE ELECTBOMAGNETIC FIELD. In determining t by experiment, it is best to make tlie alteration of resist- ance in one of the arms by means of the arrangement described by Mr Jenkin in the Eeport of the British Association for 1863, by which any value of p from 1 to 1-01 can be accurately measured. We observe (a) the greatest deflection due to the impulse of induction when the galvanometer is in circuit, when the connexions are made, and when the resistances are so adjusted as to give no permanent current. We then observe (y8) the greatest deflection produced by the permanent current when the resistance of one of the arms is increased in the ratio of 1 to p, the galvanometer not being in circuit till a little while after the con- nexion is made with the battery. In order to eliminate the effects of resistance of the air, it is best to vary p till fi — 2a nearly ; then ^<(^-^^'M <^''- If all the arms of the balance except P consist of resistance coils of very fine wire of no great length and doubled before being coiled, the induction coefficients belonging to these coils will be insensible, and t will be reduced to ^. The electric balance therefore affords the means of measuring the self- induction of any circuit whose resistance is known. (46) It may also be used to determine the coefficient of induction between two circuits, as for instance, that between P and S which we have called m; but it would be more convenient to measure this by directly measuring the current, as m (37), without using the balance. We may also ascertain the T) CI equality of ^ and ^ by there being no current of induction, and thus, when we know the value of p, we may determine that of (7 by a more perfect method than the comparison of deflections. Exploration of the Electromagnetic Field. (47) Let us now suppose the primary circuit A to be of invariable form, and let us explore the electromagnetic field by means of the secondary circuit B, which we shall suppose to be variable in form and position. A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD, 551 ^ We may begin by supposing B to consist of a short straight conductor with Its extremities sliding on two parallel conducting rails, which are put in connexion at some distance from the shding-piece. Then, if sliding the moveable conductor in a given direction increases the value of M, a negative electromotive force wUl act in the circuit B, tending to produce a negative current in B during the motion of the sliding-piece. If a current be kept up in the circuit B, then the sliding-piece wUl itself tend to move in that direction, which causes M to increase. At every point of the field there will always be a certain direction such that a conductor moved m that direction does not experience any electromotive force in whatever direc- tion its extremities are turned. A conductor carrying a current will experience no mechanical force urging it in that direction or the opposite. This direction is called the direction of the line of magnetic force through that point. Motion of a conductor across such a line produces electromotive force in a direction perpendicular to the line and to the direction of motion, and a con- ductor carrying a current is urged in a direction perpendicular to the line and to the direction of the current. (48) We may next suppose B to consist of a very small plane circuit capable of being placed in any position and of having its plane turned in any direction. The value of M wUl be greatest when the plane of the circuit is perpendicular to the line of magnetic force. Hence if a current is maintained in B it will tend to set itself in this position, and will of itself indicate, like a magnet, the direction of the magnetic force. On Lanes of Magnetic Force. (49) Let any surface be drawn, cutting the lines of magnetic force, and on this surface let any system of lines be drawn at small intervals, so as to lie side by side without cutting each other. Next, let any line be drawn on the surface cutting all these lines, and let a second line be drawn near it, its distance from the first being such that the value of M for each of the small spaces enclosed between these two lines and the lines of the first system is equal to unity. In this way let more lines be drawn so as to form a second system, so 552 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. that the value of M for every reticulation formed by the intersection of the two systems of lines is unity. Finally, from every point of intersection of these reticulations let a line be drawn through the field, always coinciding in direction with the direction of magnetic force. (50) In this way the whole field wiU be filled with lines of magnetic force at regular intervals, and the properties of the electromagnetic field will be com- pletely expressed by them. For, 1st, If any closed curve be drawn in the field, the value of M for that curve will be expressed by the number of lines of force which pass through that closed curve. 2ndly. If this curve be a conducting circuit and be moved through the field, an electromotive force will act in it, represented by the rate of decrease of the number of lines passing through the curve. 3rdly. If a current be maintained in the circuit, the conductor wUl be acted on by forces tending to move it so as to increase the number of lines passing through it, and the amount of work done by these forces is equal to the current in the circuit multiplied by the number of additional lines. 4thly. If a small plane circuit be placed in the field, and be free to turn, it wUl place its plane perpendicular to the lines of force. A small magnet will place itself with its axis in the direction of the lines of force. 5thly. If a long uniformly magnetized bar is placed in the field, each pole will be acted on by a force lq the direction of the lines of force. The number of lines of force passing through unit of area is equal to the force acting on a unit pole multiplied by a coefiicient depending on the magnetic nature of the medium, and called the coefiicient of magnetic induction. In fluids and isotropoic solids the value of this coefficient fi is the same in whatever direction the lines of force pass through the substance, but in crystallized, strained, and organized solids the value of /x. may depend on the direction of the lines of force with respect to the axes of crystallization, strain, or growth. In all bodies /x is affected by temperature, and in iron it appears to diminish as the intensity of the magnetization increases. A DYNAMICAL THEOEY OF THE ELECTROMAGNETIC FIELD. 553 On Magnetic Equipotential Surfaces. (51) If we explore the field with, a uniformly magnetized bar, so long that one of its poles is in a very weak part of the magnetic field, then the mag- netic forces will perform work on the other pole as it moves about the field. If we start from a given point, and move this pole from it to any other point, the work performed will be independent of the path of the pole between the two points; provided that no electric current passes between the different paths pursued by the pole. Hence, when there are no electric currents but only magnets in the field, we may draw a series of surfaces such that the work done in passing from one to another shall be constant whatever be the path pursued between them. Such surfaces are called Equipotential Surfaces, and in ordinary cases are perpendicular to the Lines of magnetic force. If these surfaces are so drawn that, when a unit pole passes from any one to the next in order, unity of work is done, then the work done in any motion of a magnetic pole will be measured by the strength of the pole multiplied by the number of surfaces which it has passed through in the positive direction. (52) If there are circuits carrying electric currents in the field, then there will still be equipotential surfaces in the parts of the field external to the con- ductors carrying the currents, but the work done on a unit pole in passing from one to another will depend on the number of times which the path of the pole circulates round any of these currents. Hence the potential in each surface will have a series of values in arithmetical progression, differing by the work done in passing completely round one of the currents in the field. The equipotential surfaces will not be continuous closed surfaces, but some of them will be limited sheets, terminating in the electric circuit as their common edge or boundary. The number of these will be equal to the amount of work done on a unit pole in going round the current, and this by the ordinary measurement =4777, where y is the value of the current. These surfaces, therefore, are connected with the electric current as soap- bubbles are connected with a ring in M. Plateau's experiments. Every current y has 4777 surfaces attached to it. These surfaces have the current for their common edge, and meet it at equal angles. The form of the surfaces in other, parts depends on the presence of other currents and magnets, as well as on the shape of the circuit to which they belong. VOL. 1. 70 554 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD, PART III. GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD. (53) Let US assume three rectangular directions in space as the axes of X, y, and z, and let all quantities having direction be expressed by their com- ponents in these three directions. Electrical Currents (p, q, r). (54) An electrical current consists in the transmission of electricity from one part of a body to another. Let the quantity of electricity transmitted in unit of time across unit of area perpendicular to the axis of x be called p, then p is the component of the current at that place in the direction of x. We shall use the letters p, q, r to denote the components of the current per unit of area in the directions of x, y, z. Electrical Displacements (f, g, h). (55) Electrical displacement consists in the opposite electrification of the sides of a molecule or particle of a body which may or may not be accom- panied with transmission through the body. Let the quantity of electricity which would appear on the faces dy . dz of an element dx, dy, dz cut from the body be f .dy . dz, then / is the component of electric displacement parallel to x. We shall use /, g, h to denote the electric displacements parallel to x, y, z respectively. The variations of the electrical displacement must be added to the currents p>, q, r to get the total motion of electricity, which we may call p)\ q', r, so that df P=P+dt da dh '=''-^dt> ,(A). A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 555 Electromotive Force (P, Q, E,). (56) Let P, Q, R represent the components of the electromotive force at any point. Then P represents the difference of potential per unit of length in a conductor placed in the direction of x at the given point. We may suppose an indefinitely short wire placed parallel to £c at a given point and touched, during the action of the force P, by two small conductors, which are then insulated and removed from the influence of the electromotive force. The value of P might then be ascertained by measuring the charge of the conductors. Thus if I be the length of the wire, the difference of potential at its ends will be PI, and if C be the capacity of each of the small conductors the charge on each will be ^CPl. Since the capacities of moderately large conductors, measured on the electromagnetic system, are exceedingly small, ordinary electro- motive forces arising from electromagnetic actions could hardly be measured in this way. In practice such measurements are always made with long conductors, forming closed or nearly closed circuits. Electromagnetic Momentum (F, G, H). (57) Let F, G, H represent the components of electromagnetic momentum at any point of the field, due to any system of magnets or currents. Then F is the total impulse of the electromotive force in the direction of X that would be generated by the removal of these magnets or currents from the field, that is, if P be the electromotive force at any instant during the removal of the system F = \Pdt Hence the part of the electromotive force which depends on the motion of magnets or currents in the field, or their alteration of intensity, is -=-f. <^-% -=-f (-)■ Electromagnetic Momentum of a Circuit, (58) Let s be the length of the circuit, then if we integrate ■4:+«l+4:)* <^») 70—2 556 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. round the circuit, we shall get the total electromagnetic momentum of the circuit, or the number of lines of magnetic force which pass through it, the variations of which measure the total electromotive force in the circuit. This electromag- netic momentum is the same thing to which Professor Faraday has applied the name of the Electrotonic State. If the circuit be the boundary of the elementary area dy dz, then its electro- magnetic momentum is (dH dG\ , , and this is the number of lines of magnetic force which pass through the area dy dz. Magnetic Force (a, /3, y). (59) Let a, /3, y represent the force acting on a unit magnetic pole placed at the given point resolved in the directions of x, y, and z. Coefficient of Magnetic Induction (ix). (60) Let fji be the ratio of the magnetic induction in a given medium to that in air under an equal magnetizing force, then the number of lines of force in unit of area perpendicular to x will be /xa (/a is a quantity depending on the nature of the medium, its temperature, the amount of magnetization ali-eady produced, and in crystalline bodies varying with the direction). (61) Expressing the electric momentum of small circuits perpendicular to the three axes in this notation, we obtain the following Equations of Magnetic Force, dH ^^=dy- dG dz .R ^^ dH ''^—dz- dx dG ^y^ dx - _dF dy ■(B). A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 557 Equations of Currents. (62) It is known from experiment that the motion of a magnetic pole in the electromagnetic field in a closed circuit cannot generate work unless the circuit which the pole describes passes round an electric current. Hence, except in the space occupied by the electric currents, adx + /3dy + ydz = d<]) (31) a complete differential of 4>, the magnetic potential. The quantity (f> may be susceptible of an indefinite number of distinct values, according to the number of times that the exploring point passes round electric currents in its course, the difference between successive values of ^ corre- sponding to a passage completely round a current of strength c being Aire. Hence if there is no electric current, dy dz ' but if there is a current p', dy dz Similarly, TT^ - j- = ^'^l' •^ dz dx dfi da _ , dx dy We may call these the Equations of Currents. (C). Electromotive Force in a Circuit. (63) Let ^ be the electromotive force acting round the circuit A, then f=/(^£+«S^«S)* (^^>- where ds is the element of length, and the integration is performed round the circuit. 558 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. Let the forces in the field be those due to the circuits A and B, then the electromao-netic momentum of A is /(^l+»S+^S)*=^"+* w- where u and v are the currents in A and B, and dt Hence, if there is no motion of the circuit A, i=-^{Lu + Mv) (34). p= dF dx(j' dt dx Q = dG dxjj dt dy R = dH dxlj dt dz (35), where i/; is a function of x, y, z, and t, which is indeterminate as far as regards the solution of the above equations, because the terms depending on it will disappear on integrating round the circuit. The quantity t// can always, however, be determined in any particular case when we know the actual conditions of tlie question. The physical interpretation of i/; is, that it represents the electric lootential at each point of space. Electromotive Force on a Moving Conductor. (64) Let a short straight conductor of length a, parallel to the axis of X, move with a velocity whose components are -j- , -j- , -j-, and let its ex- tremities slide along two parallel conductors with a velocity -j- . Let us find the alteration of the electromagnetic momentum of the circuit of which this arrangement forms a part. In unit of time the moving conductor has travelled distances ^ ^ ^ dt' dt' dt along the directions of the three axes, and at the same time the lengths of the parallel conductors included in the circuit have each been increased bv — ■^ dt ■ A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 559 Hence the quantity will be increased by the following increments, [dFdx dFdy dF dz\ , ''[dxdi^dydt'^^dt)'^'^^^'' "^''^'^^ ^^ conductor, dsidFdx dGdu dH dz\ , ""dty-dxTh^d^d^^^ds)' ^""^ *^^ lengthening of circuit. The total increment will therefore be ^/(l^_dG\dy_ JdH_ dF\ dz _ \d)/ dx) dt ^\dx dzjdt' or, by the equations of Magnetic Force (8), If P is the electromotive force in the moving conductor parallel to x referred to unit of length, then the actual electromotive force is Pa; and since this is measured by the decrement of the electromagnetic momentum of the circuit, the electromotive force due to motion will be dt ^^ dt .(36). (65) The complete equations of electromotive force on a moving conductor may now be written as follows : — Equations of Electromotive Force. I dij odz\ dF dxli '^^^tt~'^Jtj~~di~dx .(D). „ f dz dx\ dG d\b ^-^i^y^-dt-y^ti-ni-dy „ [ i^dx dy\ dH dxp ^==f'[^di-''di)-^t -dz The first term on the right-hand side of each equation represents the electro- motive force arising from the motion of the conductor itself This electromotive 560 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. force is perpendicular to the direction of motion and to the lines of magnetic force ; and if a parallelogram be drawn whose sides represent in direction and magnitude the velocity of the conductor and the magnetic induction at that point of the field, then the area of the parallelogram will represent the electro- motive force due to the motion of the conductor, and the direction of the force is perpendicular to the plane of the parallelogram. The second term in each equation indicates the effect of changes in the position or strength of magnets or currents in the field. The third term shews the effect of the electric potential \jj. It has no efifect in causing a circulating current in a closed circuit. It indicates the existence of a force urging the electricity to or from certain definite points in the field. Electric Elasticity. (66) When an electromotive force acts on a dielectric, it puts every part of the dielectric into a polarized condition, in which its opposite sides are oppositely electrified. The amount of this electrification depends on the electro- motive force and on the nature of the substance, and, in solids having a structure defined by axes, on the direction of the electromotive force with respect to these axes. In isotropic substances, if h is the ratio of the electromotive force to the electric displacement, we may write the Equations of Electric Elasticity, '" -(E). Q = kg Electric Resistance. {^7) When an electromotive force acts on a conductor it produces a current of electricity through it. This effect is additional to the electric displacement already considered. In solids of complex structure, the relation between the electromotive force and the current depends on their direction through the solid. A DYNAMICAL THEORY OF THE ELECTBOMAGNETIC FIELD. 561 In isotropic substances, whicli alone we shall here consider, if p is the specific resistance referred to unit of volume, we may write the Equations of Electric Resistance, P=-pp Q= -pq " E^ -pr (F). Electric Quantity. (68) Let e represent the quantity of free positive electricity contained in unit of volume at any part of the field, then, since this arises from the electri- fication of the different parts of the field not neutralizing each other, we may write the Equation of Free Electricity, df da dh ^ ax ay dz (G). (69) If the medium conducts electricity, then we shall have another con- dition, which may be called, as in hydrodynamics, the Equation of Continuity, de dp dq dr _ dt dx dy dz ■(H). (70) In these equations of the electromagnetic field we have assumed twenty variable quantities, namely, For Electromagnetic Momentum F G H „ Magnetic Intensity «• ^ „ Electromotive Force P Q „ Current due to true Conduction p q „ Electric Displacement / 9 Total Current (including variation of displacement) p' q „ Quantity of Free Electricity e ., Electric Potential ^ 7 R r h VOL. I. 71 562 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. Between these twenty quantities we have found twenty equations, viz. Three equations of Magnetic Force (B) Electric Currents (C) Electromotive Force (D) Electric Elasticity (E) Electric Resistance (F) Total Currents (A) One equation of Free Electricity (G) „ Continuity (H) These equations are therefore sufficient to determine all the quantities which occur in them, provided we know the conditions of the problem. In many questions, however, only a few of the equations are required. Intrinsic Energy of the Electromagnetic Field. (71) We have seen (33) that the intrinsic energy of any system of currents is found by multiplying half the current in each circuit into its electromagnetic momentum. This is equivalent to finding the integral E = ^t{Ep'+Gq' + Hr')dV (37) over all the space occupied by currents, where p, q, r are the components of currents, and F, G, H the components of electromagnetic momentum. Substituting the values of p, q', r from the equations of Cvirrents (C), this becomes 877 [ \dy dzj \dz dxj \dx dyj ] Integrating by parts, and remembering that a, |S, 7 vanish at an infinite distance, the expression becomes l^r (dH dG\.(dF dH\ IdG dF\\ ,„ where the integration is to be extended over all space. Referring to the equa- tions of Magnetic Force (B), p. 556, this becomes ^ = ~S{a./.a-l-^.;^^-l-y./.y}^F (38), A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 563 where a, ^, y are the components of magnetic intensity or the force on a unit magnetic pole, and ju,a, /u./8, \x,y are the components of the quantity of magnetic induction, or the number of lines of force in unit of area. In isotropic media the value of \l is the same in all directions, and we may express the result more simply by saying that the intrinsic energy of any part of the magnetic field arising from its magnetization is 877 per unit of volume, where / is the magnetic intensity. (72) Energy may be stored up in the field in a different way, namely, by the action of electromotive force in producing electric displacement. The work done by a variable electromotive force, P, in producing a variable dis- placement, /, is got by integrating from P = to the given value of P. Since P = lcf, equation (E), this quantity becomes Hence the intrinsic energy of any part of the field, as existing in the form of electric displacement, is ^tiPf+Qg + R^dV. The total energy existing in the field is therefore E = %i^{afia + ^li.p + yi^y) + l{Pf+Qg + Rh)^^dV (I). The first term of this expression depends on the magnetization of the field, and is explained on our theory by actual motion of some kind. The second term depends on the electric polarization of the field, and is explained on our theory by strain of some kind in an elastic medium. (73) I have on a former occasion* attempted to describe a particular kind of motion and a particular kind of strain, so arranged as to account for the phenomena. In the present paper I avoid any hypothesis of this kind; and m * " On Physical Lines of Force," Philosophical Magazine, 1861—62. (In this vol. p. 451.) ^ 71—2 564 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. using such words as electric momentum and electric elasticity in reference to the known phenomena of the induction of currents and the polarization of dielectrics, I wish merely to direct the mind of the reader to mechanical pheno- mena which will assist him in understanding the electrical ones. All such phrases in the present paper are to be considered as illustrative, not as explanatory. (74) In speaking of the Energy of the field, however, I wish to be under- stood literally. All energy is the same as mechanical energy, whether it exists in the form of motion or in that of elasticity, or in any other form. The energy in electromagnetic phenomena is mechanical energy. The only question is, Where does it reside 1 On the old theories it resides in the electrified bodies, conducting circuits, and magnets, in the form of an unknown quality called potential energy, or the power of producing certain effects at a distance. On our theory it resides in the electromagnetic field, in the space surrounding the electrified and magnetic bodies, as well as in those bodies themselves, and is in two different forms, which may be described without hypothesis as magnetic polarization and electric polarization, or, according to a very probable hypothesis, as the motion and the strain of one and the same medium. (75) The conclusions arrived at in the present paper are independent of this hypothesis, being deduced from experimental facts of three kinds : 1. The induction of electric currents by the increase or diminution of neighbourmg currents according to the changes in the lines of force passing through the circuit. 2. The distribution of magnetic intensity according to the variations of a magnetic potential. 3. The induction (or influence) of statical electricity through dielectrics. We may now proceed to demonstrate from these principles the existence and laws of the mechanical forces which act upon electric currents, magnets, and electrified bodies placed in the electromagnetic field. A DYNAMICAL THEORY OP THE ELECTEOMAGNETIC FIELD. 565 PART ly. MECHANICAL ACTIONS IN THE FIELD. Mechanical Force on a Moveable Conductor. (76) We have shewn (§§ 34 & 35) that the work done by the electro- magnetic forces in aiding the motion of a conductor is equal to the product of the current in the conductor multiplied by the increment of the electro- magnetic momentum due to the motion. Let a short straight conductor of length a move parallel to itself in the direction of x, with its extremities on two parallel conductors. Then the incre- ment of the electromagnetic momentum due to the motion of a will be /dFdx dGdy dHdz\^ \dx ds dx ds dx dsj That due to the lengthening of the circuit by increasing the length of the parallel conductors will be _ /dFdx dFdt/ dFdz\^ \dx ds dy ds dz ds) The total increment is \ds\dx dyj ds\dz dx)]' which is by the equations of Magnetic Force (B), p. 556, Let X be the force acting along the direction of x per unit of length of the conductor, then the work done is XaZx. Let C be the current in the conductor, and let p', q, r be its com- ponents, then Xalx = CaSx (-£l^y—£ l^P) > 566 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. or X = ii.'Yq' —fi^r' Similarly, Y=ixar — ju^p' ,■ (J). Z — [M^p' — jJiaq . These are the equations which determine the mechanical force acting on a conductor carrying a current. The force is perpendicular to the current and to the lines of force, and is measured by the area of the parallelogram formed by lines parallel to the current and lines of force, and proportional to their intensities. Mechanical Force on a Magnet. (77) In any part of the field not traversed by electric currents the dis- tribution of magnetic intensity may be represented by the differential coefficients of a function which may be called the magnetic potential. When there are no currents in the field, this quantity has a single value for each point. When there are currents, the potential has a series of values at each point, but its differential coefficients have only one value, namely, d(f) _ ^^ _ Q d^ _ dx ' dy dz ~ '' Substituting these values of a, ^, y in the expression (equation 38) for the intrinsic energy of the field, and integrating by parts, it becomes -t ^ 8Tr\dx dy dz j ] The expression t('^ + ^ + '^J^\dV^tmdV (39) \dx dy dz / ^ ' indicates the number of lines of magnetic force which have their origin within the space V. Now a magnetic pole is known to us only as the origin or termination of lines of magnetic force, and a unit pole is one which has Att lines belonging to it, since it produces unit of magnetic intensity at unit of distance over a sphere whose surface is iir. Hence if m is the amount of free positive magnetism in unit of volume, the above expression may be written 47rm, and the expression for the energy of the field becomes E=^-t{l,){m, + m,)}, and these must be equal by the principle of Conservation of Energy. Since the distribution <^i is determined by m^, and ^^ by m^, the quantities (^jW-i and (fy^m^ will remain constant. It can be shewn also, as Green has proved (Essay, p. 10), that so that we get Xdx = d (mj^i), „ d &c. &c. &c. (52). After the electromotive force has been kept up for a sufficient time the current becomes the same in each layer, and where '^ is the total difference of potentials between the extreme layers. We have then and J'~ rah' J^ r^ r ah r \ah akj' e,= -|^-^), &c. (53). These are the quantities of electricity on the different surfaces. (87) Now let the condenser be discharged by connecting the extreme surfaces through a perfect conductor so that their potentials are instantly rendered equal, then the electricity on the extreme surfaces will be altered, but that on the internal surfaces wUl not have time to escape. The total difference of potentials is now ^' = ajc/i + ah{e\ + e,) + ah{e\ + e^ + e,), &c. = (54), whence if e\ is what e^ becomes at the instant of discharge. ^ r ajc^ ah~ ^ ah' (55). A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 575 The instantaneous discharge is therefore — y, or the quantity which would be discharged by a condenser of air of the equivalent thickness a, and it is unaffected by the want of perfect insulation. (88) Now let us suppose the connexion between the extreme surfaces broken, and the condenser left to itself, and let us consider the gradual dissi- pation of the internal charges. Let '^' be the difference of potential of the extreme surfaces at any time t ; then V = ajij^ + aXf^ + &c (56); but a,Afi= -r,^, Hence f^ = A-fi~^^ , f^ = A.fi ~^\ &c. ; and by referring to the values of e\, e^, &c., we find ,(57), ' r ajcj^ ah . J^ j\ ^ ^ r ajc^ ak &c. so that we find for the difference of extreme potentials at any time, -■-{(^t)-'"-(^t>■"'-^= <=«>• (89) It appears from this result that if all the layers are made of the same substance, ^' will be zero always. If they are of different substances, the order in which they are placed is indifferent, and the effect will be the same whether each substance consists of one layer, or is divided into any number of thin layers and arranged in any order among thin layers of the other sub- stances. Any substance, therefore, the parts of which are not mathematically homogeneous, though they may be apparently so, may exhibit phenomena of absorption. Also, since the order of magnitude of the coefilcients is the same as that of the indices, the value of ^' can never change sign, but must start from zero, become positive, and finally disappear. 576 A DYNAMICAL THEORY OP THE ELECTROMAGNETIC FIELD. (90) Let US next consider the total amount of electricity which would pass from the first surface to the second, if the condenser, after being thoroughly- saturated by the current and then discharged, has its extreme surfaces connected by a conductor of resistance R. Let p be the current in this conductor; then, during the discharge, ■*■' =i>in +p^n+&G. =pR (59). Integrating with respect to the time, and calling q^, q^, q the quantities of electricity which traverse the different conductors, q^r-, + q,r^ + &G. = qR (60). The quantities of electricity on the several surfaces wUl be e\-q-qi, &c. ; and since at last all these quantities vanish, we find q^ = e\-q, g-a = e', + Cj - g ; whence qR = -I-"- + -^+&c.]-^. or ^ = ^{^'«^(^-^J + »^«(^-$J+^-} (61). a quantity essentially positive; so that, when the primary electrification is in one direction, the secondary discharge is always in the same direction as the primary discharge *. * Since this paper was communicated to the Eoyal Society, I have seen a paper by M. Gaugain in the Annales de Ghimie for 1864, in which he has deduced the phenomena of electric absorption and secondary discharge from the theory of compound condensers. A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 577 PART VI. ELECTROMAUNETIC THEORY OF LIGHT. (91) At the commencement of this paper we made use of the optical hypothesis of an elastic medium through which the vibrations of light are propagated, in order to shew that we have warrantable grounds for seeking, in the same medium, the cause of other phenomena as well as those of light. We then examined electromagnetic phenomena, seeking for their explanation in the properties of the field which surrounds the electrified or magnetic bodies. In this way we arrived at certain equations expressing certain properties of the electromagnetic field. We now proceed to investigate whether these pro- perties of that which constitutes the electromagnetic field, deduced from electro- magnetic phenomena alone, are sufficient to explain the propagation of light through the same substance. (92) Let us suppose that a plane wave whose direction cosines are I, m, n is propagated through the field with a velocity F. Then all the electro- magnetic functions will be functions of 'w — lx + my + nz— Vt. The equations of Magnetic Force (B), p. 556, will become clH dG ua = m -^ n -J— , ^ dw dw 8 = n — -l^ ^" ~ dw dw ' _ldG_^dF '^^ ~ dw dw ' If we multiply these equations respectively by I, m, n, and add, we find llJLa + 'miJi/3 + niJiy = (62), which shews that the direction of the magnetization must be in the plane of the wave. VOL. I. ^^ 578 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. (93) If we combine the equations of Magnetic Force (B) with those of Electric Currents (C), and put for brevity dF dG dH r . d' _^d' ^ d' . dx dy dz dx^ df dz- (63), AlTfJip' ■■ dJ dx -V'F (64). d T If the medium in the field is a perfect dielectric there is no true conduction, and the currents p', q. r are only variations in the electric displacement, or, by the equations of Total Currents (A), df f _<^g , dh P dt' ^~dt' ^ ~'dt (65). But these electric displacements are caused by electromotive forces, and by the equations of Electric Elasticity (E), P = Jcf, Q = kg, R^hh (66). These electromotive forces are due to the variations either of the electro- magnetic or the electrostatic functions, as there is no motion of conductors in the field ; so that the equations of electromotive force (D) are d^_d^' dt dx dG d^ dt dy (67). R- _dH_d-^ ~ dt ^ (94) Combining these equations, we obtain the following ; 7 f(^'^ t,2Z7t\ . fd'F d'^\ ^ ^ \dx df ■dHl_ df dxdtj d'^\ \dy~' -^ l^^"'^\~df^d^t)-^ '['d^.-^^l^'''^[-w'-d^l=' .(68). A DYNAMICAL THEORY OF THE ELECTKOMAGNETIC FIELD. 579 If we diiFerentiate the third of these equations with respect to y, and the second with respect to z, and subtract, J and ^ disappear, and by remem- bering the equations (B) of magnetic force, the results may be written ^V>^ = 47./.^,/.^[ (69). (95) If we assume that a, y8, y are functions of lx + my + nz-Vt = w, the first equation becomes , d'a ,jr,d'a ^^d^^=^^''^dilf (^0)' or F= ±\/^ (71). The other equations give the same value for F, so that the wave is propa- gated in either direction with a velocity V. This wave consists entirely of magnetic disturbances, the direction of mag- netization being in the plane of the wave. No magnetic disturbance whose direction of magnetization is not in the plane of the wave can be propagated as a plane wave at all. Hence magnetic disturbances propagated through the electromagnetic field agree with light in this, that the disturbance at any point is transverse to the direction of propagation, and such waves may have all the properties of polarized light. (96) The only medium in which experiments have been made to determine the value of k is air, in which yx=l, and therefore, by equation (46), V=v (72). By the electromagnetic experiments of MM. "Weber and Kohlrausch *, ■v = 310,740,000 metres per second * Leipziff Transactions, Vol. v. (1857), p. 260, or Poggendorff's Annalm, Aug. 1856, p. 10. 73—2 580 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. is the number of electrostatic units in one electromagnetic unit of electricity, and this, according to our result, should be equal to the velocity of light in air or vacuum. The velocity of light in air, by M. Fizeau's* experiments, is 7=314,858,000; according to the more accurate experiments of M. Foucault t^ F= 298,000,000. The velocity of hght in the space surrounding the earth, deduced from the coefficient of aberration and the received value of the radius of the earth's orbit, is 7=308,000,000. (97) Hence the velocity of light deduced from experiment agrees sufficiently well with the value of v deduced from the only set of experiments we as yet possess. The value of v was determined by measuring the electromotive force with which a condenser of known capacity was charged, and then discharging the condenser through a galvanometer, so as to measure the quantity of electricity in it in electromagnetic measure. The only use made of light in the experiment was to see the instruments. The value of V found by M. Foucault was obtained by determining the angle through which a revolving mirror turned, while the light reflected from it went and returned along a measured course. No use whatever was made of electricity or magnetism. The agreement of the results seems to shew that light and magnetism are affections of the same substance, and that light is an electromagnetic dis- turbance propagated through the field according to electromagnetic laws. (98) Let us now go back upon the equations in (94), in which the quantities J and '^ occur, to see whether any other kind of disturbance can be propagated through the medium depending on these quantities which disappeared from the final equations. * Comptes Eendus, Vol. xxis. (1849), p. 90. t Ibid. Vol. LV. (1862), pp. 501, 792. A DYNAMICAL THEOEY OF THE ELECTROMAGNETIC FIELD. If we determine x from the equation 581 ^'^ dx'^ df'^ df~'^' and F', G', H' from the equations then F' = F-^, G' = G-^, H' = H-^. ax ay dz dF' dG' dH' ^ dx dy dz and the equations in (94) become of the form .(73), •(74), (75), -— {^'^^(*4^)} (-)■ Differentiating the three equations with respect to x, y, and z, and adding, we find that ^=-4+<^(^, y, z) m, dt and that kV'F' = 4:TriJL JcV'G' = AiTfJi hV'H' = iTTfl d'F' 1 df d'G' df d'H' df (78). Hence the disturbances indicated by F', G', H' are propagated with the velocity V= / through the field; and since V 47rja dF' dG' dH' ^^ dx dy dz ' the resultant of these disturbances is in the plane of the wave. (99) The remaining part of the total disturbances F, G, H being the part depending on y, is subject to no condition except that expressed in the equation dt "^ df 582 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC EIELD. If we perform the operation V^ on this equation, it becomes rl T lce = ^^-hV^^{x, y, z) (79). Since the medium is a perfect insulator, e, the free electricity, is immove- dJ able, and therefore -t- is a function of x, y, z, and the value of J is either constant or zero, or uniformly increasing or diminishing with the time ; so that no disturbance depending on J can be propagated as a wave. (100) The equations of the electromagnetic field, deduced from purely experimental evidence, shew that transversal vibrations only can be propagated. If we were to go beyond our experimental knowledge and to assign a definite density to a substance which we should call the electric fluid, and select either vitreous or resinous electricity as the representative of that fluid, then we might have normal vibrations propagated with a velocity depending on this density. We have, however, no evidence as to the density of electricity, as we do not even know whether to consider vitreous electricity as a substance or as the absence of a substance. Hence electromagnetic science leads to exactly the same conclusions as optical science with respect to the direction of the disturbances which can be propagated through the field ; both affirm the propagation of transverse vibra- tions, and both give the same velocity of propagation. On the other hand, both sciences are at a loss when called on to affirm or deny the existence of normal vibrations. Relation between the Index of Refraction and the Electromagnetic Character of the suhstance. (101) The velocity of light in a medium, according to the Undulatory Theory, is ■ yo, where i is the index of refraction and V^ is the velocity in vacuum. The velocity, according to the Electromagnetic Theory, is /X V 477-//,' where, by equations (49) and (71), h = j,k„ and lc, = iTrV^. A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 583 Hence D=':. •(80), or the Specific Inductive Capacity is equal to the square of the index of refrac- tion divided by the coefficient of magnetic induction. Propagation of Electromagnetic Disturbances in a Crystallized Medium. (102) Let us now calculate the conditions of propagation of a plane v/ave in a medium for which the values of h and ju, are different in different direc- tions. As we do not propose to give a complete investigation of the question in the present imperfect state of the theory as extended to disturbances of short period, we shall assume that the axes of magnetic induction coincide in direction with those of electric elasticity. (103) Let the values of the magnetic coefficient for the three axes be X, ju,, V, then the equations of magnetic force (B) become _dH_dG] dy dz o_dF_dH vy: dz clG dx dx dF dy (81). The equations of electric currents (C) remain as before. The equations of electric elasticity (E) will be F = 4.TTa'f Q = ^7r¥g ^ (82), R^Airc'h where 47ra', iirlf, and iird' are the values of h for the axes of x, y, z. Combining these equations with (A) and (D), we get equations of the form I f^d'F , d'F , d'F\ 1 dxdtj' d (^dF dG , dm I (d'F ^ d^\ ..... dz a' df ixv\'dx' ' ^ dy' ' " dz'J iLvdx\'dx ' *" dy (104) li I, m, n are the direction-cosines of the wave, and V its velocity, and if Ix + my + nz— Vt = iv (84), 584 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC I-IELD. then F, G, H, and ^ will be functions of w ; and if we put F', G', H', ^' for the second differentials of these quantities with respect to w, the equations will be . If wis now put .(85). cnm G'-nV-^'^O V'-V'r^ in {¥ix, + cV) + mV (cV + a'\) + n'v (p?\ + 6» I +"^(l+l'+:^)('=^^-v-*)-^ we shall find F'V'U-WVU=Q with two similar equations for G' and H'. Hence either V = Q Z7=0 or .(86), (87), . (88), . (89), . (90). VF'^W, VG' = 'm^' and VH' = n^ The third supposition indicates that the resultant of F', G', H' is in the direction normal to the plane of the wave ; but the equations do not indicate that such a disturbance, if possible, could be propagated, as we have no other relation between '^' and F', G', H'. The solution F= refers to a case in which there is no propagation. ■''■ The solution ^7= gives two values for V corresponding to values of F', G', H', which are given by the equations a ^ F' + lG'^lH' = (91), -p (6> - cV) + -^ (cV - a'\) + -j^, (a'\ - by) = . (92). * [Although it is not expressly stated in the text it should be noticed that in finding equations (91) and (92) the quantity *' is put equal to zero. See § 98 and also the corresponding treat- ment of this subject in the Electricity and Magnetism ii. § 796. It may be observed that the A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 585 (105) The velocities along the axes are as follows : — Direction of propagation Direction of the electric displacements X y ■ Z X a' V a' y }f V X z X Now we know that in each principal plane of a crystal the ray polarized in that plane obeys the ordinary law of refraction, and therefore its velocity is the same in whatever direction in that plane it is propagated. If polarized light consists of electromagnetic disturbances in which the electric displacement is in the plane of polarization, then I' ■■ a' = V = c' (98). • ' If, on the contrary, the electric displacements are perpendicular to the plane of polarization, X = /A = v (94). We know, from the magnetic experiments of Faraday, Pluckei', &c., that in many crystals X., /*, v are unequal. equations referred to and the table given in § 105 may perhaps be more readily understood from a different mode of elimination.. If we write \P + ixm" + vn' = PXfJiv and MF' + fj-niG' + vnH ' = Q\[i.v, it is readily seen that F' = l Yq,'-a'X.Q V - d'kP ' with similar expressions for G', H'. From these we readily obtain by reasoning similar to that in § 104, the equation cori-esponding to (86),- viz. : This form of the equation agrees with that given in the Electricity and Magnetism ii. § 797. By means of this equation the equations (91) and (92) readily follow when *' = 0. The ratios of F' : G' : H' for any direction of propagation may also be determined.] VOL. I. '^ 586 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. The experiments of Knoblauch * on electric induction through crystals seem to shew that a, h and c may be different. The inequality, however, of X, fx,, v is so small that great magnetic forces are required to indicate their difference, and the differences do not seem of sufficient magnitude to account for the double refraction of the crystals. On the other hand, experiments on electric induction are liable to error on account of minute flaws, or portions of conducting matter in the crystal. Further experiments on the magnetic and dielectric properties of crystals are required before we can decide whether the relation of these bodies to mag- netic and electric forces is the same, when these forces are permanent as when they are alternating with the rapidity of the vibrations of light. Relation between Electric Resistance and Transparency. (106) If the medium, instead of being a perfect insulator, is a conductor whose resistance per unit of volume is p, then there will be not only electric displacements, but true currents of conduction in which electrical energy is transformed into heat, and the undulation is thereby weakened. To determine the coefficient of absorption, let us investigate the propagation along the axis of X of the transverse disturbance G. By the former equations = -4.;.(f+g) by (A), drG , ^ (Id'G ldG\ , ,„, , ,^, ^=+^'^^U^-pWJ ^y (^) ^^^ (^) (95). If G is of the form G = e~^ cos {qx + nt) (96)^ we find that ^^prpT (^n where V is the velocity of light in air, and i is the index of refraction. The proportion of incident light transmitted through the thickness x is e-'"" (98). * Philosophical Magazine, 1852. A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD, 587 Let R be the resistance in electromagnetic measure of a plate of the substance whose thickness is x, breadth b, and length I, then ^ bx' 2px = 4ni.Z^j^ (99). (107) Most transparent solid bodies are good insulators, whereas all good conductors are very opaque. Electrolytes allow a current to pass easily and yet are often very trans- parent. We may suppose, however, that in the rapidly alternating vibrations of light, the electromotive forces act for so short a time that they are unable to effect a complete separation between the particles in combination, so that when the force is reversed the particles oscillate into their former position without loss of energy. Gold, silver, and platinum are good conductors, and yet when reduced to sufficiently thin plates they allow light to pass through them. If the resistance of gold is the same for electromotive forces of short period as for those with which we make experiments, the amount of light which passes through a piece of gold-leaf, of which the resistance was determined by Mr C. Hockin, would be only 10~™ of the incident light, a totally imperceptible quantity. I find that between g-J-Q- and xoVo ^^ green light gets through such gold-leaf Much of this is transmitted through holes and cracks ; there is enough, however, transmitted through the gold itself to give a strong green hue to the transmitted light. This result cannot be reconciled with the electromagnetic theory of light, unless we suppose that there is less loss of energy when the electromotive forces are reversed with the rapidity of the vibrations of light than when they act for sensible times, as in our experiments. Absolute Values of the Electromotive and Magnetic Forces called into play in the Propagation of Light. (108) If the equation of propagation of light is F=Aco^^{z-Vt), 74—2 588 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. the electromotive force will be A A, and the energy per unit of volume will be where P represents the greatest value of the electromotive force. Half of this consists of magnetic and half of electric energy. The energy passing through a unit of area is pa w=-~ • Stt/aF' so that ' P = j8wixVW, where V is the velocity of light, and W is the energy communicated to unit of area by the light in a second. According to Pouillet's data, as calculated by Professor W. Thomson *, the mechanical value of direct sunlight at the Earth is 8 3 "4 foot-pounds per second per square foot. This gives the maximum value of P in direct sunlight at the Earth's distance from the Sun, P = 60,000,000, or about 600 Daniell's cells per metre. At the Sun's surface the value of P would be about 13,000 Daniell's cells per metre. At the Earth the maximum magnetic force would be "IQSf. At the Sun it would be 4-13. i These electromotive and magnetic forces must be conceived to be reversed twice in every vibration of hght ; that is, more than a thousand million million times in a second. ' * Transactions of the Royal Society of Edinburgh, 1854 ("Mechanical Energies of the Solar System"). . ■ , ' f The horizontal magnetic force at Kew is about 1"76 in metrical units. A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 589 PAKT VII. CALCULATION OF THE COEFFICIENTS OF ELECTROMAGNETIC INDUCTION. General Methods. (109) The electromagnetic relations between two conducting circuits, A and B, depend upon a function M of their form and relative position, as has been already shewn. M may be calculated in several different ways, which must of course all lead to the same result. First Method. M is the electromagnetic momentum of the circuit B when A carries a unit current, or ^-/(^l+«l+^l)*'. where F, G, H are the components of electromagnetic momentum due to a unit current in A, and ds' is an element of length of B, and the integration is performed round the circuit of B. To find F, G, H, we observe that by (B) and (C) d'F d'F dF ^ , with corresponding equations for G and H, p, q', and r' being the components of the current ra. A. Now if we consider only a single element ds of A, we shall have , dx , , dy J , dz , P=ds^'' 2=^^'' ^=^'^'' and the solution of the equation gives ^=^S*- -^iS*' ^=?S*- : 590 A DYNAMICAL THEOEY OF THE ELECTROMAGNETIC FIELD. ^yhere p is the distance of any point from ds. Hence ^ C Cfju /dx dx dy dy dz dz\ , T , ~ J j p \ds ds' ds ds' ds ds') = I - cos Bdsds', JJ P where 6 is the angle between the directions of the two elements ds, ds', and p is the distance between them, and the integration is performed round both circuits. In this method we confine our attention during integration to the two linear circuits alone. (110) Second Method. M is the number of lines of magnetic force which pass through the circuit B when A carries a unit current, or M= "t {p-al + jLtySm + P'jn) dS', where /^a, p,^, py are the components of magnetic induction due to unit current in A, S' is a surface bounded by the current B, and I, m, n are the direction- cosines of the normal to the surface, the integration being extended over the surface. We may express this in the form M= a'Z — sin 6 sin d' sin (bdS'ds, r p. where dS' is an element of the surface bounded by B, ds is an element of the circuit A, p is the distance between them, 9 and 6' are the angles between p and ds and between p and the normal to dS' respectively, and ^ is the angle between the planes in which 6 and 6' are measured. The integration is performed round the circuit A and over the surface bounded by B. This method is most convenient in the case of circuits lying in one plane, in which case sin ^ = 1, and sin^ = l. (111) Third Method. M is that part of the intrinsic magnetic energy of the whole field which depends on the product of the currents in the two circuits, each current being unity. A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 591 Let a, fi, y be the components of magnetic intensity at any point due to the first circuit, a, fi', y the same for the second circuit; then the intrinsic energy of the element of volume d!F of the field is The part which depends on the product of the currents is ^(aa' + ;S^' + yy')^^. Hence if we know the magnetic intensities / and /' due to the unit current in each circuit, we may obtain M by integrating over all space, where Q is the angle between the directions of / and T . Application to a Coil. (112) To find the coefficient {M) of mutual induction between two circular linear conductors in parallel planes, the distance between the curves being eveiy- where the same, and small compared with the radius of either. If r be the distance between the curves, and a the radius of either, then when r is very smaU compared with a, we find by the second method, as a first approximation, M= iira (log,— - 2 To approximate more closely to the value of M, let a and a^ be the radii of the circles, and b the distance between their planes ; then r' = {a-a,f + b\ We obtain M by considering the following conditions: — 1st. M must fulfil the differential equation da" dJf a da This equation being true for any magnetic field symmetrical with respect to the common axis of the circles, cannot of itself lead to the determination of M as a function of a, a„ and b. We therefore make use of other conditions. 592 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 2ndly. The value of M must remain the same wlaen a and a^ are exchanged. 3rdly. The first two terms of M must be the same as those given above. M may thus be expanded in the following series : — Jlf=4;ralog-|l+--^ + ^ ^, >---^ ^ + &c.| — 477a la — ai 1 5' — 3(a-ai)' J_{66' — (a — aj'^a — aj „ ^ + 2^ ■''16 "^ 48 d +'^''' (113) We may apply this result to find the coefficient of self-induction {L) of a circular coU of wire whose section is small compared with the radius of the circle. Let the section of the coil be a rectangle, the breadth in the plane of the circle being c, and the depth perpendicular to the plane of the circle being 6. Let the mean radius of the coil be a, and the number of windings n ; then we find, by integrating, ''-V ^ I j M(xy x'y) dx dy dx' dy, where M(xy x'y) means the value of M for the two windings whose coordinates are xy and x'y respectively; and the integration is performed first with respect to X and y over the rectangular section, and then with respect to x' and y' over the same space. L = 47rn^a|log.^ + _1 _ | |^^_^j cot2^-^cos26'-^cot=^logcos^-ltan=^logsin^l ^S?fg?(2-'^+lH3-45 + 27-475cos=^-3.2(|-.)^J^^ + i|g-^^ 13 sin^^. . A „ Here a= mean radius of the coil. „ r= diagonal of the rectangular section = sjlf + cl „ 6= angle between r and the plane of the circle. „ n= number of windings. The logarithms are Napierian, and the angles are in circular measure. A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 593 In the experiments made by the Committee of the British Association for determining a standard of Electrical Resistance, a double coil was used, con- sistmg of two nearly equal coils of rectangular section, placed parallel to each other, with a small interval between them. The value of L for this coU was found in the following way. The value of L was calculated by the preceding formula for six different cases, in which the rectangular section considered has always the same breadth, while the depth was A, B, C, A+B, B+C, A+B+C, and n=l in. each case. Calling the results L{A), L {B), L{C), kc, we calculate the coefficient of mutual induction M{AC) of the two coUs thus, 2ACM{AC) = {A+B+CyL{A+B + C)-{A+BYL{A+B) -{B+CYL{B+C) + B'L{B). Then if n^ is the number of windings in the coil A and % in the coil C, the coefficient of self-induction of the two coUs together is L = n,'L (A) + 2n,?i,M{AC) + n,'L (C). (114) These values of L are calculated on the supposition that the windings of the wire are evenly distributed so as to ffil up exactly the whole section. This, however, is not the case, as the wire is generally circular and covered with insulating material. Hence the current in the wire is more concentrated than it would have been if it had been distributed uniformly over the section, and the currents in the neighbouring wires do not act on it exactly as such a uniform current would do. The corrections arising from these considerations may be expressed as nu- merical quantities, by which we must multiply the length of the wire, and they are the same whatever be the form of the coU. Let the distance between each wire and the next, on the supposition that they are arranged in square order, be D, and let the diameter of the wire be d, then the correction for diameter of wire is + 2(log^ + flog2 + ^-V)- VOL. I. 75 594 A DYNAMICAL THEOKY OF THE ELECTROMAGNETIC FIELD. The correction for the eight nearest wires is + 0-0236. For the sixteen in the next row + 0'00083. These corrections being multiphed by the length of wire and added to the former result, give the true value of L, considered as the measure of the potential of the coU on itself for unit current in the wire when that current has been established for some time, and is uniformly distributed through the section of the wire. (115) But at the commencement of a current and during its variation the current is not uniform throughout the section of the wire, because the induc- tive action between different portions of the current tends to make the current stronger at one part of the section than at another. When a uniform electro- motive force P arising from any cause acts on a cylindrical wire of specific resistance p, we have where F is got from the equation d'F ldF__ dr' r dr n > r being the distance from the axis of the cylinder. Let one term of the value of F be of the form IV", where T is a function of the time, then the term of p which produced it is of the form Hence if we write P \ dt / P dT\ av d'T 1 d'T , ^f^ ^ dt) p de p r . 2^ dt 1 d'T V . 2' df The total counter current of self-induction at any point is r* — &c. / P \ , 1 ^ fjiirdT ^ uV 1 d'T ^ „ from t — to t= 3s-6, these forces will be indeterminate. This indeterminateness is got rid of by the introduction of a system of e equa- tions of elasticity connecting the force in each piece with the change in its length. In order, however, to know the changes of length, we require to assume 3s displacements of the s points; 6 of these displacements, however, are equiva- lent to the motion of a rigid body so that we have 3s -6 displacements of points, e extensions and e forces to determine from 3s -6 equations of forces, e * [Owing to an oversight this paper is out of its proper place ; it should have been immediately before the memoir on "The Electro-magnetic Field." (No. XXV.)] ON THE CALCULATION OP THE EQUILIBRIUM AND STIFFNESS OF FRAMES. 599 equations of extensions, and e equations of elasticity; so that the solution is always determinate. The following method enables us to avoid unnecessary complexity by treating separately all pieces which are additional to those required for making the frame stiff; and by proving the identity in form between the equations of forces and those of extensions by means of the principle of work. On the Stiffness of Frames. Geometrical definition of a Frame. A frame is a system of lines connecting a number of points. A stiff frame is one in which the distance between any two points cannot be altered without altering the length of one or more of the connecting lines of the frame. A frame of s points in space requires in general 3s — 6 connecting lines to render it stiff. In those cases in which stiffness can be produced with a smaller number of lines, certain conditions must be fulfilled, rendering the case one of a maximum or minimum value of one or more of its lines. The stiffness of such frames is of an inferior order, as a small disturbing force may produce a displacement infinite in comparison with itself. A frame of s points in a plane requires in general 26' — 3 connecting lines to render it stiff. A frame of s points in a line requires s — 1 connecting lines. A frame may be either simply stiff, or it may be self-strained by the intro- duction of additional connecting lines having tensions or pressures along them. In a frame which is simply stiff, the forces in each connecting line arising from the application of a force of pressure or tension between any two points of the frame may be calculated either by equations of forces, or by drawing diagrams of forces according to known methods. In general, the lines of connexion in one part of the frame may be affected by the action of this force, while those in other parts of the frame may not be so affected. Elasticity and Extensibility of a connecting piece. Let e be the extension produced in a piece by tension-unity acting in it, then e may be called its extensibility. Its elasticity, that is, the force required 600 ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES. to produce extension-unity, will be - . We shall suppose that the effect of pressure in producing compression of the piece is equal to that of tension in producing extension, and we shall use e indifferently for extensibility and com- pressibility. Work done against Elasticity. Since the extension is proportional to the force, the whole work done will be the product of the extension and the mean value of the force ; or if £c is the extension and F the force, x = eF, work = IFx = ^eF' = i-x\ When the piece is inextensible, or e = 0, then all the work applied at one end is transmitted to the other, and the frame may be regarded as a machine whose efficiency is perfect. Hence the following Theorem. If p be the tension of the piece A due to a tension-unity between the points B and C, then an extension-unity taking place in A will bring B and C nearer by a distance p. For let X be the tension and x the extension of ^, Y the tension and // the extension of the line BC ; then supposing all the other pieces inextensible, no work will be done except in stretching A, or But X=pY, therefore y= —px, which was to be proved. Problem I. A tension F is appHed between the points B and (7 of a frame which is simply stiff; to find the extension of the line joining D and F, all the pieces except A being inextensible, the extensibility of A being e. Determine the tension in each piece due to unit tension between B and C, and let p be the tension in A due to this cause. Determine also the tension in each piece due to unit tension between D and F, and let y be the tension in the piece A due to this cause. Then the actual tension of A is Fp, and its extension is eFp, and the extension of the line DF due to this cause is -Fepq by the last theorem. ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES. 601 Cor. If the other pieces of the frame are extensible, the complete value of the extension in DE due to a tension i^ in -BC is -Ft{epq), where S(epg) means the sum of the products of epq, which are to be found for each piece in the same way as they were found for A. The extension of the line BC due to a tension F m. BC itself will be -Ft{ef), tiep") may therefore be called the resultant extensibility along BC. Problem II. A tension F is applied between B and C ; to find the extension between D and E when the frame is not simply stiff, but has additional pieces R, S, T, &c. whose elasticities are known. Let p and q, as before, be the tensions in the piece A due to unit tensions in BC and DE, and let r, s, t, &c. be the tensions in A due to unit tension in R, S, T, &c. ; also let R, S, T be the tensions of R, S, T, and p, cr, t their extensibilities. Then the tension A = F2y + Rr + Ss+Tt + &c.; the extension of A = e(Fp + Rr + Ss + Tt + &c.); the extension of R - - Ft{epr) - RXer' - Sters - Ttert + &c. = Rp ; extension of S = -Ft [eps) - R%{ers) - S%es' - Tt{est) = So- ; extension of T = - Ft{ept) - Rt{ert) - St{est) - Tt{ef) = TV ; also extension of DE = - F-Z{epq) - Rt{eqr) - St(eqs) - Tt{eqt) = x, the extension required. Here we have as many equations to determine R, S, T, &c. as there are of these unknown quantities, and by the last equation we determine x the extension of DE from F the tension in BC Thus, if there is only one additional connexion R, we find tjepr) 11= — -T ;=r7 — i\— , — . S(er') + p and VOL. I. ^^ 602 ON THE CALCULATION OP THE EQUILIBEIUM AND STIFFNESS OF FKAMES. If there are two additional connexions R and S, with elasticities p and a, x=-F 2e (r' + p) %e (s^ + cr) - (2 {ers) f t (epr) % (ers) S (eqs) + 1 (eps) t {eqr) t (ers) + 1 (epq) te {r' + p) te (s' + o-)] — % {epr) % (eqr) %e (s' + cr) — S (eps) t {eqs) te (r' + p) — t {epq) (S {ers)f The expressions for the extensibility, when there are many additional pieces, are of course very complicated. It will be observed, however, that p and q always enter into the equations in the same way, so that we may establish the following general' Theorem. The extension in BC, due to unity of tension along DE, is always equal to the tension in DE due to unity of tension in BC. Hence we have the following method of determining the displacement produced at any joint of a frame due to forces applied at other joints. 1st. Select as many pieces of the frame as are sufficient to render all its points stiff. Call the remaining pieces R, S, T, &c. 2nd. Find the tension on each piece due to unit of tension in the direction of the force proposed to be applied. Call this the value of p for each piece. 3rd. Find the tension on each piece due to unit of tension in the direction of the displacement to be determined. Call this the value of q for each piece. 4th. Find the tension on each piece due to unit of tension along R, S, T, &c., the additional pieces of the frame. Call these the values of r, s, t, &c. for each piece. 5th. Find the extensibility of each piece and call it e, those of the additional pieces being p, cr, r, &c. 6th. R, S, T, &c. are to be determined from the equations Rp + Rt {er') + S{ers) + Tt {ert) + Ft {epr) = 0, So- + Rt{ers) + S{esj + Tt {est) + Ft {eps) = 0, TV + Rt{ert) + S{est) + Tt {ef) + Ft {ept) = 0, as many equations as there are quantities to be found. 7th. X, the extension required, is then found from the equation x^- Ft {epq) - Rtifirq) - St {eqs) - Tt{eqt). ON THE CALCULATION OP THE EQUILIBRIUM AND STIFFNESS OF FRAMES. 603 in structures acted on by weights in which we wish to determine the deiiection at any point, we may regard the points of support as the extremities of pieces connecting the structure with the centre of the earth; and if the supports are capable of resisting a horizontal thrust, we must suppose them connected by a piece of equivalent elasticity. The deflection is then the shortening of a piece extending from the given point to the centre of the earth. Example. Thus in a triangular or Warren girder of length I, depth d, with a load W placed at a distance a from one end, ; to find the deflection at a point distant h from the same end, due to the yielding of a piece of the boom whose extensibility is e, distant x from the same end. I — a The pressure of the support at = PT -^ ; and if a; is less than a, the W force at x wUl be —rr x{l — a), or __ x{l — a) dl If x is greater than a, Similarly, if x is less than 6, but if a; is greater than h, The deflection due to x is therefore Wepq, where the proper values of p and q must be taken according to the relative position of a, h, and x. If a, h, I, X represent the number of the respective pieces, reckoning from the beginning and calling the first joint 0, the second jomt and the piece opposite 1, &c., and if L be the length of each piece, and the extensibility of each piece =e, then the deflection of h due to W at a will be, by summation of series, = lWeL\^^^{2b{l-a)-{b-ay+l}. 76—2 t — dl a{l- x) p- dl x{l- b) 9'- dl b{l- X) ?- dl 604 ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES. This is the deflection due to the yielding of all the horizontal pieces. The greater the number of pieces, the less is the importance of the last term. Let the inclination of the pieces of the web be a, then the force on a piece between and a is TF 7-^ — . or I sm a • l-a , p =f—. — when xa. I sm a Q = T~. — when xh. L sm a If e' be the extensibility of a piece of the web, we have to sum Wte'p'q to get the deflection due to the yielding of the web, We INDEX TO VOL. I. Absorption, Electric, 573 Ampere, 193 Beam : bent into a circular ring, 55 — 57, 65 Bending, Lines of, 87, 97, 105, 106, 107, 108 Brewster, Sir David, 43, 63, 68, 263, 413 Cauchy, 32, 40, 71 Challis, 453, 503, 505 Clapeyron, 30, 32, 70, 72 Clausius, 377, 386, 387, 405 Coil, Coefficient of induction of, 592 — 597 Colour-bHndness, 119, 137, 441 Colour box, 420 ; Method of observation, 426 Coloured beams of light. Mixtures of, 143 — 147 Coloured discs, 128, 264, 265 Coloured powders, 142 Colour equations, 128, 129, 138—141, 148, 268—270, 427, 428, 442 Colour: Experiments on, 126, 263; Law of Per- ception of, 130; Theory of the Perception of, 135 Colours : History of, 411 ; Mathematical theory of Newton's diagram of, 416 ; Method of repre- senting them by straight lines, 418 ; Theory of Compound, 149, 243, 410; Three primary, Theory of, 445 Colour sensations. Relations of to the pure rays of the spectrum, 150 Colour top, 127, 147 Colour triangle, 121, 131—135, 268, 416 Compression, Equations of, 36 Condenser, Capacity of, 572 ; Theory of, 572—576 Contact, Conic of, 93, 94, 102 Cotes, 271 Current Electricity, Conduction of, 180 Currents : Action of closed, 183 ; Intrinsic energy of, 541; Mutual action between, 537; Produced by induction, 185; Quantity and Intensity as properties of, 189 Curvature, entire and specific, 89, 92 Cylinder : acted on by centrifugal force, 60 ; hollow, exposed to pressure, 45 — 50 ; hollow, dilated by heat, 62; of parallel wires twisted, 59; twisted, 42, 43, 44, 66 D'Alembert, 248 Diagrams, Conditions of indeterminateness in draw- ing reciprocal diagrams, 516; of Force, 514 Dielectrics, Theory of, 177 Dynamometer, Weber's, 546 Earth's motion, 259 Elastic bodies. Collisions of, 405 Elasticity : Axioms, 31 ; Equations of, 38 ; Co- efficients of, 41 Elastic solids : Equilibrium of, 30 ; Pressures in the interior of, determined by the action on polar- ized light, 68 Electric absorption, 573 Electrical images, 209 Electromagnet, Effect of the core, 222 Electromagnetic disturbances. Propagation of, 578, 583 Electromagnetic Eield : Dynamical theory of, 526 ; General equations of, 534, 551 — 552, 554 — 564; Mechanical actions in, 565 — 570 Electromagnetic Induction, 536 ; Calculation of coefficients, 589 606 INDEX TO VOL. I. Electromagnetic momentum, 538 : Dynamical illus- tration of, 537 Electromagnetism : Ampere's laws, 193 Electrotonic state, 188, 205, 538 Equipotential surfaces, Magnetic, 553 Euler, 29, 32, 248, 271 Induction: by motion of conductor, 540; Coefficients of, for two circuits, 539 ; Determination of coefficients of, 547 ; Electromagnetic, 536 ; Magnecrystallic, Theory of, 180 ; of one cur- rent by another, 540 ; Paramagnetic and dia- magnetic, Theory of, 178 Instrument, perfect optical, 274 Isinglass, optical properties of, 43, 67 Earaday, 155, 188, 205, 241, 504, 529, 531, 535^ 542, 573, 585 Eelici, 538 Figures : General relation between the numbers of Joule, 377 points lines and polygons in, 515 ; Reciprocal, 514 Kohlrausch, 492, 499, 535, 569, 579 Eizeau, 500, 580 Fluid : Application to Lines of Force, 175; Theory Lame, 30, 32, 70, 72 of motion, 160; through a resisting medium, 163 Laplace, 292, 293, 294, 369 Foramen Centrale, Sensibility to light, 242 Forbes, Professor, 1, 124, 142, 145, 146, 243 Force : Diagrams of, 514 ; Lines of, 155, 158, 241 ; Magnetic Lines of, 551 ; Physical lines of, 451 Forces : Absolute values of, concerned in propagation of light, 588 ; Electromotive, 181 Foucault, 229, 248, 580 Fourier, 361 Frames, Equilibrium and stiffness of, 598 Leslie, Sir John, 16 Light : distinction between optical and chromatic properties of, 411 ; Electromagnetic theory of, 502, 577 ; propagation of, forces called into play, 587 ; propagation of, in a crystallized medium, 583 Liquids, Compressibility of, 50 Listing, 271 Magnetic field of variable intensity, 214 Magnetic lines of force, 552 Magnetism, Quantity and Intensity as indicated by lines of force, 192 Magnets, Permanent, Theory of, 178 Momentum, Electromagnetic, 538 Gases : Conductivity of, 403 — 405 ; Diffusion of, 392, 403; Dynamical Theory of, 377; Fric- tion, 390 ; Ratio of specific heats of, 409 Gauss, 81, 88—90, 271 Graham, 403 Grassman, 125, 414, 419 Gravitation, arising from the action of surrounding Nasmyth, J., 57 medium, 571 Navier, 30, 31, 32, 72 Green, 196 Neumann, 208, 512, 527 Gregory, Dr, 126 Newton, 3, 124, 135, 142, 143, 144, 145, 146, 149, Gutta percha, optical properties of, 43 151, 249, 267, 410, 411, 412 Heavy body, descent of, in a resisting medium, 115 (Ersted, 30, 33, 50 Helmholtz, 125, 141, 144, 145, 146, 152, 204, 243, Optical Instruments, General laws of, 271 ; Mathe- 271, 414, 415, 488, 533 matical treatment of, 281—285 Herapath, 377 Oval Curves, 1 Hersohel, Sir J. F. W., 142 Plate, bent by pressures, 57 Image, perfect defined, 273 Plateau, 243, 295 Images, Electrical, Theory of, 209 Platometer, New form of, 230 INDEX TO VOL. I. 607 Pliicker, 585 Poinsot, 248, 250 Poinsot's Theory of Rotation, Instrument to illus- trate, 246 Poisson, 30, 32, 72 Polyhedron, inscribed in a surface, 94, 98, 99 Pressures, Equations of, 37 Problems, Solutions of, 74 Ray, Reduced path of, 280 Reciprocal Figures, 514; Application to Statics, 522; Possibility of drawing, 516; Relation between the number of points lines and polygons in, 515 Refraction, index of, how related to specific in- ductive capacity, 583 Regnault, 33, 71 Resistance, Electric, how related to transparency, 586 Resisting medium, descent of a heavy body in, 115 Rigid body, Stability of the steady motion about a fixed centre of force, 374 ; motion of about a sphere, 296 Ring : motion of, when rigid, about a sphere, 296— 310 ; motion of, when the parts are not rigidly connected, 310; of equal satellites, 360 Rings, Efiect of long continued disturbances on a system of, 352; mutual perturbations of two, 345 ; Fluid, Loss of energy due to friction, 354 Rolling Curves, 4 ; Examples of, 22 — 29 Rolling of Curves on themselves, 19 Rotation, Theory of, 249 Saturn's Rings, Stability of, 286, 288—376 Sources and sinks defined, 163 Spectrum, Relation of the Colours of, to Compound Colours, 410 Sphere, hollow, dilated by heat, 64; exposed to pressures, 51- — 55 Sphere : magnecrystallic, 217 ; magnetic in uniform field of magnetic force, 212 Spherical shell : Electromagnetic, 220, 222 ; Per- manent magnetism in, 220 ; Revolving in magnetic field, 226 Spherical electromagnetic coil machine, 224 Spheres perfectly elastic : Motions and Collisions of, 378 ; Boyle's law, 389; Mean distance between collisions, 386 ; Mean Velocity, 381 ; Mean Velo- city-square, 381 ; Two systems in same vessel, 383 Spheres : Two, between poles of a magnet, 216 ; Two, in uniform magnetic field, 215 Stokes, 32, 33, 71, 72, 143, 209, 391, 410 Struve, 292 Surfaces : Applicability of, 95 ; Conjugate systems of curves on, 95, 96 ; Transformation of, by bending, 80 Telescopes, perfect, 275—279 Thomson, Sir W., 157, 196, 199, 209, 212, 374, 453, 503, 505, 528, 529, 533, 588 Top, Dynamical, 248 ; Instantaneous axis, 255 ; Invariable axis, 252 — 255 ; Method of observ- ing the motion, 257 Tractory of a curve, 13 Tractory of circle, 15, 17 Tubes, unit, 161 Unit, Electrostatic and Electromagnetic, 569 Verdet, 504, 506, 507, 513, 529 Vortices, molecular : Applied to Electric Currents, 467 ; Applied to Magnetic phenomena, 451 ; Applied to Statical Electricity, 489 Wave length. Method of determining, 423 Weber, 208, 492, 499, 507, 527, 535, 545, 569, 579 Wheatstone, 434 Wilson, Dr George, 137, 415 Young, 32, 60, 124, 136, 137, 412, 419, 447 CAMBRIDGE : PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS.