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THE SCIENTIFIC PAPERS OF
JAMES CLERK MAXWELL
Edited by W. D. NIVEN, M.A., F.R.S.
Two Volumes Bound As One
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Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London W. C. 2.
This Dover edition, first published in 1965, is an unabridged and unaltered republication of the work first pubhshed by Cambridge University Press in 1890. This edition is published by special arrange- ment with Cambridge University Press.
The work was originally pubhshed in two separate volumes, but is now published in two volumes bound as one.
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THE SCIENTIFIC PAPERS OF
JAMES CLERK MAXWELL
Edited by W. D. NIVEN, M.A., F.R.S,
Volume One
TO HIS GRACE
THE DUKE OF DEVONSHIRE 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
SHORTLY after the death of Professor James Clerk Maxwell a Committee was 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 publish 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 EiicyclopcEdia Biitannica, 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 Joui-nal, to Messrs Metcalfe and Co., Publishers of the Quarterly Journal of Pure and Applied Mathematics, and to the Lords of the Conmiittee 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.
Numerous and important Papers, contributed by Clerk Maxwell to the Transactions or Proceedings of the Royal Societies of London and of Edinburgh, of the Cambridge Philosophical Society, of the Royal Scottish Society of Arts, and of the London Mathematical Society; Lectures delivered by Clerk Maxwell at the Royal 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 diagrams 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 remams to add that the footnotes inserted by the Editor are enclosed between square brackets.
Cambridge, Augv^t, 1890.
PEEFACE.
CLERK 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 IGth 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 mth any great success in that profession. At all events, a quiet life in the country
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 married 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 bom 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 ccirageous 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 caupjs 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 grotesquely 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. aa
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, in 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 gratifpng 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 abready 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 eai'lier for the sake of the social intercourse which he would have found it difficult 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.
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 in 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 Bums 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 poetry, — 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.
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 diagrams, 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 carry 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 presentel 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 occupied 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 Tiinity College, but was afterwards elected an honorary Fellow, at the same time as Professor Cayley.
During the yeai*s 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 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 and in part aeriform ; (3) 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 modem 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 aiTangement 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 Jenkin ; 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 memoii' 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, ofiBces which entailed a few weeks' residence at the University in winter. His chief employment during those years was the prepai-ation 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. XVU
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 great 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 their 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 1873 ; 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.
XVIU 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 Researches" of the Hon. Henry Cavendish. This work, published in 1879, has had the eflfect 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 spirits 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 oflf 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 admired.
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 Encyclopoedia 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 modem 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.
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 draw aside from the subject in the pursuit of analytical 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 afready 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 vnth 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 T 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 number 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 cuiTents and magnets and the induction produced in a circuit by any variation of the strength of the field m 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 transmittmg force and motio^i, 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
teotions being communicated by forces arising from the relative displacement of their
connected parts, in virtue of their 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
currehts 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 formulae for the mechanical force on currents, 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 this 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 current as well as the current 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 ai-e 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 d3niamical 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 h}^othesis 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 coefl&cient 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 Encyclopcedia Britannica. Some of these contain more or less popular expositions of this subject which Maxwell bad himself in great 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 Arts 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 diagrammatical ly 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
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 (1875) 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 efiects of the entire instrument without examining the mechanism by which these effects are produced, and then, as in the paper in 1858, 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 1867, 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 applied 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. XXvii
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 otjiers 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 suflScient 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 analogous 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 diflScult 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, (b) Essays or Short Treatises, (c) Biographical Sketches, (d) Criticisms and Reviews.
XXVIU 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 Britanrdca and several others of a kindred character to Nature.
Class (c) contains such articles as " Fai-aday " 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 pages.
In some of these writings, particularly those in class (b), the author allowed himself a gi-eater 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 fi-om 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" appeai-ed 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 laboui-s 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 akic 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 memoirs 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 Dj-namics.
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 Navax College, Greenwich,
August, 1890.
TABLE OF CONTENTS.
II. Ill
IV.
V.
VI.
VII.
IX.
X. XI.
XII. XIII.
XIV. XV.
On the Description of Oval Curves and those having a plurality of Foci; with
remarks by Professor Forbes
On the Theory of Rolling Curves ■*
On the Equilibrium of Elastic Solids ^^
Solutions of Problems
On the Transformation of Surfaces by Bending 80
On a paHicular case of the descent of a heavy body in a resisting medium . 115
On the Theory of Colours in relation to Colour- Blindness 119
Experiments on Colour as perceived by the Eye, with remarks on Colour -Blindness 126
On Faraday's Lines of Force ^"^^
Description of a New Form of the Platometer, an Instrument for measuring the
areas of Plane Figures drawn on paper 230
On the elementary theory of Optical Instruments 238
On a method of drawing the Theoretical Form3 of Faraday's Lines of Force
without calculation
On the unequal sensibility of the Foramen Centrale to Light of different Colours 242 On the Theory of Compound Colours with reference to mixtures of Blue and
Yellow Light ^^'^
On an instrument to illustrate Poimot's TJieory of Rotation . . • .246 On a Dynamical Top, for exhibiting the phenomena of the motions of a body of invariable form about a fixed point, with s&ine suggestions as to the Earth's
motion
Account of Experiments on the Perception of Colour 263
97 1 On the general laius of Optical Instruments
On Theories of the Constitution of Saturn's Rings 286
On the stability of the motion of Saturn s Rings 288
Illustrations of the Dynamical Theory of Gases
On the Theory of Compound Colours and the Relations of the Colours of the Spectrum
On the Theory of Three Primary Colours ***^
451
On Physical Lines of Force
On Reciprocal Figures and Diagrams of Forces °^*
A Dynamical Theory of the Electromagnetic Field 526
On the Calculation of the EquilibHum and Stiffness of Frames .... 598
ERRATA.
Page 40. In the first of equations (12), second group of terms, read (hP dy' d^
instead of
d^^^d^^^d^^
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
d^M d2M_ldM da? "^ db' a da~
Page 592, in the first line of the expression for L change
- K cos 26 into - ^ cosec 26.
[From the Proceedings of the Royal Society of Edinburgh, Vol, li. April, 1846.]
I. On the Description of Oval Curves, and those having a plurality of Foci; ivith remarks by Professor Forbes. Communicated by Professor Forbes.
Mr Clerk Maxwell ingeniously suggests the extension of the common theory of the foci of the conic sections to curves of a higher degree of com- plication in the following manner : —
(1) As in the ellipse and hyperbola, any point in the curve has the sum or difference of two lines drawn from two points or foci = a. constant quantity, so the author infers, that curves to a certain degree analogous, may be described and determined by the condition that the simple distance from one focus pliLS a multiple distance from the other, may be = a constant quantity; or more generally, m times the one distance + n times the other = constant.
(2) The author devised a simple mechanical means, by the wrapping of a thread round pins, for producing these curves. See Figs. 1 and 2. He
Fig. 1. Two FocL Katios 1,
Fig. 2. Two Foci Ratios 2, 3.
then thought of extending the principle to other curves, whose property should be, that the sum of the simple or multiple distances of any point of
DESCRIPTION OF OVAL CURVES.
the curve from three or more points or foci, should be = a constant quantity ; and this, too, he has effected mechanically, by a very simple arrangement of a string of given length passing round three or more fixed pins, and con- straining a tracing point, P. See Fig. 3. Farther, the author regards curves
Fig. 3. Three Foci. Eatios of Equality.
of the first kind as constituting a particular class of curves of the second kind, two or more foci coinciding in one, a focus in which two strings meet being considered a double focus; when three strings meet a treble focus, &c.
Professor Forbes observed that the equation to curves of the first class is easily found, having the form
V^+7= a-VhJ{x- c)' + y\
which is that of the curve known under the name of the First Oval of Descartes*. Mr Maxwell had already observed that when one of the foci was at an infinite distance (or the thread moved parallel to itself, and was confined in respect of length by the edge of a board), a curve resembling an ellipse was traced ; from which property Professor Forbes was led first to infer the identity of the oval with the Cartesian oval, which is well known to have this property. But the simplest analogy of all is that derived from the method of description, r and r being the radients to any point of the curve from the two
foci ;
mr + nr — constant,
which in fact at once expresses on the undulatory theory of light the optical character of the surface in question, namely, that light diverging from one focus F without the medium, shall be correctly convergent at another point /
* Herschel, On Light, Art. 232 ; Lloyd, On Light and Vision, Chap. vii.
DESCRIPTION OF OVAL CURVES. J
within it ; and in this case the ratio — expresses the index of refraction of
the medium*.
If we denote by the power of either focus the number of strings leading to it by Mr Maxwell's construction, and if one of the foci be removed to an infinite distance, if the powers of the two foci be equal the curve is a parabola ; if the power of the nearer focus be greater than the other, the curve is an eUipse; if the power of the infinitely distant focus be the greater, the curve is a hyperbola. The first case evidently corresponds to the case of the reflection of parallel rays to a focus, the velocity being unchanged after reflection; the second, to the refraction of parallel rays to a focus in a dense medium (in which light moves slower) ; the third case to refraction into a rarer medium.
The ovals of Descartes were described in his Geometry, where he has also given a mechanical method of describing one of themt, but only in a particular case, and the method is less simple than Mr Maxwell's. The demonstration of the optical properties was given by Newton in the Principia, Book i., prop. 97, by the law of the sines; and by Huyghens in 1690, on the Theory of Undu- lations in his Traite de la Lumiere. It probably has not been suspected that so easy and elegant a method exists of describing these curves by the use of a thread and pins whenever the powers of the foci are commensurable. For instance, the curve. Fig. 2, drawn with powers 3 and 2 respectively, give the proper form for a refracting surface of a glass, whose index of refraction is 1'50, in order that rays diverging from f may be refracted to F.
As to the higher classes of curves with three or more focal points, we cannot at present invest them with equally clear and curious physical properties, but the method of drawing a curve by so simple a contrivance, which shall satisfy the condition
mr + nr +pr" + &c. = constant,
is in itself not a little interesting; and if we regard, with Mr Maxwell, the ovals above described, as the limiting case of the others by the coalescence of two or more foci, we have a farther generalization of the same kind as that so highly recommended by Montucla^ by which Descartes elucidated the conic sections as particular cases of his oval curves.
♦ This was perfectly well shewn by Hnyghens in his Traite de la Lumiere, p. 111. (1690.)
+ Edit. 1683. Geometria, Lib. ii. p. 54.
X Histoire dea Mathematiqties. First Edit IL 102.
[From the Transactions of the Royal Society of Edinburgh, Vol. xvi. Part v.]
II. On the Theory of Rolling Curves. Communicated by the Eev. Professor
Kelland.
There is an important geometrical problem which proposes to find a curve having a given relation to a series of curves described according to a given law. This is the problem of Trajectories in its general form.
The series of curves is obtained from the general equation to a curve by the variation of its parameters. In the general case, this variation may change the form of the curve, but, in the case which we are about to consider, the curve is changed only in position.
This change of position takes place partly by rotation, and partly by trans- ference through space. The roUing of one curve on another is an example of this compound motion.
As examples of the way in which the new curve may be related to the series of curves, we may take the following : —
1. The new curve may cut the series of curves at a given angle. When this angle becomes zero, the curve is the envelope of the series of curves.
2. It may pass through correspondiug points in the series of curves. There are many other relations which may be imagined, but we shall confine our attention to this, partly because it aSbrds the means of tracing various curves, and partly on account of the connection which it has with many geometrical problems.
Therefore the subject of this paper will be the consideration of the relations of three curves, one of which is fixed, while the second rolls upon it and traces the third. The subject of rolling curves is by no means a new one. The first idea of the cycloid is attributed to Aristotle, and involutes and evolutes have been long known.
THE THEORY OF ROLLING CURVES. 0
In the Histmy of the Royal Academy of Sciences for 1704, page 97, there is a memoir entitled "Nouvelle formation des Spirales," by M. Varignon, in which he shews how to construct a polar curve from a curve referred to rectangular co-ordinates by substituting the radius vector for the abscissa, and a circular arc for the ordinate. After each curve, he gives the curve into which it is " unrolled," by which he means the curve which the spiral must be rolled upon in order that its pole may trace a straight line; but as this 18 not the principal subject of his paper, he does not discuss it very fully.
There is also a memoir by M. de la Hire, in the volume for 1706, Part ii., page 489, entitled "Methode generale pour r^duire toutes les Lignes courbes ^ des Roulettes, leur generatrice ou leur base ^tant donnde telle qu'on voudra."
M. de la Hire treats curves as if they were polygons, and gives geome- trical constructions for finding the fixed curve or the rolling curve, the other two being given; but he does not work any examples.
In the volume for 1707, page 79, there is a paper entitled, "Methode generale pour determiner la nature des Courbes form^es par le roulement de toutes sortes de Courbes sur une autre Courbe quelconque." Par M. Nicole.
M. Nicole takes the equations of the three curves referred to rectangular co-ordinates, and finds three general equations to connect them. He takes the tracing-point either at the origin of the co-ordinates of the rolled curve or not. He then shews how these equations may be simplified in several particular cases. These cases are —
(1) When the tracing-point is the origin of the roUed curve.
(2) When the fixed curve is the same as the rolling cxirve.
(3) When both of these conditions are satisfied.
(4) When the fixed line is straight.
He then says, that if we roll a geometric curve on itself, we obtain a new geometric curve, and that we may thus obtain an infinite number of geometric curves.
The examples which he gives of the application of his method are all taken from the cycloid and epicycloid, except one which relates to a parabola, rolling on itself, and tracing a cissoid with its vertex. The reason of so small a number of examples being worked may be, that it is not easy to eliminate the co-ordinates of the fixed and rolling curves from his equations.
The case in which one curve roUing on another produces a circle is treated of in Willis's Principles of Mechanism. Class C. Boiling Contact.
6 THE THEORY OP ROLLHiTO CURVES.
He employs the same method of finding the one curve from the other which is used here, and he attributes it to Euler (see the Acta Petropolitana, Vol. v.).
Thus, nearly all the simple cases have been treated of by different authors; but the subject is still far from being exhausted, for the equations have been applied to very few curves, and we may easily obtain new and elegant proper- ties from any curve we please.
Almost all the more notable curves may be thus linked together in a great variety of ways, so that there are scarcely two curves, however dissimilar, between which we cannot form a chain of connected curves.
This will appear in the list of examples given at the end of this paper.
Let there be a curve KAS, whose pole is at C.
THE THEORY OF ROLLING CURVES. 7
Let the angle DCA = 6, and CA=r, and let
Let this curve remain fixed to the paper.
Let there be another curve BAT, whose pole is B.
Let the angle MBA = 0t, and BA=r^, and let
Let this curve roll along the curve KAS without slipping. Then the pole B will describe a third curve, whose pole is C. Let the angle DCB = 0^, and CB = r„ and let
We have here six unknown quantities 0,dAr,r^r^; but we have only three equations given to connect them, therefore the other three must be sought for in the enunciation.
But before proceeding to the investigation of these three equations, we must premise that the three curves will be denominated as follows : — The Fixed Curve, Equation, e^ = ^^{r^. The Rolled Curve, Equation, 0. = <f>,{r,). Tlie Traced Curve, Equation, 6^ = 4>.,{r^.
When it is more convenient to make use of equations between rectangular co-ordinates, we shall use the letters x^^, x^^, x^ij^. We shall always employ the letters s^s^^ to denote the length of the curve from the pole, p.p^p^ for the per- pendiculars from the pole on the tangent, and q^q/i^ for the intercepted part of the tangent.
Between these quantities, we have the following equations: —
r = ^/^T?, ^ = tan-|,
a? = r cos ^, y = r sin 6,
r" ydx — xdy
jm'S ""^w+w'
THE THEORY OF ROLLING CURVES.
rdr
dS _ xdx + ydy
2=-r=7x!fi' r-
J{dxy + (dyY'
' "^ W '^d^ daf
We come now to consider the three equations of rolling which are involved in the enunciation. Since the second curve rolls upon the first without slipping, the length of the fixed curve at the point of contact is the measure of the length of the rolled curve, therefore we have the following equation to connect the fixed curve and the rolled curve —
«! = Sj.
Now, by combining this equation with the two equations
it is evident that from any of the four quantities 6{r^6^r^ or x^^x^^, we can obtain the other three, therefore we may consider these quantities as known functions of each other.
Since the curve rolls on the fixed curve, they must have a common tangent.
Let PA be this tangent, draw BP, CQ perpendicular to PA, produce CQ, and draw BR perpendicular to it, then we have CA=r^, BA = r^, and CB = r,; CQ=p„ PB=p,, and BN=p,; AQ = q„ AP = q„ and CN=q,.
Also r,'=CR=CR + RR = (CQ + PBY+(AP-AQf
=p,' + 2p,p, +p,' + r,' -p,' - 2q,q, + r," -p,' fz = n' + n' + 2piPa - 2q,q^. Since the first curve is fixed to the paper, we may find the angle 6,. Thus e, = DCB = DCA + ACQ + RCB
= e?. + tan-| + tan-|§ ^, = ^, + tan--^ + tan-^ ^^^^
TjdO^ Pi +pi
THE THEORY OF ROLLING CURVES. »
Thus we have found three independent equations, which, together with the equations of the curves, make up six equations, of which each may be deduced from the others. There is an equation connecting the radii of curvature of the three curves which is sometimes of use.
The angle through which the rolled curve revolves during the description of the element ds„ is equal to the angle of contact of the fixed curve and the rolling curve, or to the sum of their curvatures,
ds^ ds^ ds.
But the radius of the rolled curve has revolved In the opposite direction through an angle equal to dO,, therefore the angle between two successive posi- tions of r, is equal to -^-dd,. Now this angle is the angle between two
successive positions of the normal to the traced curve, therefore, if 0 be the centre of curvature of the traced curve, it is the angle which ds^ or ds^ subtends at 0. Let OA^T, then
ds^ r4d^ ds, ,^ _ ds^ ds, ,.
^J__J_ 1 _^ •*• '^'ds, T~ R, R, ds/
-tAt^tJ RJR.'
As an example of the use of this equation, we may examine a property of the logarithmic spiral.
In this curve, p = mr, and R = — , therefore if the rolled curve be the ■^ m
logarithmic spiral
/I 1\ 1 ^m
"^[t^tJ-r^v/
m_ 1
t~r:,*
AO therefore ^0 in the figure = ?ni2i, and -^ = m.
Let the locus of 0, or the evolute of the traced curve LYBH, be the curve OZY, and let the evolute of the fixed curve KZAS be FEZ, and let us consider FEZ as the fixed curve, and OZF as the traced curve.
10 THE THEORY OF ROLLING CURVES.
Then in the triangles BPA, AOF, we have OAF=PBA, and ^='^ = ^y
therefore the triangles are similar, and FOA = APB = - , therefore OF is perpen- dicular to OA, the tangent to the curve OZY, therefore OF is the radius of the curve which when roUed on FEZ traces OZY, and the angle which the curve makes with this radius is OFA=PAB = %mr^m, which is constant, there- fore the curve, which, when rolled on FEZ, traces OZY, is the logarithmic spiral. Thus we have proved the following proposition : " The involute of the curve traced by the pole of a logarithmic spiral which rolls upon any curve, is the curve traced by the pole of the same logarithmic spiral when rolled on the involute of the primary curve."
It follows from this, that if we roll on any curve a curve having the property _2:»i — Wjri, and roll another curve having Pi = 'm^r^ on the curve traced, and so on, it is immaterial in what order we roll these curves. Thus, if we roll a logarithmic spiral, in which jp = mr, on the nth involute of a circle whose radius is a, the curve traced is the w+lth involute of a circle whose radius is Jl-m\
Or, if we roll successively m logarithmic spirals, the resulting curve is the n + mth involute of a circle, whose radius is
aJl—m^ sll- m/, Jkc.
We now proceed to the cases in which the solution of the problem may be simplified. This simplification is generally effected by the consideration that the radius vector of the rolled curve is the normal drawn from the traced curve to the fixed curve.
In the case in which the curve is rolled on a straight line, the perpen- dicular on the tangent of the rolled curve is the distance of the tracing point from the straight line ; therefore, if the traced curve be defined by an equation in iCg and y„
'^.°p.= / "'„... (1)'
and
'••=^'^©^ ^'^-
THE THEORY OF ROLLING CURVES. 11
By substituting for r, in the first equation, its value, as derived from the second, we obtain
-■©■[©■-]=©'■
If we know the equation to the rolled curve, we may find (-7-^') in
terms of r,, then by substituting for r, its value in the second equation, we
dx (1 X
have an equation containing x^ and -^, from which we find the value of -t— '
dy, du,
in terms of x^; the integration of this gives the equation of the traced curve.
As an example, we may find the curve traced by the pole of a hyperbolic spiral which rolls on a straight line.
a
fdrA' _ rl ,ddj ~ a'
The equation of the rolled curve is 6^ =
- •■©■-■[(IJ-]'
dx^ _ ^3 '* dy,~Ja'-x,''
This is the differential equation of the tractory of the straight line, which is the curve traced by the pole of the hyperbolic spiral. By eliminating x^ in the two equations, we obtain
dr^_ /dxA
This equation serves to determine the rolled curve when the traced cuive is given.
As an example we shall find the curve, which being rolled on a straight line, traces a common catenary.
Let the equation to the catenary be
'l(e' + e-^.
12 THE THEORY OF ROLLING CURVES.
Then
dy,~N a' '
dr
then by integration ^ =cos'^ ( 1 j
2a r =
1+COS0'
This is the polar equation of the parabola, the focus being the pole ; there- fore, if we roll a parabola on a straight line, its focus will trace a catenary.
The rectangiilar equation of this parabola is af = Aay, and we shall now consider what curve must be rolled along the axis of y to trace the parabola.
By the second equation (2),
n = ^9 /-4- + l> but x^^Pi, V ^»
.-. r/=^/ + 4a", .-. 2a = Vr/-jp/ = g'„ but q^ is the perpendicular on the normal, therefore the normal to the curve always touches a circle whose radius is 2a, therefore the curve is the involute of this circle.
Therefore we have the following method of describing a catenary by con- tinued motion.
Describe a circle whose radius is twice the parameter of the catenary; roll a straight line on this circle, then any point in the line will describe an involute
THE THEORY OF ROLLING CURVES. 13
of the circle ; roll this curve on a straight line, and the centre of the circle will describe a parabola ; roll this parabola on a straight line, and its focus will trace the catenary required.
We come now to the case in which a straight line rolls on a curve.
When the tracing-point is in the straight line, the problem becomes that of involutes and evolutes, which we need not enter upon ; and when the tracmg- point is not in the straight line, the calculation is somewhat complex; we shall therefore consider only the relations between the curves described in the first and second cases.
Definition. — The curve which cuts at a given angle all the circles of a given radius whose centres are in a given curve, is called a tractory of the given curve.
Let a straight line roll on a curve A, and let a point in the straight line describe a curve B, and let another point, whose distance from the first point is b, and from the straight line a, describe a curve C, then it is evident that the curve B cuts the circle whose centre is in C, and whose radius is b,
at an angle whose sine is equal to r, therefore the curve 5 is a tractory of
the curve C.
When a = b, the curve B is the orthogonal tractory of the curve C. If tangents equal to a be drawn to the curve B, they will be terminated in the curve C; and if one end of a thread be carried along the curve C, the other end will trace the curve B.
When a = 0, the curves B and C are both involutes of the curve A, they are always equidistant from each other, and if a circle, whose radius is 6, be rolled on the one, its centre will trace the other.
If the curve A is such that, if the distance between two points measured along the curve is equal to 6, the two points are similarly situate, then the curve B is the same with the curve C. Thus, the curve A may be a re- entrant curve, the circumference of which is equal to 6.
When the curve -4 is a circle, the curves B and C are always the same.
The equations between the radii of curvature become
1 1 _ r
14 THE THEORY OF ROLLING CURVES.
When a = 0, T=0, or the centre of curvature of the curve B is at the point of contact. Now, the normal to the curve C passes through this point, therefore —
"The normal to any curve passes through the centre of curvature of its tractory,"
In the next case, one curve, by rolling on another, produces a straight line. Let this straight line be the axis of y, then, since the radius of the rolled curve is perpendicular to it, and terminates in the fixed curve, and since these curves have a common tangent, we have this equation,
If the equation of the rolled curve be given, find -j-^ in terms of r^, sub- stitute Xi for r^, and multiply by x^, equate the result to -^ , and integrate.
Thus, if the equation of the rolled curve be
d = Ar-"" + &c. + Kr-^ + Lr'^ + if log r + iVr + &c. + Zr"",
^ = - n^r-(»+^) - &c. - 2Kr-' - I/p-' + Mr'' + N+ &c. + wZr"-^ dr
-r-= - nAx~'* - &c. - 2Kx~"- - Lx~^ + M+ Nx + &c. + nZx", ax
y = -^ Aa^-"" + &c. + 2Kx-' -L\ogx + Mx + ^Naf + &c. + -^ Zx""^',
which is the equation of the fixed curve.
If the equation of the fixed curve be given, find -^ in terms of cc, sub- stitute r for X, and divide by r, equate the result to -t-, and integrate.
Thus, if the fixed curve be the orthogonal tractory of the straight line, whose equation is
y = a log . + Ja^
a + \la^ — x^
dy _ Jo' — af dx~ X
THE THEORY OF ROLUNG CURVES.
15
de _Ja?-7* dr r*
0 = cos"^
this is the equation to the orthogonal tractory of a circle whose diameter is equal to the constant tangent of the fixed curve, and its constant tangent equal to half that of the fixed curve.
This property of the tractory of the circle may be proved geometrically, thus — Let P be the centre of a circle whose radius is PD, and let CD be a line constantly equal to the radius. Let BCP be the curve described by the point C when the point D is moved along the circumference of the circle, then if tangents equal to CD be drawn to the curve, their extremities will be in the circle. Let ACH be the curve on which BCP rolls, and let OPE be the straight line traced by the pole, let CDE be the common tangent, let it cut the circle in D, and the straight line in E.
Then CD = PD, .'. LDCP^ LDPC, and CP is perpendicular to OE, .'. L CPE= LDCP+ LDEP. Take away LDCP-^ L DPC, and there remains DPE=DEP, .-. PD=^DE, .-. CE=2PD.
16 THE THEORY OF ROLLING CURVES.
Therefore the curve ACH haa a constant tangent equal to the diameter of the circle, therefore ACH is the orthogonal tractorj of the straight line, which is the tractrix or equitangential curve.
The operation of finding the fixed curve from the rolled curve is what Sir John Leslie calls " divesting a curve of its radiated structure."
The method of finding the curve which must be rolled on a circle to trace a given curve is mentioned here because it generally leads to a double result, for the normal to the traced curve cuts the circle in two points, either of which may be a point in the rolled curve.
Thus, if the traced curve be the involute of a circle concentric with the given circle, the rolled curve is one of two similar logarithmic spirals.
If the curve traced be the spiral of Archimedes, the rolled curve may be either the hyperbolic spiral or the straight line.
In the next case, one curve rolls on another and traces a circle.
Since the curve traced is a circle, the distance between the poles of the fixed curve and the rolled curve is always the same; therefore, if we fix the rolled curve and roll the fixed curve, the curve traced will still be a circle, and, if we fix the poles of both the curves, we may roU them on each other without friction.
Let a be the radius of the traced circle, then the sum or difference of the radii of the other curves is equal to a, and the angles which they make with the radius at the point of contact are equal,
.♦. n-=±(a±r,)andn^^ = r,^\
dO, _ ±(a±r^ dS, drt~ r, dvi'
If we know the equation between ^j and r,, we may find ^— in terms of r„
substitute ± (a ± r,) for r„ multiply by ^ \ and integrate.
Thus, if the equation between 6^ and r^ be
r, = a sec $,,
TEU: THEORY OF ROLLING CURVES. 17
which is the polar equation of a straight line touching the traced circle whose equation is r = ay then
dd _ a
dr, ~ r, -Jr.'-a' a
{r,±a)Jr,'±2r,a dO^ r^±a a
dr, r, (r,±a) Jrf±2r^
a
_ 2a _ 2a
Now, since the rolling curve is a straight line, and the tracing point is
not in its direction, we may apply to this example the observations which
have been made upon tractories.
2a Let, therefore, the curve ^ = ^ — 7 be denoted by A, its involute by B, and
the circle traced by C, then B is the tractory of C; therefore the involute
2a of the curve ^ = ^ — r is the tractory of the circle, the equation of which is
^ = cos"' /— — I. The curve whose equation is ^'=s — ; 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 m 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
18 THE THEORY OF ROLLING CURVES.
tangents is the same as in the catenary, namely, by describing a circle 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 wiU be similar to the locus of the intersection of the tangent with the perpendicular from the pole ; the magnitude, however, of the traced curve will be double that of the other curve; therefore, if we call n = <^o^o the equation to the fixed curve, r, = <f>,6, that of the traced curve, we have
also, £^ = f.
SimUarly, r, = 2p, = 2r,f = A^ Ar, (^J, 0,^6,-2 cos- ^ .
Similarly, r„ = 2p„., = 2r„_, ^ &c. = 2^ (^^J , and ^^f.
^„ = ^„-7lC0S-f-\
'o
V 0n = 6. — ncos~^ -^ .
Let e, become 6^'; 0„ 6,' and ^ , ^. Let ^„^-^„ = a,
^„^ = ^;-ncos- ^, » «.
a = ^„^- e„ = ^.^-^o-ncos-^ ^' +n cos-^ ^
-1 Pn -1 Pn O- , ^0 ~ ^0
\ cos ^ ^^-^ — COS * -^— = - 4 .
THE THEORY OF ROLLING CURVES. 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 6^ varies by an angle equal to a. Let this variation = (^ ; then if 6^ — 6 J = fi,
^ n n Now, if n increases, <f> will diminish ; and if n becomes infinite,
<^ = ^ + ^ = 0 when a and )8 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 rolled on itself, traces itself.
For the curve which has this property, if rolled on itself, and the operation repeated an infinite number of times, will 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
n=2"a(cos|)". -1=2". (sing (oosf-; 2"a'(cos^")'"
.'. r^ = 2p,= 2
r, = 2
^2-a'(cosg%2-a^(sing (cosg"^'^
2"a cos — / n\ „+i
^^cos-j+(sm-j
20 THE THEORY OF ROLLING CURVES.
Now ^1-^0= -cos-^^"= -cos-' cos -" = -^,
" n+1 substituting this value of 6^ in the expression for r^,
r. = 2-'a^cos--J ,
similarly, if the operation be repeated ni times, the resulting curve is
*afcos— ^^y \ n + mj
When n=l, the curve is
r = 2a cos 9,
the equation to a circle, the pole being in the circumference.
When n = 2, it is the equation to the cardioid
r = 4a (cos -J .
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, bei6g performed on a circle, and again on the resulting curve, will produce a cardioid, and the intermediate curve between the circle and cardioid is
r = 2
> / 20\i
As the operation of rolling a curve on itself is represented by changing n into (n + 1) in the equation, so this operation may be represented by changing n into (w + i).
Similarly there may be many other fractional operations performed upon the curves comprehended under the equation
r = 2"a(cos-j.
We may also find the curve, which, being rolled on itself, will produce a given curve, by making 7i= — 1.
THE THEORY OF 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 d^, then P^^p,^2a^2a_
therefore suppressing the suflSx,
= ar,
* • \d0j a '
dr r
7i-''
■■&-')
- 2a ^~l-cos^'
the polar equation of the parabola whose parameter is 4rt.
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 given function of the angle described by the second.
Let 0^ = (f}0i, then r^ + r^ = a, and -j^ = — ;
d0^ a-r^'
Let us take as an example, the pair of wheels which will represent the angular motion of a comet in a parabola.
THE THEORY OF ROLLING CURVES.
Here 6^ = tan -^ ,
. ^_
2 cos' -^
a 2 + cos ^1 '
therefore the first wheel is an ellipse, whose major axis is equal to | 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"' B^ and r^ = a - r„
'• 1+ 1
a ^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 C. Let the curve A, when rolled 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 Rolling 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 THEORY OF ROLLING CURVES. 23
Ex. 3. When the rolled curve is the parabola whose parameter is 4a, the traced curv^e 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
r 2a
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
y' = d' + a^.
Second. In the examples of a straight Hne I'olling 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 cui-ve 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
X I
y = a log , + JcL' — -f^,
a + V a' - ar"
and C is a straight line at a distance a from the vertex of the catenary.
24 THE THEORY OF ROLLING CURVES.
Ex. 3. When tKe curve A is the polar catenaxy the curve B is the tractory of the circle
and the curve (7 is a circle of which the radius is - .
Third. Examples of one curve rolling on another, and tracing a straight line.
Ex. 1. The curve whose equation is
0 = Ar-"* + &c. + Kr-' + Lr'^ + Jf log r + iVr + &c. + Zt^, when rolled on the curve whose equation is
n — 1 71+ L
traces the axis of y.
Ex. 2. The circle whose equation is r = a cos ^ rolled on the circle whose radius is a traces a diameter of the circle.
Ex. 3. The curve whose equation is
^=J'i-
1 — versm - , a
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 roll on it the spiral of Archimedes r = ad, 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 ^ = t^ t^® P^^® "^^ ^^^^ ^^® tangent at the vertex of the parabola.
THE THEORY OF ROLLING CURVES. 25
Ex. 7. If we roll the curve whose equation is
r = a cos (t^) on the ellipse whose equation is
the pole will trace the axis h.
Ex. 8. K we roll the curve whose equation ia
on the hyperbola whose equation is
the pole will trace the axis h.
Ex, 9. If we roll the lituus, whose equation is
on the hyperbola whose equation is
the pole will trace the asymptote.
Ex. 10. The cardioid whose equation is
r = a(H- cos ^), rolled on the cycloid whose equation is
12 = a versin"' - + J2ax - ic*, ^ a
traces the base of the cycloid.
Ex. 11. The curve whose equation is
0 = versm-'- + 2^/ 1,
rolled on the cycloid, traces the tangent at the vertex.
26 THE THEORY OF ROLLING CURVES.
Ex. 12. The straight line whose equation is
r = a sec B, 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
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
'■=5'
rolled on the logarithmic curve whose equation is
1 ^ 2/ = alog-,
traces the axis of y or the asymptote.
Ex. 17. The involute 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
THE THEORY OF ROLLING CURVES. 27
rolled on the witch, whose equation is
traces the asymptote.
Ex. 19. The curve whose equation is
r — a tan Q, rolled on the curve whose equation is
traces the axis of y.
Ex. 20. The curve whose equation is
2r
e=
rolled on the curve whose equation is
y = / , or r = a tan $,
traces the axis of y.
Ex. 21. The curve whose equation is
r = a (sec d — tan 0), rolled on the curve whose equation is
2/ = alogg+l), traces the axis of y.
Fourth. Examples of pairs of rolling curves which have their poles at a fixed distance = a.
Ce straight line whose equation is ^=sec"'- ..„ , .
r
2a
The polar catenary whose equation is 0= ±fj I ±
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.
Curve whose equation is ^-/J^ — l + versin"^-.
Ex. 4.
28 THE THEORY OF ROLLING CURVES.
fCaxdioid whose equation is r=2a(l+co8^).
Ex. 5.
Ex. 6.
Ex. 7.
[Curve whose equation is ^ = sin"*- + log ,— — — .
(Conchoid, r = a (secg- 1).
Icurve, ^ = >A-?
Spiral of Archimedes, r = a0.
T T
Curve, ^ = - + log
+ sec"^ - a
a ° a
f Hyperbolic spiral, r=-Q
Ex. 8. -!
ICurve,
a
e'+l
1
Cpse whose equation is ^"^^2+ ~Q'
Ex. 10.
(Involute of circle, ^~Ja^^^ ®®^"^ a '
'curve, e^J^±2l±log(-±l+J^.±2'^.
Fifth. Examples of curves rolling on themselves. Ex. 1. When the curve which rolls on itself is a circle, equation
r = a cos 6, the traced curve is a cardioid, equation r = a(l+cos^). Ex. 2. When it is the curve whose equation is
r = 2"a (cos-j , the equation of the traced curve is
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
J
2a , . ., r
1 — versin - .
r a
Ex. 7. When it is the curve whose equation is the equation of the traced curve is r = a (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 Nicolfe 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 ; 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 fertihty 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 experiment 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 (Ersted 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 kept 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 which 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 EQUILIBRITJM OF ELASTIC SOLIDS. 31
ence between elastic solids and fluids. Both tend to recover their vohirne, 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 solid, —
1. TTie sum of the three pressures is proportional to the sum of the com- pressions ichich they produce.
2. The difference between two of the pressures is propo7'tional 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 will 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 Lam6 and Clapeyrou. St^t- note A.
t See note B.
32 THE EQUIUBRIUM 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, Lam^ 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 AnnoUes de Chimie et de Physique, 2^ Sdrie, xv. 264, and xxxviii. 435 ; L'AppUcati(m de la Micanique, Tom. I.
Those of Poisson in Mem. de I'lnstitut, vm. 429 ; Annales de Chimie, 2" S^rie, XXXVI, 334 ; xxxvii. 337 ; xxxvtil 338 ; xlu. Journal de VEcole Polytechnique, cahier xx., with an abstract in Annales de Chimie for 1829.
The memoir of MM. Lam^ 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. in. 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 linear 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 h, and K^fi — ^m, k = m.
The theory of Professor Stokes is contained in Vol. vin. 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 OF ELASTIC SOLIDS. 33
Having assumed the proportionality of pressure to compression, he proceeds to define his coefficients.— "Let -^8 be the pressures corresponding to a uniform linear dilatation 8 when the solid is in equilibrium, and suppose that it becomes mA8, 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 8x = kx, Sy= - hj, and hz = 0, calls into action a pressure - Bk 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 ^ = 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 cylinder 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 jylasticity or the tendency which a constrained body has to relieve itself from a state of constraint, by its molecules assuming new positions of equi- librium. This property is opposed to Hnear elasticity ; and these two properties exist in all bodies, but in variable ratio.
M. Wertheim, in Annales de Chimie, 3« Sdrie, xxiii., has given the results of some experiments on caoutchouc, from which he finds that K=k, or fi = ^m; and concludes that k = K in all substances. In his equations, fi is therefore made equal to f 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, CoUadon 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 the 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 diflference 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 Zx, By, Bz. They are generally denoted by a, /8, 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 Bx, By, Bz, these variations being functions of
X, y, 2.
* See note C.
THE EQUILIBRIUM OF ELASTIC SOLIDS. 35
The quantities Sx, Sy, 8z, 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
dhx
dx '
dy'
dz'
dSy dx '
dhy dy'
dSij
dz '
dSz dx'
dhz
dy-
dBz
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 Id y
axes a, yS, y.
2. The nine direction-codnes of these axes, with the six connecting equa- 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, ^, y are
cos(aOa-)=aj, cos {^Ox) = \, co%(yQ)x) = c,, cos (aOy) = tto, _cos {fiOy) = h„, cos (yO^/) = c., cos (aOz) =a3, cos (/SOz) =63, cos {yOz) = c,.
These direction-cosines are connected by the six equations, a^ + h{ + Ci' = 1 , «i«s + ^h + CjC, = 0,
a./ -I- h^ + c,' = 1 , a^a^ + h.h^ + cx^ = 0,
a; + 63' + Gj' = 1 , a/t, + bj), + c^c, = 0.
The rotation of the system of axes a, 13, y, round the axis of x, from y to z, =B0^, y, from z to x, =S^j, z, from x to y, =^0/,
36
THE EQUILIBRIUM OF ELASTIC SOLIDS.
By resolving the displacements 8a, h/S, By, B6„ B9.„ Z6„ in the directions of the axes x, y, z, the displacements in these axes are found to be hx = a,8a + h,Bp + c3y -Be^ + Bd,y, By = aM + h,Bl3 -f c,By - Bd,x + Bd.z, Bz = a,Ba + hM + CsBy - BO^ + Bd,x. Sa .^ ^Si8
But
B^^rf, and 8y = y^,
and Q. = a^x + a^ + a.^, /3 = b,x + h^ + h.^, and y = c,x + c,y -h c^z.
Substituting these values of Sa, Sy8, and By in the expressions for Bx, By, Bz, and differentiating with respect to x, y, and z, in each equation, we obtain the equations
dBx Ba, ,. 8/8,2 , ^y
dy a ^ y
dBz _ Ba dz
a p y
(1)-
dBx Ba B^ T J By ,5s/,
dy a ' ^ y
a
dBx Ba dz a
Ba
BI3
J'
8^
dz a p y
dBy Ba BB T ^ By
dx a p y
Be,
c.f^ + Bdi
Be,
-J— = — ctjCti + -^ 6361 + -^ C3C1 + 8^2
dZz dx
dBz
8^ a
Sa
8^ S/8
r
Be,
Equations of compression.
{2).
Equations of the equilibnum 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 EQUILIBRIUM OF ELASTIC SOLIDS.
37
3. Two tangential actions on each face.
Let the six faces of the small parallelopiped be denoted by x^, 3/,, z„ x^ y„ and z,, then the forces acting on x^ are : —
1. A normal pressure jp, acting in the direction of x on the area dydz,
2. A tangential force g, 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
a; 2/ z
On the face a:,
— 'p^dydz
- q^dydz
-q.'dydz
^.
{P^'r J^dx)dydz
(^3 + 7^ ^^) c?yc?x
(q.'+-^^dx)dydz
2/1
— q^dzdx
—p^dzdx
— q.dzdx
y-x
{q\ + ^dy)dzdx
{p.+^dy)dzdx
(q, + ^dy)dzdx
Zi
— q^dxdy
— q^dxdy
—p^dxdy
^2
fe+ -4^dz)dxdy
(q^+^dz)dxdy
(p. + ^dz)dxdy
Attractions,
pXdxdydz
p Ydxdydz
pZdxdydz
Taking the moments of these forces round the axes of the particle, we find
?i' = ?i, q^=q.^ qz=qz',
and then equating the forces in the directions of the three axes, and dividing by dx, dy, dz, we find the equations of pressures,
dy dz dx ^ dz dx dy '^
Equations of Pressures.
(3).
38
THE EQUILIBRIUM OF ELASTIC SOLIDS.
The resistance which the sohd opposes to these pressures is called Elasticity, and is of two kinds, for it opposes either change of volume or change of Jigure. These two kinds of elasticity have no necessary connection, for they are possessed in very different ratios by different substances. Thus jelly has a cubical elas- ticity little different from that of water, and a linear elasticity as small as we please ; while cork, whose cubical elasticity is very small, has a much greater Imear elasticity than jelly.
Hooke discovered that the elastic forces are proportional to the changes that excite them, or as he expressed it, " Ut tensio sic vLs."
To fix our ideas, let us suppose the compressed body to be a parallelepiped, and let pressures Pi, Pj, P3 act on its faces in the direction of the axes a> A y, which will become the principal axes of compression, and the com-
pressions will be
So. 8^ Sy a' ^' 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,(^ + f + ^
(4).
II.
(P,-P,) = m
(P._p.) = „,g_^
(P.-P,) = m
rv ^rts T Equations of Elasticity.
h 7
By Ba
(5).
The quantity fj. is the coefiicient of cubical elasticity, and m that of linear elasticity.
THE EQUILrBRIUM OF ELASTIC SOLmS.
39
By solving these equations, the values of the pressures P„ P,, P„ and the
8a 8^ Sy , r J
compressions — ' ~S ' ^^7 ^^ found.
a \9/x 3m/ ^ ^ m
! = (!_ M(p. + P, + p.) + lp,
j3 \9/x 3 m/ ^ * ^ ?7i '
?r = (_L_ i\(P_+P_+P_) + ip_
y \9/z 3m/ ^ ^ m
(6).
(7).
From these values of the pressures in the axes a, )8, y, may be obtained.. the equations for the axes x, y, z, by resolutions of pressures and compressions*.
For
and
q = aaP^ + hhP, + ccP, ; , . . IdZx , d%y , d8z\ . d8x'
, . V IdZx . d8y , d8z\ dBy
, , , fdSx , d8y , rfSj\ , dSz
m /c?Sz c?Sx
(8)-
2 Vo?a; c?2
.(9).
See the Memoir of Lame and Clapeyron, and note A.
40
THE EQUIUBRIUM OP ELASTIC SOLIDS.
d$X /I 1 \ , , , N , 1
(10).
dy * ax ' m^ dz dy m ^
d^ dx
dz m^
(11).
By substituting in Equations (3) the values of the forces given in Equa- tions (8) and (9), they become
(12).
These are the general equations of elasticity, and are identical with those of M. Cauchy, in his Exercices d' Analyse, Vol. ni., p. 180, published in 1828,
where h stands for m, and K for ft - o" > and those of Mr Stokes, given in the
Cambridge Philosophical Transactions, Vol. viii., part 3, and numbered (30);
in his equations ^ = 3/x, B = — .
If the temperature is variable from one part to another of the elastic
soHd, the compressions -y- , -r^, -J^ , at any point will be diminished by a
quantity proportional to the temperature at that point. This prmciple is applied in Cases X. and XI. Equations (10) then become
THE EQUILIBRIUM OF ELASTIC SOLIDS.
41
dy
^ = fe - 3mj (P^-^P^+P^) + '^^^^P^
(13).
CfV being the linear expansion for the temperature v.
Having found the general equations of the equilibrium of elastic solids, I proceed to work some examples of their application, which afford the means of determining the coefficients /t, m, and o), and of calculating the stiffness of solid figures. I begin with those cases in which the elastic soHd is a hollow cylinder exposed to given forces on the two concentric cylindric surfaces, and the two parallel terminating planes.
In these cases the co-ordinates x, y, z are replaced by the co-ordinates x = x, measured along the axis of the cylinder. 2/ = r, the radius of any point, or the distance from the axis. z — rd, the arc of a circle measured from a fixed plane passing through the axis.
Px = o, are the compression and pressure in the direction of the axis at any point.
-^ = -J— , Pi =p, are the compression and pressure in the direction of the
radius.
dBz dhrd Br . . _ . , ,. . - 1
~dz~'db¥~l^' JP8 = ?, are the compression and pressure m the direction of the
tangent.
Equations (9) become, when expressed in terms of these co-ordinates —
m doO
dZx dx
dSx dx
m dB0
m dSx dr
*=2
.(14).
The length of the cylinder is h, and the two radii a, and a, in every VOL. I. G
42 THE EQUIUBRnJM 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 Bd, 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
m „ dhd
Therefore for the whole cylindric surface
ar Whence 8,=_^^ (1,_1.) ,
^^ "' = 2^&-i) ('«>■
The optical effect of the pressure of any point is expressed by
I=<oq,b = <o.^^ (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 will be inversely proportional to the square of the distance fi-om the axis of the cylinder, and the planes of polarization of these lays 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 OF 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 appUed to the interior cylinder.
By continuing the force of torsion while the jeUy 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 gla^s ; 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 uncrystallised 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 TransoLctions 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 jpositive 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 readdy 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.
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 imcrystallised 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 hB 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).
?2 = 1^^
dx '
This force acts at the distance r from the axis ; therefore its resistance to torsion will be q.r, and the resistance in a circular annulus will be
q^r^Ttrdr = mirr' -r- dr
and the whole resistance for the hollow cylinder will be expressed by
„, mn dS6 , ^ ,. /,^v
720 M
^(-1-] (17).
In this equation, m is the coefl&cient of linear elasticity; a^ and a^ are the radii of the exterior and interior surfaces of the hollow cyUnder in inches ; M is the moment of torsion produced by a weight acting on a lever, and is expressed
THE EQUILIBRIUM OF ELASTIC SOLIDS. 45
bj 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 cylinder whose angular motion is measured by means of indices, or more accurately by small mirrors attached to the cylinder ; 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 /x, and it seems to answer best with thick cylinders 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 hollow cylinder exposed to normal pressures only. When the pressures parallel to the axis, radius, and tangent are substituted for p^, p^, and pt, Equations (10) become
S = (i-34)(^+^-^^) + ^ (^«)-
^^t^(±-±]io+p + q) + :^q (20).
By multiplying Equation (20) by r, differentiating with respect to r, and
comparing this value of —j— with that of Equation (19),
p-q _(J__ _1\ /^ . ^ . ^\ _ i ^ rm " \9/x 3m/ \dr dr drj 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 rnwarde, we find the equation of the equiHbrium of the element,
ir£ = 4 (21).
r dr
46 THE EQUILIBRIUM OF ELASTIC SOLIDS.
By comparing this equation witli the last, we find
\9fi Zmj dr \9/i ^ 3m/ \dr ^ drj Integrating,
Since o, the longitudinal pressure, is supposed constant, we may assume
c -(^-^]o ' \9u, 3m/ . , .
c. = 12 =(^ + g)-
9/x, 3 m Therefore q—p = c^ — 2p, therefore by (21),
a linear equation, which gives
1 ^c, ^ = ^3^ + 2-
The coefficients Cj and Cj must be found from the conditions of the surface of the soHd. If the pressure on the exterior cylindric surface whose radius is a, be denoted by A,, and that on the interior surface whose radius is a^ by A,,
then p = h^ when r = ai and p = h.j when r = a^ and the general value of p is
_a^h^ — a^\ a^a^ h^ — h^ /22\
^" a,' -a,' ^ oT^^ ^ ^'
2-i'=2i^ ^73^- ''y (21).
*= «.'-«.' +^^57::^' (^^^■
/=5<.(^-2)=-26<.^"A^. (24).
This last equation gives the optical eflfect 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 OF ELASTIC SOLIDS. 47
Case I. ; but the direction of the principal axes ia different, as in this case they are parallel and perpendicular to the radius. The dark bands seen by polarized Ught wiU therefore be parallel and perpendicular to the plane of polarisation, 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,.
hx (l\( ^aX-ct'h-X . 2 / a,%-a,%\ ]
X \9/x
= o(^ + ~] + 2{Ka,^-Ka,^)
1/1 1
.(25).
,9/x 3m/ ' ^ ' ' ' 'Ui,'-a,'\9fj, 3mJ r 9/x \ a/ — a/ / 3?
When r = a., - ^ ^ fo4-2 ^4-^) + ^^^ ( - ^._^. ' '-o
(26).
~ VSft 3my "^ ' a; - a,' \ 9/x ^ 3m / ^ cv - a,' 1,9/x "^ 3m/ J
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^ = h^.
2. When the external pressure is to the internal pressure as the square of tlie interior diameter is to that of the exterior diameter, or when a^-h^ = a^-h^.
3. When the cylinder is soHd, or when a. = 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 in a
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.
X
48
THE EQUILIBRIUM OF ELASTIC SOLIDS.
1. When ^ji = A„ the equations of compression become
\9fi'*"3mj"'"^ '\9ij. 3m
(27).
7 = i('>+2^) + 3i(^-<')
When hi = hi = o, then
Zx _hr _ \ X ~ r " Sfi'
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^%
^ \9yx "^ 3m/
Bx
X
7 = |w + 3l(3^--»)^
(28).
In this case, when o = 0, the compressions are independent of /x. 3. In a solid cylinder, aj = 0,
The expressions for — and — are the same as those in the first case, when h^ — hf
When the lon^tudinal pressure o vanishes,
Bx
X
r ' \9/x 3m/ '
THE EQUILIBRIUM OF ELASTIC SOLIDS.
49
When the cylinder ia pressed on the plane sides only,
8x
r \9fi dmj
4. When the solid is infinite, or when a, is infinite,
p = K--._a-(\-K)
I=<o{p-q)=-^a.;{h,-h,) r 9/x ^ ' 3m ^ '
(29).
5. When 8r = 0 in a solid cylinder,
Zx Zo
6. When
X 2m + 3/A
So; _ hr _ 2>h x~ * r ~ m + 6iM
.(30).
Since the expression for the efiect of a longitudinal strain is Bx
if we make
VOL. I.
-=o(— + —) X \9/i, 3m/ '
r, 9mu, ^, 8x 1
E = ^ , then — = o ^^
m + 6/x cc E
(31).
50 THE EQUILIBRIUM OF ELASTIC SOLIDS.
The quantity E may be deduced from experiment on the extension of wires or rods of the substance, and /x is given in terms of m and E by the equation,
„ = _^!!L_ (32),
^^^ ^ = S (^^)'
P being the extending force, h the length of the rod, s the sectional area, and Bx 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 compressibihty of liquids is that employed by Canton, (Ersted, Perkins, Aime, &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 difference 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 compressibihty 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 appUed 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 compressibihty was greater in lead than in glass. The quantity found
THE EQUILIBRrcrM OF ELASTIC SOLIDS. 51
by Tredgold from the extension of rods was that denoted by E, and the value of ft 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 compressibihty 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 compressibility 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 h^ and K, 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 r.
The compressions at any point are -.— in the radial direction, and — in the tangent plane, the values of these compressions are : —
fr=[h-^^P^''i)*h^ ('")•
T = fe-3fJ(^ + 2,) + l5 (35).
Multiplying the last equation by r, differentiating with respect to r, and equating the result with that of the first equation, we find
52
THE EQUILIBRITTM OF ELASTIC SOLIDS.
Since the forces whicli act on the particle in the direction of the radius must balance one another, or
2qdrde +p (rdey =(^p + ^d7^(r + dry 6,
_r dp
therefore ^""-^ = 2 37 ^^^^'
Substituting this value of q -p in the preceding equation, and reducing,
therefore
^ + 2^ = 0. dr dr
Integrating, But and the equation becomes
therefore
p-\-2q = c,. r dp ,
dp dr
+ 3^-^-i = 0,
1 c.
Since p = h, when r = a.,, and p = K when r = a,, the value of p at any distance is found to be
^~ a^-af r' a^-a,' 9- a,'-ai "^^ 7^ <-a/
(37). .(38).
When r = a„ -y = -^r:^^ - + t ^^ ^^737^3 ^
~ a,' - a/ U 2»i/ a/ - «/ \jx 2wi/ _ When the external and internal pressures are equal
.(39).
h^ = h.,=p = q, and -y-
SV K
.(40),
THE EQUILIBRIUM OF ELASTIC SOLIDS. 53
the change of internal capacity depends entirely on the cubical elasticity of the vessel, and not on its thickness or linear elasticity.
When the external and internal pressures are inversely as the cubes of the radii of the surfaces on which they act,
aX = a,%, p = ^ K q= -i^K
when r = r- — ^ '
(41).
V 2 ^^
In this case the change of capacity depends on the linear elasticity alone.
M. Regnault, in his researches on the theory of the steam engine, has given an account of the experiments which he made in order to determine with accuracy the compressibility of mercury.
He considers the mathematical formulae very uncertain, because the theories of molecular forces from which they are deduced are probably far from the truth ; and even were the equations free from error, there would be much uncertainty in the ordinary method by measuring the elongation of a rod of the substance, for it is diflScult to ensure that the material of the rod is the same as that of the hollow sphere.
He has, .therefore, availed himself of the results of M. Lam6 for a hollow sphere in three different cases, in the first of which the pressure acts on the interior and exterior surface at the same time, while in the other two cases the pressure is applied to the exterior or interior surface alone. Equation (39) becomes in these cases, —
1. When ^1 = /ij, -^ = — and the compressibility of the enclosed liquid being /x,, and the apparent diminution of volume S'F,
v-.£-;) «■
2. When /i, = 0,
54 THE EQUILIBRIUM OF ELASTIC SOLIDS.
3. When h,^0,
8V_ h K , 9^\
V a^-a^ \ii ^ m ^ ' V2 J
M. Lamp's equations differ from these only in assuming that fi, = |-m. If this assumption be correct, then the coefficients /u,, m, and jMj, may be found from two of these equations ; but since one of these equations may be derived from the other two, the three coefficients cannot be found when /u, is supposed independent of m. In Equations (39), the quantities which may be varied at pleasure are \ and h^, and the quantities which may be deduced from the apparent compressions are,
'■=G+4)^°<^S-i)=^"
therefore some independent equation between these quantities must be found, and this cannot be done by means of the sphere alone; some other experiment must be made on the liquid, or on another portion of the substance of which the vessel is made.
The value of /x^, the elasticity of the liquid, may be previously known.
The linear elasticity m of the vessel may be found by twisting a rod of the material of which it is made ;
Or, the value of E may be found by the elongation or bending of the
We have here five quantities, which may be determined by experiment.
on sphere.
, audi:
-i^
2 3m
We have here
fiv
(43)
1.
(42)
2.
(31)
3.
(17)
4. 5.
+ — ) by external pressure
Cj = ( j equal pressures.
m by twisting the rod.
/Xj the elasticity of the liquid.
THE EQUILIBRIUM OF ELASTIC SOLIDS.
55
When the elastic sphere is solid, the internal radius a, vanishes, and
fh=p = q, and -y = ^-
When the case becomes that of a spherical cavity in an infinite solid, the external radius a^ becomes infinite, and
P=K-f{K-K)
r-
= K+i
^i'h-h,)
r
= ^^>i+^^(^>-^^)
1 m
v =
■'-!
(44).
The effect of pressure on the surface of a spherical cavity on any other part of an elastic solid is therefore inversely proportional to the cube of its distance from the centre of the cavity.
When one of the surfaces of an elastic hollow sphere has its radius rendered invariable by the support of an incompressible sphere, whose radius is Oj, then
— = 0, when r = a^,
therefore
2771
q=h
2a^m + 3«//x
3a,V 2a>i + 3a//x
r* 2a^m + 3a//x
IK
W hen r = a,, j-y — lu r-—. ~— ,-
" V -2a>2 + 3a.//i,
K^
1
r* 2a/m + 3a//i
(45).
Case V.
On the equilibrium of an elastic beam of rectangular section uniformly bent.
By supposing the bent beam to be produced till it returns into itself, we may treat it as a hollow cylinder.
66 THE EQUILIBRIUM OF ELASTIC SOLIDS.
Let a rectangular elastic beam, whose length is 2irc, be bent into a circular form, so as to be a section of a hollow cylinder, those parts of the beam which lie towards the centre of the circle will be longitudinally compressed, while the opposite parts will be extended.
The expression for the tangential compression is therefore
Br _ r — c r ~ c '
r
Sr Comparing this value of — with that of Equation (20),
V=(^-4)<''+-p+«)+^'''
dr
,,. , /I 2\ .,
ion
and by (21), q=p + r
By substituting for q its value, and dividing by r (q- + ^) • the equat:
becomes
dp 2m + 3/x j9 _ 9?n/i. — {m — 3/x) o 9m/x c dr m + 6fx r~ (m + 6fi) r (m + 6/x) r' *
a linear differential equation, which gives
^ ^ m — 3fir 2m + 3/x
Ci may be found by assumiQg that when r^a^, p = \, and q may be found from p by equation (21).
As the expressions thus found are long and cumbrous, it is better to use the following approximations : —
_/_9m^\ y ( )
l^\llcl^ \ (48).
In these expressions a is half the depth of the beam, and y is the distance of any part of the beam from the neutral surface, which in this case is a cylin- dric surface, whose radius is c.
These expressions suppose c to be large compared with a, since most sub- stances break when - exceeds a certain small quantity.
THE EQUILIBRIUM OF ELASTIC SOLIDS. 57
Let b be the breadth of the beam, then the force with which the beam resists flexure = M
M=lhyq = ^^^-^ = Ef (49),
which is the ordinary expression for the stiffness of a rectangular beam.
The' stiffness of a beam of any section, the form of which is expressed by an equation between x and y, the axis of x being perpendicular to the plane of flexure, or the osculating plane of the axis of the beam at any point, is ex- pressed by
Mc = E{ifdx (50),
M being the moment of the force which bends the beam, and c the radius of the circle into which it is bent.
Case YI.
At the meeting of the British Association in 1839, Mr James Nasmyth described his method of making concave specula of silvered glass by bending.
A circular piece of silvered plate-glass was cemented to the opening of an iron vessel, from which the air was afterwards exhausted. The mirror then became concave, and the focal distance depended on the pressure of the air.
Buffon proposed to make burning- mirrors in this way, and to produce the partial vacuum by the combustion of the air in the vessel, which was to be effected by igniting sulphur in the interior of the vessel by means of a burn- ing-glass. Although sulphur evidently would not answer for this purpose, phos- phorus might; but the simplest way of removing the air is by means of the air-pump. The mirrors which were actually made by Buffon, were bent by means of a screw acting on the centre of the glass.
To find an expression for the curvature produced in a flat, circular, elastic plate, by the difference of the hydrostatic pressures which act on each side of it,—
Let t be the thickness of the plate, which must be small compared with its diameter.
Let the form of the middle surface of the plate, after the curvature is produced, be expressed by an equation between r, the distance of any point from the axis, or normal to the centre of the plate, and x the distance of the point from the plane in which the middle of the plate originally was, and let
ds=-^{dxY + {dr)\
VOL I. 8
58 THE EQUILIBRIUM OF ELASTIC SOLIDS.
Let A, be the pressure on one side of the plate, and h^ that on the other.
Let p and q be the pressures in the plane of the plate at any point, p acting in the direction of a tangent to the section of the plate by a plane passing through the axis, and q acting in the direction perpendicular to that plane.
By equating the forces which act on any particle in a direction parallel to the axis, we find
^ drdx , ^ dpdx , ^ d^x ^ ,, j^dr
By making p = 0 when r = 0 in this equation, when integrated,
p-l^l^^--'^-) ("^-
The forces perpendicular to the axis are
[drV . dpdr , ^ d^r .^ i\dx ^ .
Substituting for p its value, the equation gives
_ (^1 - h^ idr dr dx\ (h^ - h^ /dr ds^d^^ds ^r\ , . ^" t ''[d'sdi'^d^)'^ 2t "^^[didxd^ dxd^)""^ ^'
The equations of elasticity become
dSs (\ 1 \ / ^ h, + h\^p
Differentiating -j- = -^ (""''')' ^^^ ^ ^^ *^^^®
dhr dr dr dSs
dr ~ ds ds ds '
By a comparison of these values of -t— ,
ds
dr\ ds) \9iJ,
, t^rwl 1\/ , ,K + h\,qdrp^ (I l\fdp,dq\
w dr as
THE EQUILIBRIUM OF ELASTIC SOUDS. 59
To obtain an expression for the curvature of the plate at the vertex, let a
be the radius of curvature, then, as an approximation to the equation of the
plate, let
r» 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
a =
18m/x, \-\-h^ m- 3/x
. 1 c 1 "T" ' T 7~ ~T~z ; — TT"
.(53).
^i-A, lOm + 51/x A,-^2 lOw + 51/t '
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 cylinder 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, hS the angle of torsion, M the force producing torsion, hx the change of length, and P the longitudinal force. Each of the wires becomes a helix whose radius is r, its angular rotation Zd, and its length along the axis x-Zx.
Its length is therefore {rZey
— IJ
and the tension is = jE; 1 1 - /[ 1 - - ] V r^ (-]'] .
This force, resolved parallel to the axis, is
60 THE EQUIUBRTCM OF ELASTIC SOUDS.
and since — and r — are small, we may assume
XX
-"-{-l-n?)'} <">■
The force, when resolved in the tangential direction, is approximately
"-■^m'i-m '">
By eliminating — between (54) and (55) we have
X
M: ^^'
^ip.E.^m (56).
X 24 \ a?/
When P = 0, 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 cylinder 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 difference 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 \s 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 EQUILIBBIUM OF ELASTIC SOLIDa.
61
The equation of the equilibrium of a particle [see Equation (21)], becomes
dp Air'k ,
where q and p are the tangential and radial pressures, k 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 ^ in Equations (19), (20), and neglect- ing 0,
-(i-3^)(«|-?-g)-M^S-f-^.^)
which gives
1 TT^k
2gt^\
1 , Tj'k
2+^K + ^«
(-"?)
TT'k
2gf
^=-V + 2^»(-2 + f)^ + c.
(57).
If the radii of the surfaces of the hollow cylinder be a, and cu„ and the pressures actmg on them h^ and h^, then the values of c^ and c, are
(58).
-f^'-(«--.')S(^-S.
When o, = 0, as in the case of a solid cylinder, c, = 0, and
« = *'+0 {2('^ + «.') + |(3'^-«,')} (59).
When A, = 0, and r^a^,
^ = ^U-2) (60).
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.
g2 THE EQUILIBRIUM OP ELASTIC SOLIDS.
Since I=ho>(q-p) = '^ (^-2\br', a transparent revolving cylinder, when
polarized light is transmitted parallel to the axis, will 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 hollow 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 till 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 = Ci\ogr + Ci (61).
Let the temperatures at the surfaces be 0^ and 0^, and the radii of the surfaces a, and a^, then
^ 0^-0^ loga,0^-logaA
^'""logaj-loga/ '~ loga^-loga^ Let the coeflBcient of linear dilatation of the substance be c,, then the proportional dilatation at any point will be expressed by c,v, and the equations of elasticity (18), (19), (20), become
r \,9/x 3m/ ^ ^ ^' m The equation of equHibrivuu is
2-P+r'^ (21),
and since the tube is supposed to be of a considerable length
-J— =c^ a constant quantity.
CL2C
THE EQUILIBRIUM OF ELASTIC SOLIDS. 63
From these equations we find ttat
9/x 3m and hence v = c^\ogr + Cz, p may be found in terms of r.
Hence ? = (|l + 4) " ^.«' •«§ '- ^. ^ + <'• + (|l + ^) ''.^-
Since I—hco (q —p) = ho)i— + - — ) CjCg — 260)05 -^ ,
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 ^j = ^„ = 0, and
/2 . J_V^.^ Aoo-r I ^i'^/log^i-log«2 , a/logct,-a/logaA
/^ 1_\- I a^a^ log g, - log ct, a^ log a, - a/ log a \
^-1,9,. + 3m/ ^^^3^^^S^ r^ a'-a^ + <-a,^ +V'
\9/x 3m/ ' ' \ r" a{-a^ J
...(62).
There will, therefore, be no action on polarized light for the ring whose radius is r when
r" = 2 „ log - .
Case X.
Sir David Brewster has observed {Edinburgh Transacticms, Vol. viii.), that when a solid cylinder of glass is suddenly heated at the cylindric siuface 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 polari^ied rays of %ht is proportional to the square of the radius r, or
/= bCj^cor' = h(o {q —p) = hayr -^ , Since if a be the radius of the cylinder, ^ = 0 when r^a,
Hence ?=J(3r'-o").
2
By substituting these values of p and q in equations (19) and (20), and , . d h' dhr T ^ ,
^=|(4 + li)'-' + »" (««)•
c^ being the temperature of the axis of the cylinder, and c, 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,
dr ^' * r
Ci = aM,— ?, c- = -5-2 —,
o, — o, o, — a,
1 /2 .1 \-^ 1 .
When h, = h, = 0 the expression for p becomes
p = /2 ly- r_aXLl _^A.l^ a.'-a» |
^ \9/t* 3m/ '^ ' ''[a/-a/7^ a,-o^r {0,-0,) (o^-o^)] ^ '
From this value of p the other quantities may be found, as in Case IX., from the equations of Case IV.
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.), all 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 sHding of the planks ; and since each of these deflections is small compared with the length of the beam, the total deflection will be the sum of the deflections due to bending and sliding.
Let
A=Mc = E\xi/'dy (65).
A is the stiffiiess of the beam as found in Case Y., 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 h^, and let this quantity be small compared with the length of the beam.
At the middle of the beam, 8,y is found by the usual methods to be
% = ^ {-h^w + ^^l'W) (66).
Let
B = — \xdy = — (sectional area) (jo7).
B is the resistance of the beam to the sliding of the planks. The de- flection of the beam arising from this cause is
% = 2]b(^'+^^ (68).
VOL. I. 9
66 THE EQXnUBRnJM OF ELASTIC SOLIDS.
This quantity is small compared with S^y, when the depth of the beam is small compared with its length.
The whole deflection ^y = B^ + S^
A3/ = - (^.Z-^iS + ^ {U +^l) (^^)-
Case XIII.
When the values of the compressions at any point have been found, when two difierent 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 will 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 A^ and A^ (Fig. 1) be the centres of the twisted cylinders. Join ^1^25 and bisect A^A, in 0. 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 ^1 and A^; it vanishes at B^^ and jBj ; and if the retardation at 0 be taken for unity, the isochromatic curves 2, 4, surround Aj^ 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 EQUILIBEIUM OF ELASTIC SOLIDS.
67
.4,(7, = -4,^,, and two loops surrounding B^ and B^', the other curves, for which /=l4-» -gS-j ^c-» consist each of two ovals surrounding B^ and jB,, and an exterior portion surrounding all the former curves.
Fig. 1.
I have produced these curves in the jelly of isinglass described in Case I. They are best seen by using circularly polarised 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 / for the different curves, not in an arithmetical, but in a geometrical progression, ascending by powers of 2.
68
THE EQUILrBRTOM 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 determuiing the pressures in a general case, applying it to the case of a triangle of unannealed plate-glass.
D D
Fig. 3.
The lines of equal intensity of the action on Hght 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 (^. 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 dilatation (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
?-?>=^^ (21).
70 THE EQUILIBRIUM OF EliASTIC SOLIDS.
Let R denote the radius of curvature of the curve of compression at any point, and let S denote the length of the curve of dilatation at the same point,
and since {q -p), R and S are known, and since at the surface, where (^^ {x, y) = 0, j9 = 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 ^i and <j>^ are known, to resolve the pressures in the direction of the axes.
The new quantities are p^, p„ and ^3 ; and the equations are
tan^=-^, {p-qY = q.' + (p.-p.y, Pi+P.=P + q- Pi Pi
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 Herschell'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. Lamd 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 EQUILIBRIUM OF 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.
Note 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. Regnault'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, Avhile 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 tiiose of Stokes, using the symbols m and fi, and the fundamental equations (4) and (5), which define them.
72
THE EQUILIBRIUM OF ELASTIC SOLIDS.
Note C.
As the coefficient <w, 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 insufficiency 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 Lam^ 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:
(P. + P..i'J = 3.(^%|4).
(^.-^.)=-(l
(P,
p,)=r,j'y.
(P,-P^ = m
fZz Bx
fi being the modulus of auhical, 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 EQUILIBRIUM OF 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 ^~dm-6E'
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 a (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.
Note. 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 the following geometrical theorems.
(1) 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 hne, 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 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
p'P = 2AB.hp.
Hence, if a line be drawn touching one circle in T, and cutting the other in P and p, then
(PTY : {pT)' :: 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 wiU 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 will 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 line PTp, the square of the velocity of P will be to the square of the velocity of p as PH to ph, that is, as (PTf to (pTf. Hence, the velocities of P and p are in the proportion of PT to pT, and therefore, by the corollary to (2), the line joining them will continue a tangent to the circle B during each instant, and will therefore remain a tangent during the motion.
76 SOLUTIONS OF PROBLEMS.
The cb'cle 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 jB 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 will 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 directions, B will 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 ?i 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 circles 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 circiunference 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 such 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 witJdn 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 OF PROBLEMS. 77
dicular from the fixed point on the tangent to the path of the ray at any point will 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 /x, from another where it is /Aj, 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 ir the ratio of /i, to fi^. 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, 0 the fixed point; then OY.FE=OP' + 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=OP' + AO.OB. If we put in this expression AO . OB = a^,
PO = r, OY=p, PR = 2p, it becomes 2pp = ?'*+■ a*,
78 SOLUTIONS OF PROBLEMS.
To find the law of the index of refraction of the medium, so that a ray from A may describe the circle APB, /x must be made to vary inversely as p by Lemma I.
Let AO = r^, and let the refractive index at A=fii; then generally
h'
_c p
_ 2C7p . a' + r''
/^1 =
. 2Cp
a' + r:'
a' + r,'
but at A
therefore
The value of /n at any point is therefore independent of p, the radius of
the given circle; so that the same law of refractive index will cause any other
ray to describe another circle, for which the value of a' is the same. The
a^ . . .
value of OB is — , which is also independent of p ; so that every ray which
proceeds from A must pass through B.
Again, if we assume /x^ as the value of /x when r = 0,
ar + r,'
therefore h' — H-o
d' + r'^-'
a result independent of r^. This shews that any point A' may be taken as the origin of the ray instead of A, and that the path of the ray will still be circular, and will pass through another point B' on the other side of 0, such that
Next, let CP be a ray from C, a point without the medium, falling at P on a spherical surface whose centre is C.
Let 0 be the fixed point in the medium as before. Join PO, and produce
to Q till OQ = jyp. Through Q draw a circle touching CP in P, and cutting
CO in A and B ; then PBQ is the path of the ray within the medium.
SOLUTIONS OF PROBLEMS. 79
Since CP touches the circle, we have CP'^CA. CB,
= {CO-OA){CO-\-OB);
but 0A= -^;
therefore CF' = CQ + CO (oB - ^^
an equation whence OB may be found, B being the point in the medium through which all rays from C pass.
Note. The possibility of the existence of a medium of this kind possessing remarkable optical properties, was suggested by the contemplation of the structure of the crystalline lens in fish; and the method of searching for these properties was deduced by analogy from Newton's Principia, Lib. L Prop. vii.
It would require a more accurate investigation into the law of the refractive index of the different coats of the lens to test its agreement with the supposed medium, which is an optical instrument theoretically perfect for homogeneous light, and might be made achromatic by proper adaptation of the dispersive power of each coat.
On the other hand, we find that the law of the index of refraction which would give a minimum of aberration for a sphere of this kind placed in water, gives results not discordant with facts, so far as they can be readily ascertained.
[From the Transactions of the Cambridge Philosophical Society, Vol. ix. Part iv.]
IV. On the Transformation of Surfaces by Bending.
Euclid has given two definitions of a surface, which may be taken as examples of the two methods of investigating their properties.
That in the first book of the Elements is —
"A superficies is that which has only length and breadth."
The superficies difiers from a line in having breadth as well as length, and the conception of a third dimension is excluded without being expHcitly introduced.
In the eleventh book, where the definition of a soHd is first formally given, the definition of the superficies is made to depend on that of the solid — " That which bounds a soHd is a superficies."
Here the conception of three dimensions in space is employed in forming a definition more perfect than that belonging to plane Geometry.
In our analytical treatises on geometry a surface is defined by a function of three independent variables equated to zero. The surface is therefore the boundary between the portion of space in which the value of the function is positive, and that in which it is negative; so that we may now define a surface to be the boundary of any assigned portion of space.
Surfaces are thus considered rather with reference to the figures which they limit than as having any properties belonging to themselves.
But the conception of a surface which we most readily form is that of a portion of matter, extended in length and breadth, but of which the thick-
TKANSFORMATION OF SURFACES BY BENDING. 81
ness may be neglected. By excluding the thickness altogether, we arrive at Euclid's first definition, which we may state thus —
" A surface is a lamina of which the thickness is diminished so as to become evanescent."
We are thus enabled to consider a surface by itself, without reference to the portion of space of which it is a boundary. By drawing figures on the surface, and investigating their properties, we might construct a system of theorems, which would be true independently of the position of the surface in space, and which might remain the same even when the form of the solid of which it is the boundary is changed.
When the properties of a surface with respect to space are changed, while the relations of lines and figures in the surface itself are unaltered, the surface may be said to preserve its identity, so that we may consider it, after the change has taken place, as the same surface.
When a thin material lamina is made to assume a new form it is said to be hent. In certain cases this process of bending is called development, and when one surface is bent so as to coincide with another it is said to be applied to it.
By considering the lamina as deprived of rigidity, elasticity, and other mechanical properties, and neglecting the thickness, we arrive at a mathemati- cal definition of this kind of transformation.
" The operation of bending is a continuous change of the form of a surface, without extension or contraction of any part of it."
The following investigations were undertaken with the hope of obtaining more definite conceptions of the nature of such transformations by the aid of those geometrical methods which appear most suitable to each particular case. The order of arrangement is that in which the different parts of the subject presented themselves at first for examination, and the methods employed form parts of the original plan, but much assistance in other matters has been derived from the works of Gauss*, Liouvillef, Bertrand^, Puiseux§, &c., references to which will be given in the course of the investigation.
* Disquisitiones generalea circa superficies curvas. Presented to the Royal Society of Gottingen, 8th October, 1827. Commentationes Recentiores, Tom. vi.
t Liouville's Journal, xii. X ^^'^- ^^'^' § ^^"^■
VOL, I. 11
82
TRANSFORMATION OF SURFACES BY BENDING.
On the Bending of Surfaces generated hy the motion of a straight line in space.
If a straight line can be drawn in any surface, we may suppose that part of the surface which is on one side of the straight line to be fixed, while the other part is turned about the straight line as an axis.
In this way the surface may be bent about any number of generating lines as axes successively, till the form of every part of the surface is altered.
The mathematical conditions of this kind of bending may be obtained in the following manner.
Let the equations of the generating line be expressed so that the constants involved in them are functions of one independent variable u, by the variation of which we pass from one position of the line to another.
If in the equations of the generating line Aa, u = u^, then in the equations of the line Bh we may put u = U2, and from the equations of these lines we may find by the common methods the equations of the shortest line PQ between Aa and Bb, and its length, which we may call S^. We may also find the angle between the directions of ^a and Bb, and let this angle be SO.
In the same way from the equations of Cc, in which u = u^, we may deduce the equa- tions of RS, the shortest line between Bb and Cc, its length 8^5 and the angle hd^ between the directions of Bb and Cc. We may also find the value of QR, the distance between the points at which PQ and RS cut Bb. Let QR = h(T, and let the angle between the directions of PQ and RS be S^.
Now suppose the part of tlie surface between the lines Aa and Bb to be fixed, while the part between Bb and Cc is turned round Bb as an axis. The line RS wiU then revolve round the point R, remaining perpendicular to Bhy and Cc will still be at the same distance from Bb, and wiU make the same angle with it. Hence of the four quantities S4j S^2> ^cr and 8</>, 8^ alone will be changed by the process of bending. 8<^, however, may be varied in a perfectly arbitrary manner, and may even be made to vanish.
,•?_..
TRANSFORMATION OF SURFACES BY BENDING. 83
For, PQ and RS being both perpendicular to Bh, RS may be turned about Bh till it is parallel to PQ, in which case 8^ becomes = 0.
By repeating this process, we may make all the " shortest lines" parallel to one another, and then all the generating lines will be parallel to the same plane.
We have hitherto considered generating lines situated at finite distances from one another ; but what we have proved will be equally true when their distances are indefinitely diminished. Then in the limit
du
B0
dO
u,-u,
" du
Str
da-
" du
8(f>
d(f>
Uj — Wi '* du '
All these quantities being functions of u, ^, 0, a- and (f), 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 ^, 0, a- and <j).
When the surface is bent in any manner about the generating lines, C> ^, and a- remain unaltered, but cf) 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 <f) = 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 0 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 o-.
84 TRANSFORMATION OF SURFACES BY BENDING.
Since ia this case the quantities C> ^, and cr are represented bj distinct geometrical quantities, we may simplify 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,
-T- = 0, and ^ constant. If these surfaces be bent so that <j> = 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 0, a- 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 a, its absolute
curvature at any point by -j- , and its torsion at the same point by ■— .
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 that 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 lines 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.
TRANSFORMATION 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 FV from the vertex, measured along a generating line, and the angle AVO which the plane of the generating line makes with a fixed plane through the axis. Let FV=s and AVO = 6. 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. Its length PR = Ss, and its breadth PQ = rhd, 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 hs and rB$', and if
r = /xr, and 0' = -0,
rze'=rze,
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 /x 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 6" and r, then putting / = /ir, the equation between s and r will give the form of the generating line after bending.
86 TRANSFORMATION OF SURFACES BY BENDING.
When /x is greater than 1 it may happen that for some values of 5, y- is
greater than -. In this case
-j- = fi-j- is greater than 1 ;
a result which indicates that the curve becomes impossible for such values of s and ft.
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 difficulty, 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 imderstood.
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 aU 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 soHd angle of the polyhedron. It will be shewn that
TRANSFORMATION 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 point, 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 kind may be inscribed, one in each surface.
Since the kind 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 will 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 tlie 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 tliis condition mav ccntinue
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 lines 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 Hne 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 diminished 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 different 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 off equal lengths from the curv^es 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 line 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. 22
90 TRANSFORMATION OF SURFACES BY BENDING.
tying 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 that 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 result 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 Inextensible Surfaces. It is to be found in the Transactions of the Royal Irish Academy, Vol. XXII. Science, &c., and was read May 23, 18.53.
Professor Jellett has obtained a system of three partial differential equations which express the conditions to which the displacements of a continuous inextensible 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 flexibihty of the surface is limited by certain conditions, and he has obtained the following results : —
If the displacements of an inextensible surface he all parallel to the same plane, the mrface moves as a rigid body.
Or, more generally,
If the movement of an inextensible 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 :—
// any curve be traced upon an inextensible surface whose principal radii of curvature are finite and of the same sign, and if this curve he rendered immoveable, the entire surface will become immoveable also.
In a developable surface composed of an inextensible membrane, any one of its rectilinear sections may be fixed without destroying the fiexibility 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 fiexure may he fi^ed without destroying the fiexibility of the surface.
If an arc of a curve traced upon an inextensible surface be rendered fixed or rigid, the entire of the quadrilateral, formed by drauring the two curves of fiexure through each extremity of the curve, become fixed or rigid also.
Professor Jellett has also investigated the properties of partially inextensible 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 inextensible, 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, 1851
92 TRANSFORMATION 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, aU whose sides shall he triangles, and all whose solid angles shall he 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 all 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 surface.
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 \7ill 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.
1)4
TRANSFORMATION OF SURFACES BY BENDING.
To Jind tJie 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 0 its centre of curvature, then
= 1.^ in the limit, when CR and PR coincide, ^ CP
-s CP' or calling CP the "sa,gitta," 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 insciihe a polyhedron in a given surface, all ivhose sides shcdl he plane quadrilaterals, and all whose solid angles shall he tetraliedral.
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 dc 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. ©5
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 will be referred to as "conju- gate systems."
5. On the elementary conditions of the applicahilitij of two surfaces.
It is evident, that if one surface is capable of being appUed 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 eflfected 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.
Therefore 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 corresponcling 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 when tJie 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
TRANSFORMATION OF SURFACES BY BENDING.
second at the same point. Then let the first surface be turned about the normal as an axis till 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 will have a common tangent, and will 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, FQES is a parallelo- 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 P 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 difierential 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, fulfil 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 Hnes of bending is completely determined. Supposing the problem reduced to its analyticid expression, the equations of these curves would appear under the form of double solutions of differential equations of the first order and second degree, each of which 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 lo 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 2 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 TRANSFORMATION OF SURFACES 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 It.
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 cwvature 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 cw^ature 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 foi-med 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 have 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 will 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 will be ultimately the same as the angle of intersection of the first and second systems, which we may call <f> ; 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 will difier 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 inclination of two adjacent facets will depend on the normal curvature of the lines of bending, and will 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 be 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 u^ and u\.
100 TRANSFORMATION OF 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 (w, 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 (wj to P (u), be expressed by s, and PM the length of the curve {u) from {u\) to (u), by s\ then s and s will be functions of u and u.
Let (w + Sm) 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 (w + Sw), will be
ds ^ du
Similarly PR may be expressed by
ds\ ,
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 angle of intersection of the system ; but when the values of 8w and hu are considered as finite though small, the angles a, 6, c, d of the facets which form a soHd angle will depend on the tangential curvature of the two systems of lines.
Let T be the tangential curvature of a curve of the first system at the given point measured in the direction in which u increases, and let r\ that of the second system, be measured in the direction in which xC increases.
Then we shall have for the values of the four plane angles which meet at P,
, \ ds ^ , 1 ds^
1 _, 1 c?/ ^ . 1 ds ^
~^ It du It du '
, \ ds rs , \ ds ^ J . I ds' , 1 ds ^
TRANSFORMATION OF SURFACES BY BENDING. 101
These values are correct as far as the first order of small quantities. Those corrections which depend on the curvature of the surface are of the second order.
Let p be the normal curvature of a curve of the first system, and p that of a curve of the second, then the inclination I of the plane facets a and 6, separated by a curve of the second system, will be
p sin ^ du as far as the first order of small angles, and the inclination V of h and c will be
7/ 1 0^ ^
/ = -7—. — 7 -J- ou p Bin.<f> du
to the same order of exactness.
12. On the corresponding polygon on the surface of the sphere of reference.
By the method described in Art. (2) we may find a point on the sphere corresponding to each facet of the polyhedron.
In the annexed figure, let a, b, c, d be the points on the sphere corresponding to the four facets which meet at the solid angle P. Then the area of the spherical quadrilateral a, h, c, d will be the measure of the entire curvature of the solid angle P.
This area is measured by the defect of the sum of the exterior angles from four right angles ; but these exterior angles are equal to the four angles a, h, c, d, which form the solid angle P, therefore the entire curvature is measured by
k = 2'rr-{a + h + c-{-d).
Since a, h, c, d are invariable, it is evident, as in Art. (2), that the entire curvature at P is not altered by bending.
By the last article it appears that when the facets are small the angles b and d are approximately equal to <j), and a and c to (tt — ^), and since the sides of the quadrilateral on the sphere are small, we may regard it as approximately a plane parallelogram whose angle bad = <f).
The sides of this parallelogram will be I and I', the supplements of the angles of the edges of the polyhedron, and we may therefore express its area as a plane parallelogram
k = IV sin <f>.
102
TRANSFORMATION OF SURFACES BY BENDING.
By the expression for I and V in the last article, we find
, 1 ds ds\ ^ ,
k = — r-. — 7 J- J-/ ou du pp sm<^ du du
for the entire curvature of one solid angle.
Since the whole number of solid angles is equal to the whole number of facets, we may suppose a quarter of each of the facets of which it is composed to be assigned to each solid angle. The area of these will be the same as that
of one whole facet, namely,
, ds ds' o ^ , sm 9 -J- T-> ou ou ;
therefore dividing the expression for k by this quantity, we find for the value
of the specific curvature at P
1 ■^ pp sm'<^ which gives the specific curvature in terms of the normal curvatures of the lines of bending and their angle of intersection.
13. Further reduction of this expression by rmans of the " Conic of Con- tact" as defined in Art. (3).
Let a and b be the semiaxes of the conic of contact, and h the sagitta or perpendicular to its plane from the centre to the surface.
Let CP, CQ be semidiameters parallel to the lines of bending of the first and second systems, and therefore conjugate to each other.
By (Art. 3),
, CP"
p=^-hr
and p=i-j^; and the expression for p in Art. (12), becomes
^~{CP.CQsm(t>)''
But CP .CQbukJ) is the area of the parallelogram CPRQ, which is one
quarter of the circumscribed parallelogram, and therefore by a well-known
theorem
CP .CQsm4> = ah,
TRANSFORMATION OF SURFACES BY BENDING. 103
and the expression for p becomes
or if the area of the circumscribing parallelogram be called A,
The principal radii of curvature of the surface are parallel to the axes of the conic of contact. Let H and i^ denote these radii, then
and therefore substituting in the expression for p,
1
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 80 elementary, is more direct than that here given, and wUl 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. 0)1 the variation of normal curvature of the lines of bending as we pa^s 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 hne of the first system to another, along a line of the second.
In analytical language we have to find the value of
du \pj
Referring to the figure in Art. (11), we shall see that this may be done if we can determine the difierence between the angle of inclination of the facets a and h, and that of c and d : for the angle I between a and b is
J 1 ds 5. , psiJKp du
104 TRANSFORMATION OF SURFACES BY BENDING.
and therefore the difference between the angle of a and b and that of c and d is
~ du ~ du \psm<f> du'j whence the differential of p with respect to u may be found
We must therefore find U, and this is done by means of the quadrilateral on the sphere described in Art. (12).
15. To find the values of hi and U\
In the annexed figure let ahcd repre- sent the small quadrilateral on the surface of the sphere. The exterior angles a, h, c, d are equal to those of the four facets which meet at the point P of the surface, and the sides represent the angles which the planes of those facets make with each other ; so that
ah = l, lc = l\ cd = l + U, da = l' + Br,
and the problem is to determine Bl and hi" in terms of the sides I and V and the angles a, h, c, d.
On the sides ha, he complete the parallelogram ahcd.
Produce ad to p, so that ap = aS. Join Bp. Make eq = cd and join dq. then Bl = cd- ah, = cq — ch, = -(qo + oB),
Now qo = qd tan qdo
= cd sin qcd cot qod, but cd = I nearly, sin qcd = qcd==(e + h-7r) and qod = <f>; .'. qo^l (c + h- it) cot <f>.
TRANSFORMATION OF SURFACES BY BENDING. 105
Also oS = -—-^ — Sin bop
= aB (Bap) — — 7 ^ ^' 8m<f>
= l'(a+h-7T)J-r.
Substituting the values of a, h, c, d from Art. (11), Sl= — (qo + 08)
= —I —, ^- cot <i>Su — V — T—, - — r Bu. r du ^ r du sm0
Finally, substituting the values of I, V, and Bl from Art. (14),
d ( \ ds"\ sj 5 , cot (/) cZs' 1 (i5 5. ^ , 1 ds I ds' ^ ,
du \p sin <p du / p sm <f> du r du p sm <j> du r du
which may be put under the more convenient form
— n ^ = — 1 / 1 ^^'\ 1 ds , p I ds 1 du^ °'^'~du ^ \sin <j> du) r du ^ p' r du sin <^ '
and from the value of Bl' we may similarly obtain
d ,, '\ _ _^ 1 / 1 ^\ ,i^ +^j_^i^ ^ du ^ ^ ^ ' du' ° \sin <f> du) r du' ^ p r du sin (ft '
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 = — / . , ., and q = —,, ^ pp sm=<^' ^ p
whence p' = q — ^^-r , p'^ = t-tj y
^ ^ p sin <f) 9. P s^ Y
and the equations become
d ,. \ d , ( ^Tl'X 1 ds , , 2 ds 1
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. 14
d_
du'
106 TRANSFORMATION OF SURFACES BY BENDING.
16. To obtain the conditio of Instantaneous lines of bending.
We have now obtained tlie 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 difierent quantities which enter into our equations, none are changed by the operation of bending except q, so that in differentiating with respect to t all the rest may be considered constant, q being the only variable.
Differentiating the equations of last article with respect to t, we obtain d" ,, . 2 ds 1 d ,, .
Whence
c?" ,, . 2 ds' 1 I d ,. .
A^t'^^'^^^ =
{.4 1- 1 si^)-'^ Tu ^, ii'^^H^'o^'^' 1 1 ii^^ 3^.<(">^*)-
and
(log l)
dududt
( d /2ds 1 \ 2 ds 1 d , } ^ d ,, 2 ds 1 1 d ,, .
{M?d^^^'r-di7^^d^^'^'irqdt^^'^^^
two independent values of the same quantity, whence the requiied 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 , ,
ds du
sin
*)
+ - , \, cot <!> y
2 ds r du
\l ds ( d , f ,ds . A 2 ds . ,\
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.
17. 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. In that case we have the two conditions
(!)■
d , / ds . ,\ 2ds ^ , ^^ ^,log(i^r^^sm<^j + -^,cotc^ = 0,
y
1 ds ^ or - -J- = 0. r du
|^log(i>r'^,siD<^)+|^cot<^ = 0,'
1 d/ ^ or -, -J-, = 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 compHcated, 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 conceptions rather than to exhibit the results of mathematical labour.
18. On the application of the ordinary unethods 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 familiar 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 foimd in the memoir of Gauss.
Let the equation to the surface be put under the form
then the value of the specific curvature is
d\ dh d^
dot? dif dx
~dJz'^ dz^ dx 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 BENDINO. 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 (x, y) let a be the angle which a projected curve of the first system makes with the axis of x, and /8 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 y3 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
72
^ = Pg sin' a + PQ- sin' A
d'z
, J = — PQ sin a cos a — PQ~^ sin y3 cos ^,
^ = PQ cos' a + P^-^ cos' /8.
It will be seen that these values satisfy the condition last given. Near the origin we have
d*z dh d\ I* n- . , / n\
and q=Q'*.
110 TRANSFORMATION OF SURFACES BY BENDING.
Differentiating these values of -y-^ , &c., we shall obtain two values of , , and of 1—7—3, which being equated will give two equations of condition.
Now if s' be measured along a curve of the first system, and R be any
function of x and y, then
dE dR dR .
-^j-y = -^j- cos a + -7- sm a, as dx ay
, dR _ dR ds' du' ds du '
We may also show that -=-^ = - ,
, ,, , da . da d . (ds' . ,\
and that cos a ;i — sm a ;t- = t- log ( -j—, sm 0 1 . cty (j/X cLs \ci/U I
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
(^.) *{rf-j-©'
This surface may be generated by the motion of a straight line whose equation is of the form
= acosnl — j, 2/ = asinni-f-
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, ar' + y = a', and the straight lines z = c, cc=^0, 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 + Bt, 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^,
ac
8C= ,/^ Bin 2tBt,
Ja' + c and the co-ordinates of the point of intersection are
x = 2a cos' t, y = 2a sin* t, z= —c cos 2t. The angle 80 between the consecutive lines is
Ja- + c The distance So- between consecutive shortest lines is
^ 3a'-F-2c*
and the angle S<^ between these latter lines is
sin 2t8t,
'Ja' + c
Hence if we suppose ^, 6, cr, (f), and t to vanish together, we shall have by integration
(T = ~—, ( 1 — cos 2t),
Ja' + c'
By bending the surface about its generating lines we alter the value of (ft in any manner without changing 4, 0, or or. For instance, making <^ = 0, all the generating lines become parallel to the same plane. Let this plane be that of xy, then ^ is the distance of a generating line from that plane. The projections
o- =
112 TRANSFORMATION OF 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 a, and the angle which its tangent makes with the axis of x hj 0, 6 and o- being connected by the equation
^ I 1 - cos 6 ,
which shows the curve to be an epicycloid.
The generating lines of the surface when bent into this form are therefore tangents to a cylindrical 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
Putting r = Jaf + f, then the equation of the generating line is
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 z, then,
s = |<jV\ wherefore, r = (|)' c " * 5^.
Let 6 be the angle which the plane of any generating line makes with that of xz, then s and 6 determine the position of any point on the surface. The length and breadth of an element of the surface will be Ss and rB$.
Now let the surface be bent in the manner formerly described, so that 0 becomes 0^, and r, r, when
0^ = 1x0 and r' = -ry then r' = (f)'c-V"'s'
provided o' = /u,'c. The equation between r' and s being of the same form as that between r and ^ shows that the surface when bent is similar to the original surface, its dimensions being multiphed by fi*.
TRANSFORMATION OF SURFACES BY BENDING. 113
This, however, is true only for one half of the surface when bent. The other half is precisely symmetrical, but 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 efifect 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 (3) be infinite at eveiy 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 be fixed, while the remainder is bent, the boundary of the fixed portion is a curve of fiexure.
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 rebroussement, 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 <^=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 fiexure be fi^ed, 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 rectilinear 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 have 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 aji 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 efiects 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 OF A HEAVY BODY IN A RESISTING 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 sHp 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 wOl 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 will act on the whole in the positive direction, and will therefore cause the general path of the body to incline 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 wUl 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 amphtude 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 incHned 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 Transactions of the Royal Scottish Society of Arts, Vol. iv. Part in]
VI. On the Theory of Colours in relation to Colour-Blindness. A letter to Dr G. Wilson.
Dear Sir, — As you seemed to think that the results which I have obtained in the theory of colours might 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 refranglbility, 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 refranglbility 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 THEORY OF COLOURS IN RELATION TO COLOUR-BLINDNESS.
do SO by artificial means ; and our reasonings on tKem 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 difierent 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 numher 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 readily 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 difference 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 wiU 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 different 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 will depend : — (1) 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 differ not only in intensity and Ime, 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 theoiy 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 light 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 tliree primitive sensations.
In this way, every possible colour may have its position and intensity
VOL. I. 16
122 THE THEORY OF 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 Opticks (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 foimd at some point, w, within the triangle. If lines be drawn through w 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 within 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 C^^ J
part of each is exposed. By slipping one disc over another,
we can expose any given portion of each colour. These >^ — ~^ j:«^« „i J „ ^:^.^.^^ j. j.^^i.^4. ,'4.; ^v ( <=> )
discs are placed on a little top or teetotum, consisting of \^ y
a flat disc of tin-plate and a vertical axis of ivory. This
axis passes through the centre of the discs, and the quantity of each colour exposed
is measured by a graduation on the rim of the disc, which is divided into 100 parts.
* [See the author's Memoir in the Philosophical Transactions, 1860, on the Theory o£ Compound Colours, and on the relations of the Colours of the Spectrum.]
THE THEORY OF COLOURS IN RELATION TO COLOUR-BLINDNESS. 123
By spinning the top, each colour is presented to the eye for a time pro- portional to the angle of the sector exposed, and I have found by independent experiments, that the colour produced by fast spinning is identical with that produced by causing the light of the different colours to fall on the retina at once.
By properly arranging the discs, any given colour may be imitated and afterwards registered by the graduation on the rim of the top. The principal use of the top is to obtain colour-equations. These are got by producing, by two different combinations of colours, the same mixed tint. For this purpose there is another set of discs, half the diameter of the others, which lie above them, and by which the second combination of colours is formed.
The two combinations being close together, may be accurately compared, and when they are made sensibly identical, the proportions of the different colours in each is registered, and the results equated.
These equations in the case of ordinary vision, are always between four colours, not including black.
From them, by a very simple rule, the different colours and compounds have their places assigned on the triangle of colours. The rule for finding the position is this : — Assume any three points as the positions of your three standard colours, whatever they are ; then form an equation between the three standard colours, the given colour and black, by arranging these colours on the inner and outer circles so as to produce an identity when spun. Bring the given colour to the left-hand side of the equation, and the three standard colours to the right hand, leaving out black, then the position of the given colour is the centre of gravity of three masses, whose weights are as the number of degrees of each of the standard colours, taken positive or negative, as the case may be.
In this way the triangle of colours may be constructed by scale and compass from experiments on ordinary vision. I now proceed to state the results of experiments on Colour-Blind vision.
If we find two combinations of colours which appear identical to a Colour- Blind person, and mark their positions on the triangle of colours, then the straight line passing through these points will pass through all points corre- sponding to other colours, which, to such a person, appear identical-with the first two.
We may in the same way find other lines passing through the series of
IG— 2
124 THE THEORY OF COLOURS IN RELATION TO COLOUR-BLINDNESS.
colours wMch appear alike to the Colour-Blind. All these lines either pass through one point or are parallel, ac- cording to the standard colours which we have assumed, and the other arbitrary assumptions we may have made. Knowing this law of Colour-Blind vision, we may predict any number of equations which will be true for eyes having this defect.
The mathematical expression of the difference between Colour-BUnd and ordinary vision is, that colour to the former is a function of two independent variables, but to an ordinary eye, of three ; and that the relation of the two kinds of vision is not arbitrary, but indicates the absence of a determinate sensation, depending perhaps upon some undiscovered structure or organic arrangement, which forms one-third of the apparatus by which we receive sensations of colour.
Suppose the absent structure to be that which is brought most into play when red light falls on our eyes, then to the Colour-Blind red light will be visible only so far as it affects the other two sensations, say of blue and green. It will, therefore, appear to them much less bright than to us, and will excite a sensation not distinguishable from that of a bluish-green light.
I cannot at present recover the results of all my ^periments ; but I recollect that the neutral colours for a Colour-Blind person may be produced by com- bining 6 degrees of ultramarine with 94 of vermiUon, or 60 of emerald-green with 40 of ultramarine. The first of these, I suppose to represent to our eyes the kind of red which belongs to the red sensation. It excites the other two sensations, and is, therefore, visible to the Colour-BHnd, but it appears very dark to them and of no definite colour. I therefore suspect that one of the three sensations in perfect vision will be found to correspond to a red of the same hue, but of much greater purity of tint. Of the nature of the other two, I can say nothing definite, except that one must correspond to a blue, and the other to a green, verging to yellow.
I hope that what I have written may help you in any way in your experiments. I have' put down many things simply to indicate a way of thinking about colours which belongs to this theory of triple sensation. We are indebted to Newton for the original design ; to Young for the suggestion of the means of working it out; to Prof. Forbes'' for a scientific history of its application
*Phil. Mag. 1848.
THE THEORY OF COLOURS IN RELATION TO COLOUR-BLINDNESS. 125
to practice; to Helmholtz for a rigorous examination of the facts on which it rests; and to Prof Graasman (in the Phil. Mag, for 1852), for an admirable theoretical exposition of the subject. The colours given in Hay's Nomenclature of Colours are illustrations of a similar theory applied to mixtures of pigments, but the results are often different from those in which the colours are combined by the eye alone. I hope soon to have results with pigments compared with those given by the prismatic spectrum, and then, perhaps, some more definite results may be obtained. Yours truly,
J. C. MAXWELL.
Edinburgh, 4tli Jan. 1855.
[From the Transactions of the Royal Society of Edinburgh, Vol xxi. Part ii.]
VII. Experiments on Colour, as perceived hy the Eye, with remarks on Colour- Blindness. Communicated by Dr Gregory.
The object of tbe following communication is to describe a method by which every variety of visible colour may be exhibited to the eye m such a form as to admit of accurate comparison ; to shew how experiments so made may be registered numerically; and to deduce from these numerical results certain laws of vision.
The different tints are produced by means of a combination of discs of paper, painted with the pigments commonly used in the arts, and arranged round an axis, so that a sector of any required angular magnitude of each colour may be exposed. "When this system of discs is set in rapid rotation, the sectors of the different colours become indistinguishable, and the whole appears of one uni- form tint. The resultant tints of two different combinations of colours may be compared by using a second set of discs of a smaller si^e, and placing these over the centre of the first set, so as to leave the outer portion of the larger discs exposed. The resultant tint of the first combination will then appear in a ring round that of the second, and may be very carefully compared with it.
The form in which the experiment is most manageable is that of the com- mon top. An axis, of which the lower extremity is conical, carries a circular plate, which serves as a support for the discs of coloured paper. The circumfer- ence of this plate is divided into 100 equal parts, for the purpose of ascertainmg the proportions of the different colours which form the combination. When the discs have been properly arranged, the upper part of the axis is screwed down, so as to prevent any alteration in the proportions of the colours.
The instrument used in the first series of experiments (at Cambridge, in November, 1854) was constructed by myself, with coloured papers procured from
EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE.
127
Mr D. R Hay. The experiments made in the present year were with the improved top made by Mr J. M. Bryson, Edinburgh, and coloured papers pre- pared by Mr T. Purdie, with the unmixed pigments used in the arts. A number of Mr Bryson's tops, with Mr Purdie's coloured papers has been prepared, so as to afford different observers the means of testing and comparing results inde- pendently obtained.
The colour used for Mr Purdie's papers were —
Vermilion
V
Ultramarine
U
Emerald Green
EG
Carmine .
C
Prussian Blue .
PB
Brunswick Green
BG
Red Lead
RL
Verditer Blue .
VB
Mixture of Ultramarine
Orange Orpiment
00
and Chrome
uc
Orange Chrome
OC
Chrome Yellow
CY
Gamboge
Gam
Pale Chrome .
PC
Ivory Black . Snow White .
Bk SW
White Paper (Pirie, Aberdeen),
The colours in the first column are reds, oranges, and yellows; those in the second, blues ; and those in the third, greens. Vermilion, ultramarine, and emerald green, seem the best colours to adopt in referring the rest to a uniform standard. They are therefore put at the head of the Hst, as types of three convenient divisions of colour, red, blue, and green.
It may be asked, why some variety of yellow was not chosen in place of green, which is commonly placed among the secondary colours, while yellow ranks as a primary? The reason for this deviation from the received system is, that the colours on the discs do not represent primary colours at all, but are simply specimens of different kinds of paint, and the choice of these was deter- mined solely by the power of forming the requisite variety of combinations. Now, if red, blue, and yellow, had been adopted, there would have been a difficulty in forming green by any compound of blue and yellow, while the yellow formed by vermilion and emerald green is tolerably distinct. This will be more clearly perceived after the experiments have been discussed, by referring to the diagram.
As an example of the method of experimenting, let us endeavour to form a neutral gray by the combination of vermilion, ultramarine, and emerald green. The most perfect results are obtained by two persons acting in concert., when
128 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE.
the operator arranges the colours and spins the top, leaving the eye of the observer free from the distracting effect of the bright colours of the papers when at rest.
After placing discs of these three colours on the circular plate of the top, and smaller discs of white and black above them, the operator must spin the top, and demand the opinion of the observer respecting the relation of the outer ring to the inner circle. He will be told that the outer circle is too red, too blue, or too green, as the case may be, and that the inner one is too light or too dark, as compared with the outer. The arrangement must then be changed, so as to render the resultant tint of the outer and inner circles more nearly alike. Sometimes the observer will see the inner circle tinted with the complementary colour of the outer one. In this case the operator must interpret the observation with respect to the outer circle, as the inner circle contains only black and white.
By a little experience the operator will learn how to put his questions, and how to interpret their answers. The observer should not look at the coloured papers, nor be told the proportions of the colours during the experiments. When these adjustments have been properly made, the resultant tints of the outer and inner circles ought to be perfectly indistinguishable, when the top has a sufficient velocity of rotation. The number of divisions occupied by the different colours must then be read off on the edge of the plate, and registered in the form of an equation. Thus, in the preceding experiment we have ver- milion, ultramarine, and emerald green outside, and black and white inside. The numbers, as given by an experiment on the 6th March 1855, in dayhght without sun, are —
•37 V + -27 U + '36 EG = -28 SW+-72 Bk (1).
The method of treating these equations will be given when we come to the theoretical view of the subject.
In this way we have formed a neutral gray by the combination of the three standard colours. We may also form neutral grays of different intensities by the combination of vermilion and ultramarine with the other greens, and thus obtain the quantities of each necessary to neutralize a given quantity of the proposed green. By substituting for each standard colour in succession one of the colours which stand under it, we may obtain equations, each of which contains two standard colours, and one of the remaining colours.
EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 129
Thus, in the case of pale chrome, we have, from the same set of experiments, •34 PC + -55U + -12 EG = '37 SW + -63Bk (2).
"We may also make experiments in which the resultiag tint is not a neutral gray, but a decided colour. Thus we may combine ultramarine, pale chrome, and black, so as to produce a tint identical with that of a compound of vermilion and emerald-green. Experiments of this sort are more difficult, both from the inability of the observer to express the difference which he detects in two tints which have, perhaps, the same hue and intensity, but differ in purity ; and also from the complementary colours which are produced in the eye after gazing too long at the colours to be compared.
The best method of arriving at a result in the case before us, is to render the hue of the red and green combination something like that of the yellow, to reduce the purity of the yellow by the admixture of blue, and to diminish its intensity by the addition of black. These operations must be repeated and adjusted, till the two tints are not merely varieties of the same colour, but absolutely the same. An experiment made 5th March gives —
•39 PC-I--21 U + -40 Bk = ^59 V-f41 EG (3).
That these experiments are really evidence relating to the constitution of the eye, and not mere comparisons of two things which are in themselves identical, may be shewn by observing these resultant tints through coloured glasses, or by using gas-light instead of day-light. The tints which before appeared identical will now be manifestly different, and will require alteration, to reduce them to equality.
Thus, in the case of carmine, we have by day-light,
•44 C-h-22 JJ + 'U EG= •I? SW-f-^83 Bk, while by gas-light (Edinburgh)
•47 C-l-^08 U-1-^45 EG = ^25 SW-|-^75 Bk, which shews that the yellowing effect of the gas-light teUs more on the white than on the combination of colours. If we examine the two resulting tints which appeared identical in experiment (3), observing the whirling discs througli a blue glass, the combination of yellow, blue, and black, appears redder than- the other, while through a yellow glass, the red and green mixture appears redder. So also a red glass makes the first side of the equation too dark, and a green glass makes it too light.
VOL. I. 17
130 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE.
The apparent identity of the tints in these experiments is therefore not real, but a consequence of a determinate constitution of the eye, and hence arises the importance of the results, as indicating the laws of human vision.
The first result which is worthy of notice is, that the equations, as observed by different persons of ordinary vision, agree in a remarkable manner. If care be taken to secure the same kind of light in all the experiments, the equations, as determined by two independent observers, will seldom shew a difference of more than three divisions in any part of the equation containing the bright standard colours. As the duller colours are less active in changing the resultant tint, their true proportions cannot be so well ascertained. The accuracy of vision of each observer may be tested by repeating the same experiment at different times, and comparing the equations so found.
Experiments of this kind, made at Cambridge in November 1854, shew that of ten observers, the best were accurate to within 1^ division, and agreed within 1 division of the mean of all ; and the worst contradicted themselves to the extent of 6 degrees, but still were never more than 4 or 5 from the mean of all the observations.
We are thus led to conclude —
1st. That the human eye is capable of estimating the likeness of colours with a precision which in some cases is very great.
2nd. That the judgment thus formed is determined, not by the real identity of the colours, but by a cause residing in the eye of the observer.
3rd. That the eyes of different observers vary in accuracy, but agree with each other so nearly as to leave no doubt that the law of colour-vision is identical for all ordinary eyes.
Investigation of the Law of the Perception of Colour.
Before proceeding to the deduction of the elementary laws of the perception of colour from the numerical results previously obtained, it will be desirable to point out some general features of the experiments which indicate the form which these laws must assume.
Betuming to experiment (1), in which a neutral gray was produced from red, blue, and green, we may observe, that, while the adjustments were incom-
EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 131
plete, the difference of the tints could be detected only by one circle appearing more red, more green, or more blue than the other, or by being lighter or darker, that is, having an excess or defect of all the three colours together. Hence it appears that the nature of a colour may be considered as dependent on three things, as, for instance, redness, blueness, and greenness. This is con- firmed by the fact that any tint may be imitated by mixing red, blue, and green alone, provided that tint does not exceed a certain brilliancy.
Another way of shewing that colour depends on three things is by con- sidering how two tints, say two lilacs, may differ. In the first place, one may be lighter or darker than the other, that is, the tints may differ in shade. Secondly, one may be more blue or more red than the other, that is, they may differ in hue. Thirdly, one may be more or less decided in its colour ; it may vary fi*om purity on the one hand, to neutrality on the other. This is sometimes expressed by saying that they may differ in tint.
Thus, in shade, hue, and tint, wo have another mode of reducing the elements of colour to three. It will be shewn that these two methods of con- sidering colour may be deduced one from the other, and are capable of exact numerical comparison.
On a Geographical Method of Exhibiting the Relations of Colours.
The method which exhibits to the eye most clearly the results of this theory of the three elements of colour, is that which supposes each colour to be repre- sented by a point in space, whose distances from three co-ordinate planes are proportional to the three elements of colour. But as any method by which the operations are confined to a plane is preferable to one recLuiring space of three dimensions, we shall only consider for the present that which has been adopted for convenience, founded on Newton's Circle of colours and Mayer and Young's Triangle.
Vermilion, ultramarine, and emerald-green, being taken (for convenience) as standard colours, are conceived to be represented by three points, taken (for con- venience) at the angles of an equilateral triangle. Any colour compounded of these three is to be represented by a point found by conceiving masses propor- tional to the several components of the colour placed at their respective angular points, and taking the centre of gravity of the three masses. In this way, each
17—2
132 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE.
colour will indicate by its position the proportions of the elements of which it is composed. The total intensity of the colour is to be measured by the whole number of divisions of V, U, and EG, of which it is composed. This may be indicated by a number or coefficient appended to the name of the colour, by which the number of divisions it occupies must be multiplied to obtain its mass in calculating the results of new combinations.
This will be best explained by an example on the diagram (No. 1). We have, by experiment (l),
•37 Y+-27 U + -36 EG= -28 SW4- 72 Bk.
To find the position of the resultant neutral tint, we must conceive a mass of -37 at V, of -27 at U, and of '36 at EG, and find the centre of gravity. This may be done by taking the line UV, and dividing it in the proportion of •37 to ^27 at the point a, where
aV : aU :: ^27 : '37.
Then, joining a with EG, divide the joining line in W in the proportion of ^36 to ("37 + "27), W will be the position of the neutral tint required, which is not white, but 0*28 of white, diluted with 0^72 of black, which has hardly any effect whatever, except in decreasing the amount of the other colour. The total in- tensity of our white paper will be represented by oi = 3'57; so that, whenever white enters into an equation, the number of divisions must be multiplied by the coefficient 3-57 before any true results can be obtained.
We may take, as the next example, the method of representing the relation of pale chrome to the standard colours on our diagram, by making use of ex- periment (2), in which pale chrome, ultramarine, and emerald-green, produced a neutral gray. The resulting equation was
•33PC + -55U + -12EG = -37SW + -63Bk (2).
In order to obtain the total intensity of white, we must multiply the number of divisions, -37, by the proper coefficient, which is 3*57. The result is 1-32, which therefore measures the total intensity on both sides of the equation.
Subtracting the intensity of •55U + -12EG, or '67 from 1-32, we obtain '65 as the corrected value of -33 PC. It will be convenient to use these corrected values of the different colours, taking care to distinguish them by small initials instead of capitals.
EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 133
Equation (2) then becomes
•65 pc + -55 U + -12 EG = 1 -32 w.
Hence pc must be situated at a point such that w is the centre of gravity of •65pc + -55U + '12EG.
To find it, we begin by determining ^ the centre of gravity of -55 U + '12EG, then, joining /8w, the point we are seeking must lie at a certain distance on the other side of w from c This distance may be found from the proportion,
•65 : (-55 + -12) :: ^ : w pc, which determines the position of pc. The proper coefficient, by which the ob- served vakies of PC must be corrected, is ^, or 1-97.
We have thus determined the position and coefficient of a colour by a single experiment, in which it was made to produce a neutral tint along with two of the standard colours. As this may be done with every possible colour, the method is applicable wherever we can obtain a disc of the proposed colour. In this way the diagram (No. l) has been laid down from observations made in daylight, by a good eye of the ordinary type.
It has been observed that experiments, in which the resultant tint is neutral, are more accurate than those in which the resulting tint has a decided colour, as in experiment (3), owing to the effects of accidental colours produced in the eye in the latter case. These experiments, however, may be repeated till a very good mean result has been obtained.
But since the elements of every colour have been already fixed by our previous observations and calculations, the agreement of these results with those calculated from the diagram forms a test of the correctness of our method.
By experiment (No. 3), made at the same time with (l) and (2), we have •39PC + -2lU + -40Bk = -59V + -4lEG (3).
Now, joining XJ with pc, and V with EG, the only common point is that at which they cross, namely y.
Measuring the parts of the line V EG, we find them in the proportion of •58 V and "42 EG = 1*00 7.
Similarly, the line U pc is divided in the proportion 78 pc and •22U=r00y.
134
EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE.
But -78 pc must be divided by 1-97, to reduce it to PC, as was previously explained. The result of calculation is, therefore,
•39 PC + -22 U + -39 Bk = -58 V + "42 EG, the black being introduced simply to fill up the circle.
This result differs very little from that of experiment (3), and it must be recollected that these are single experiments, made independently of theory, and chosen at random.
Experiments made at Cambridge, with all the combinations of five colours, shew that theory agrees with calculation always within 0-012 of the whole, and sometimes within 0*002. By the repetition of these experiments at the numerous opportunities which present themselves, the accuracy of the results may be rendered still greater. As it is, I am not aware that the judgments of the human eye with respect to colour have been supposed capable of so severe a test.
Further consideration of the Diagram of Colours.
We have seen how the composition of any tint, in terms of our three standard colours, determines its position on the diagram and its proper coefficient. In the same way, the result of mixing any other colours, situated at other points of the diagram, is to be found by taking the centre of gravity of their reduced masses, as was done in the last calculation (experiment 3).
We have now to turn our attention to the general aspect of the diagram.
The standard colours, V, U, and EG, occupy the angles of an equilateral triangle, and the rest are arranged in the order in which they participate in red, blue, and green, the neutral tint being at the point w within the triangle. If we now draw lines through w to the different colours ranged round it, we shall find that, if we pass from one line to another in the order in which they lie from red to green, and through blue back again to red, the order will be —
Carmine . Vermilion . Red Lead . Oi-ange Orpiment Orange Chrome Chrome Yellow Gramboge .
Coefficient.
0-4
Pale Chrome
1-0
Mixed Green (U C)
1-3
Brunswick Green
10
Emerald Green .
1-6
Verditer Blue .
1-5
Prussian Blue .
1-8
Ultramarine
Coefficient. 2 0 0-4 0-2 10 0-8 01 10
EXPERIMENTS ON COLOUR, AS PERCEITED BY THE EYE. 135
It may be easily seen that this arrangement of the colours corresponds to that of the prismatic spectrum ; the only difference being that the spectrum is deficient in those fine purples which lie between ultramarine and vermilion, and which are easily produced by mixture. The experiments necessary for deter- mining the exact relation of this list to the lines in the spectrum are not yet completed.
If we examine the colours represented by different points in one of these lines through w, we shall find the purest and most decided colours at its outer extremity, and the faint tints approaching to neutrality nearer to w.
If we also study the coefficients attached to each colour, we shall find that the brighter and more luminous colours have higher numbers for their coefficients than those which are dark.
In this way, the qualities which we have already distinguished as hue, tint, and shade, are represented on the diagram by angular position with respect to ir, distance from w, and coefficient; and the relation between the two methods of reducing the elements of colour to three becomes a matter of geometry.
Theory of the Perception of Colour.
Opticians have long been divided on this point ; those who trusted to popular notions and their own impressions adopting some theory of three primary colours, while those who studied the phenomena of light itself proved that no such theory could explain the constitution of the spectrum. Newton, who was the first to demonstrate the actual existence of a series of kinds of light, countless in number, yet all perfectly distinct, was also the first to propound a method of calculating the effect of the mixture of various coloured light ; and this method was substantially the same as that which we have just verified. It is true, that the directions which he gives for the construction of his circle of colours are somewhat arbitrary, being probably only intended as an indication of the general nature of the method, but the method itself is mathematically reducible to the theory of three elements of the colour- sensation*.
♦ See Note III. For a confirmation of Newton's analysis of Light, see Helmholtz, Pogg. Ann, 1852; and Phil. Mag. 1852, Part ii.
136 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE.
Youno", who made the next great step in the establishment of the theory of light, seems also to have been the first to follow out the necessary conse- quences of Newton's suggestion on the mixture of colours. He saw that, since this tripUcity has no foundation in the theory of light, its cause must be looked for in the constitution of the eye; and, by one of those bold assumptions which sometimes express the result of speculation better than any cautious trains of reasoning, he attributed it to the existence of three distinct modes of sensation in the retina, each of which he supposed to be produced in different deo-rees by the different rays. These three elementary effects, according to his view, correspond to the three sensations of red, green, and violet, and would separately convey to the sensorium the sensation of a red, a green, and a violet picture ; so that by the superposition of these pictures, the actual variegated world is represented*.
In order fully to understand Young's theory, the function which he attributes to each system of nerves must be carefully borne in mind. Each nerve acts, not, as some have thought, by conveying to the mind the knowledge of the length of an undulation of light, or of its periodic time, but simply by being Quore or less affected by the rays which fall on it. The sensation of each elementary nerve is capable only of increase and diminution, and of no other change. We must also observe, that the nerves corresponding to the red sensation are affected chiefly by the red rays, but in some degree also by those of every other part of the spectrum ; just as red glass transmits red rays freely, but also suffers those of other colours to pass in smaller quantity.
This theory of colour may be illustrated by a supposed case taken from the art of photography. Let it be required to ascertain the colours of a land- scape, by means of impressions taken on a preparation equally sensitive to rays of every colour.
Let a plate of red glass be placed before the camera, and an impression taken. The positive of this will be transparent wherever the red light has been abundant in the landscape, and opaque where it has been wanting. Let it now be put in a magic lantern, along with the red glass, and a red picture will be thrown on the screen.
Let this operation be repeated with a green and a violet glass, and, by
* Young's Lectures, p. 345, Kelland's Edition. See also Helmholtz's statement of Young's Theory, in his Paper referred to in Note I. ; and Herschel's LigJU, Art. 518.
EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 137
means of three magic lanterns, let the three images be superimposed on the screen. The colour of any point on the screen will then depend on that of the corresponding point of the landscape; and, by properly adjusting the intensities of the lights, &c., a complete copy of the landscape, as far as visible colour is concerned, will be thrown on the screen. The only apparent difference will be, that the copy will be more subdued, or less pure in tint, than the original. Here, however, we have the process performed twice — first on the screen, and then on the retina.
This illustration will shew how the functions which Young attributes to the three systems of nerves may be imitated by optical apparatus. It is therefore unnecessary to search for any direct connection between the lengths of the undulations of the various rays of light and the sensations as felt by us, as the threefold partition of the properties of light may be effected by physical means. The remarkable correspondence between the results of experiments on different individuals would indicate some anatomical contrivance identical in all. As there is little hope of detecting it by dissection, we may be content at present with any subsidary evidence which we may possess. Such evidence is furnished by those individuals who have the defect of vision which was described by Dalton, and which is a variety of that which Dr G. Wilson has lately investigated, under the name of Colour-Blindness.
Testimony of the Colour- Blind with respect to Colour.
Dr George Wilson has described a great number of cases of colour- bhndness, some of which involve a general indistinctness in the appreciation of colour, while in others, the errors of judgment are plainly more numerous in those colours which approach to red and green, than among those which approach to blue and yellow. In these more definite cases of colour-blindness, the phenomena can be tolerably well accoimted for by the hypothesis of an insensibility to red light; and this is, to a certain extent, confirmed by the fact, that red objects appear to these eyes decidedly more obscure than to ordinary eyes. But by experiments made with the pure spectrum, it appears that though the red appears much more obscure than other colours, it is not wholly invisible, and, what is more curious, resembles the green more than any other colour. The spectrum to them appears faintly luminous in the red;
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138 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE.
bright yellow from orange to yellow, bright but not coloured from yellow- green to blue, and then strongly coloured in the extreme blue and violet, after which it seems to approach the neutral obscure tint of the red. It is not easy to see why an insensibihty to red rays should deprive the green rays, which have no optical connection with them, of their distinctive appearance. The phenomena seem rather to lead to the conclusion that it is the red serisation which is wanting, that is, that supposed system of nerves which is affected in various degrees by all light, but chiefly by red. We have fortunately the means of testing this hypothesis by numerical results.
Of the subjects of my experiments at Cambridge, four were decided cases of colour-bHndness. Of these two, namely, Mr E. and Mr S., were not suflficiently critical in their observations to afford any results consistent within 10 divisions of the colour-top. The remaining two, Mr N. and Mr X., were as consistent in their observations as any persons of ordinary vision can be, while the results shewed all the more clearly how completely their sensations must differ from ours.
The method of experimenting was the same as that adopted with ordinary eyes, except that in these cases the operator can hardly influence the result by yielding to his own impressions, as he has no perception whatever of the similarity of the two tints as seen by the observer. The questions which he must ask are two, Which circle appears most blue or yellow ? Which appears lightest and which darkest ? By means of the answers to these questions he must adjust the resulting tints to equality in these respects as it appears to the observer, and then ascertain that these tints now present no difference of colour whatever to his eyes. The equations thus obtained do not require five colours including black, but four only. For instance, the mean of several obser- vations gives —
•19 G+'05 B + -76 Bk=100R (4).
[In these experiments R, B, G, Y, stand for red, blue, green, and yellow papers prepared by Mr D. R. Hay. I am not certain that they are identical with his standard colours, but I beUeve so. Their relation to vermihon, ultra- marine, and emerald-green is given in diagram (1). Their relations to each other are very accurately given in diagram (2).]
It appears, then, that the dark blue-green of the left side of the equation is equivalent to the full red of the right side.
EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 139
Hence, if we divide the line BG in the proportion 19 to 5 at the point y8, and join R)8, the tint at ^ will differ from that at R (to the colour-blind) only in being more brilliant in the proportion of 100 to 24, and all inter- mediate tints on the line R^ will appear to them of the same hue, but of intermediate intensities.
Now, if we take a point D, so that RD is to R^ in the proportion of 24 to 100 — 24, or 76, the tint of D, if producible, should be invisible to the colour-blind. D, therefore, represents the pure sensation which is unknown to the colour-blind, and the addition of this sensation to any others cannot alter it in their estimation. It is for them equivalent to black.
Hence, if we draw lines through D in different directions, the colours belonging to any line ought to differ only in intensity as seen by them, so that one of them may be reduced to the other by the addition of black only. If we draw DW and produce it, all colours on the upper side of DW will be varieties of blue, and those on the under side varieties of yellow, so that the line DW is a boundary line between their two kinds of colour, blue and yellow being the names by which they call them.
The accuracy of this theory will be evident from the comparison of the experiments which I had an opportunity of making on Mr N. and Mr X. with each other, and with measurements taken from the diagram No. 2, which was constructed from the observations of ordinary eyes only, the point D alone being ascertained from a series of observations by Mr N.
Taking the point y, between R and B, it appears, by measurement of the lines Ry and By, that y corresponds to
•07 B + -93R.
By measurement of Wy and Dy, and correction by means of the coeflScient of W, and caUing D black in the colour-blind language, y corresponds to
•105 W-f895 Bk. Therefore
By measurement -93 R+ '07 B = ^105 W + •sgs Bk 1
By observation N. & X. together "94 R-f -06 B = •lO W-f-^90 Bk I (5).
By X. alone -93 R-h-07 B = -10 W + -90 Bk J
The agreement here is as near as can be expected.
18—2
140
EXPERIMENTS ON COLOira, AS PERCEIVED BY THE EYE.
By a similar calculation with respect to the point 8, between B and G,
By measurement -43 B + -57 G = -335 W + *665 Bk 1
Observed by N. and X '41 B + '59 G = '34 W + -66 Bk I (6).
By X. alone -42 B + -58 G = -32 W + -68 Bk J
We may also observe, that the line GD crosses RY. At the point of inter- section we have —
By calculation '87 B + 'IS Y = -34 G + -66 Bk
Observed by N. and X -86 R + -14 Y = -40 G + 'GO Bk
X •84R + '16 Y=-31 G + '69 Bk
X -QOR + 'IO Y = -27 G + 73Bk
.(7).
Here observations are at variance, owing to the decided colours produced affecting the state of the retina, but the mean agrees well with calculation.
Drawing the line BY, we find that it cuts lines through D drawn to every colour. Hence all colours appear to the colour-blind as if composed of blue and yellow. By measurement on the diagram, we find for red
Measured -138 Y+-123 B + 749 Bk = 100 R'
Observed by N..., -15 Y + 'll B-1--74 Bk = 100RJ- (8).
X....-13 Y + 'll B + -76 Bk = 100R
.(9).
For green we have in the same way —
Measured 705 Y + -295 B = '95 G + -05 Bkl
Observed by N.... 70 Y + -30 B = -86 G + -14 Bk i .... X.... 70 Y+-30 B = '83 G+-17BkJ
For white —
Measured '407 Y + -593 B = '326 W + "674 Bk
Observed by N.... -40 Y+-60 B = -33 W+-67 Bk X.... -44 Y+-56 B=-33 W+-67 Bk
The accuracy of these results shews that, whether the hypothesis of the want of one element out of three necessary to perfect vision be actually true or not, it affords a most trustworthy foundation on which to build a theory of colour-blindness, as it expresses completely the observed facts of the case. They also furnish us with a datum for our theory of perfect vision, namely, the point D, which points out the exact nature of the colour-sensation, which must be added to the colour-blind eye to render it perfect. I am not aware
EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 141
of any method of determining by a legitimate process the nature of the other two sensations, although Young's reasons for adopting something like green and violet appear to me worthy of attention.
The only remaining subject to which I would call the attention of the Society is the effect of coloured glasses on the colour-blind. Although they can- not distinguish reds and greens from varieties of gray, the transparency of red and green glasses for those kinds of light is very different. Hence, after finding a case such as that in equation (4), in which a red and a green appear iden- tical, on looking through a red glass they see the red clearly and the green obscurely, while through a green glass the red appears dark and the green light.
By furnishing Mr X. with a red and a green glass, which he could dis- tinguish only by their shape, I enabled him to make judgments in previously doubtful cases of colour with perfect certainty. I have since had a pair of spectacles constructed with one eye-glass red and the other greeiL These Mr X. intends to use for a length of time, and he hopes to acquire the habit of discri- minating red from green tints by their different effects on his two eyes. Though he can never acquire our sensation of red, he may then discern for himself what things are red, and the mental process may become so familiar to him as to act unconsciously like a new sense.
In one experiment, after looking at a bright light, with a red glass over one eye and a green over the other, the two tints in experiment (4) appeared to him altered, so that the outer circle was lighter according to one eye, and the inner according to the other. As far as I could ascertain, it appeared as if the eye which had used the red glass saw the red circle brightest. This result, which seems at variance with what might be expected, I have had no opportunity of verifying.
This paper is already longer than was originally intended For further information I would refer the reader to Newton's Optich, Book i. Part ii., to Young's Lectures on Natural Philosophy, page 345, to Mr D. R. Hay's works on Colours, and to Professor Forbes on the "Classification of Colours" (Phil. Mag., March, 1849).
The most remarkable paper on the subject is that of M. Helmholtz, in the Philosophical Magazine for 1852, in which he discusses the different theories of primary colours, and describes his method of mixing the colours of the spectrum. An examination of the results of M. Helmholtz with reference to the theory
142 EXPERIMENTS OX COLOUR, AS PERCBIV^ED BY THE EYE.
of three elements of colour, by Professor Grassmann, is translated in the Phil. Mag., April, 1854.
References to authors on colour-blindness are given in Dr G. Wilson's papers on that subject. A valuable Letter of Sir J. F. W. Herschel to Dalton on his peculiarity of vision, is to be found in the Life of Dalton by Dr Henry.
I had intended to describe some experiments on the propriety of the method of mixino- colours by rotation, which might serve as an extension of Mr Swan's experiments on instantaneous impressions on the eye. These, together with the explanation of some phenomena which seem to be at variance with the theory of vision here adopted, must be deferred for the present. On some future occasion, I hope to be able to connect these simple experiments on the colours of pigments with others in which the pure hues of the spectrum are used. I have already constructed a model of apparatus for this purpose, and the results obtained are sufficiently remarkable to encourage perseverance.
Note I. On different Methods of Exhibiting the Mixtures of Colours.
(1) Mechanical Mixture of Coloured Powders. By grinding coloured powders together, the differently- coloured particles may be so intermingled that the eye cannot distinguish the colours of the separate powders, but receives the impression of a uniform tint, depending on the nature and proportions of the pigments used. In this way, Newton mixed the powders of orpiment, purple, bise, and viride ceris, so as to form a gray, which, in sun- light, resembled white paper in the shade. (Newton's Opticks, Book i. Part n., Exp. XV.) This method of mixture, besides being adopted by all painters, has been employed by optical writers as a means of obtaining numerical results. The specimens of such mixtures given by B. R. Hay in his works on Colour, and the experiments of Professor J. D. Forbes on the same subject, shew the importance of the method as a means of classifying colours. There are two objections, however, to this method of exhibiting colours to the eye. When two powders of unequal fineness are mixed, the particles of the finer powder cover over those of the coarser, so as to produce more than their due effect in influencing the resultant tint. For instance, a small quantity of lamp-black.
EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 143
mixed with a large quantity of chalk, will produce a mixture which is nearly black. Although the powders generally used are not so different in this respect as lamp-black and chalk, the results of mixing given weights of any coloured powders must be greatly modified by the mode in which these powders have been prepared.
Again, the light which reaches the eye from the surface of the mixed pow- ders consists partly of light which has fallen on one of the substances mixed without being modified by the other, and partly of light which, by repeated reflection or transmission, has been acted on by both substances. The colour of these rays will not be a mixture of those of the substances, but will be the result of the absorption due to both substances successively. Thus, a mixture of yellow and blue produces a neutral tint tending towards red, but the remainder of white light, after passing through both, is green; and this green is generally sufficiently powerful to overpower the reddish gray due to the separate colours of the substances mixed. This curious result has been ably investigated by Professor Helmholtz of Konigsberg, in his Memoir on the Theory of Compound Colours, a translation of which may be found in the Annals of Philosophy for 1852, Part 2.
(2) Mixture of differently-coloured Beams of Light by Superposition
on an Opaque Screen. When we can obtain light of sufficient intensity, this method produces the most beautiful results. The best series of experiments of this kind are to be found in Newton's Opticks, Book i. Part ii. The different arrangements for mixing the rays of the spectrum on a screen, as described by Newton, form a very complete system of combinations of lenses and prisms, by which almost every possible modification of coloured light may be produced. The principal objections to the use of this method are— (1) The difficulty of obtaining a con- stant supply of uniformly intense light; (2) The uncertainty of the effect of the position of the screen with respect to the incident beams and the eye of the observer; (3) The possible change in the colour of the incident light due to the fluorescence of the substance of the screen. Professor Stokes haa found that many substances, when illuminated by homogeneous light of one refrangi- bility, become themselves luminous, so as to emit light of lower refrangibility. This phenomenon must be carefully attended to when screens are used to exhibit light.
144 EXPERIMENTS ON COLOUK, AS PERCEIVED BY THE EYE.
(3) Union of Coloured Beams hy a Piism so as to form one Beam.
The mode of viewing the beam of light directly, without first throwing it on a screen, was not much used by the older experimenters, but it possesses the advantage of saving much light, and admits of examining the rays before they have been stopped in any way. In Newton's 11th proposition of the 2nd Book, an experiment is described, in which a beam is analysed by a prism, concentrated by a lens, and recombined by another prism, so as to form a beam of white light similar to the incident beam. By stopping the coloured rays at the lens, any proposed combination may be made to pass into the emergent beam, where it may be received directly by the eye, or on a screen, at pleasure.
The experiments of Helmholtz on the colours of the spectrum were made with the ordinary apparatus for directly viewing the pure spectrum, two oblique slits crossing one another being employed to admit the light instead of one vertical sht. Two pure spectra were then seen crossing each other, and so exhibiting at once a large number of combinations. The proportions of these combinations were altered by varying the inclination of the slits to the plane of lefraction, and in this way a number of very remarkable results were obtained, — for which see his Memoir, before referred to.
In experiments of the same kind made by myself in August 1852 (inde- pendently of M. Helmholtz), I used a combination of three moveable vertical slits to admit the light, instead of two cross shts, and observed the compound ray through a slit made in a screen on which the pure spectrum is formed. In this way a considerable field of view was filled with the mixed light, and might be compared with another part of the field illuminated by light proceeding from a second system of slits, placed below the first set. The general character of the results agreed with those of M. Helmholtz. The chief difficulties seemed to arise from the defects of the optical apparatus of my own eye, which ren- dered apparent the compound nature of the light, by analysing it as a prism or an ordinary lens would do, whenever the lights mixed differed much in refrangibility.
EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 145
(4) Union of two beams by means of a transparent surface, which reflects the first and transmits the second.
The simplest experiment of this kind is described by M. Helmholtz. He places two coloured wafers on a table, and then, taking a piece of transparent glass, he places it between them, so that the reflected image of one apparently coincides with the other as seen through the glaas. The colours are thus mixed, and, by varying the angle of reflection, the relative intensities of the reflected and transmitted beams may be varied at pleasure.
In an instrument constructed by myself for photometrical purposes two re- flecting plates were used. They were placed in a square tube, so as to polarize the incident light, which entered through holes in the sides of the tubes, and was reflected in the direction of the axis. In this way two beams oppositely polarized were mixed, either of which could be coloured in any way by coloured glasses placed over the holes in the tube. By means of a Nicol's prism placed at the end of the tube, the relative intensities of the two colours as they entered the eye could be altered at pleasure.
(5) Union of two coloured beams by means of a doubly -refracting Prism.
I am not aware that this method has been tried, although the opposite polarization of the emergent rays is favourable to the variation of the experiment.
(6) Successive presentation of the different Colours to the Retina.
It has long been known, that light does not produce its full effect on the eye at once, and that the effect, when produced, remains visible for some time after the light has ceased to act. In the case of the rotating disc, the various colours become indistinguishable, and the disc appears of a imiform tint, which is in some sense the resultant of the colours so blended. This method of com- bining colours has been used since the time of Newton, to exhibit the results of theory. The experiments of Professor J. D. Forbes, which I witnessed in 1849, first encouraged me to think that the laws of this kind of mixture might be discovered by special experiments. After repeating the well-known experiment in which a series of colours representing those of the spectrum are combined
VOL. I. ^^
146 EXPERIMENTS ON COLOUB, AS PERCEIVED BY THE EYE.
to form gray, Professor Forbes endeavoured to form a neutral tint, by the combination of three colours only. For this purpose, he combined the three so-called primary colours, red, blue, and yellow, but the resulting tint could not be rendered neutral by any combination of these colours ; and the reason was found to be, that blue and yellow do not make green, but a pinkish tint, when neither prevails in the combination. It was plain, that no addition of red to this, could produce a neutral tint.
This result of mixing blue and yellow was, I beUeve, not previously known. It directly contradicted the received theory of colours, and seemed to be at variance with the fact, that the same blue and yellow paint, when ground together, do make green. Several experiments were proposed by Professor Forbes, in order to eliminate the effect of motion, but he was not then able to under- take them. One of these consisted in viewing alternate stripes of blue and yellow, with a telescope out of focus. I have tried this, and find the resultant tint pink as before*. I also found that the beams of light coloured by trans- mission through blue and yellow glasses appeared pink, when mixed on a screen, while a beam of light, after passing through both glasses, appeared green. By the help of the theory of absorption, given by Herschelf, I made out the complete explanation of this phenomenon. Those of pigments were, I think, first explained by Helmholtz in the manner above referred to J.
It may still be asked, whether the effect of successive presentation to the eye is identical with that of simultaneous presentation, for if there is any action of the one kind of light on the other, it can take place only in the case of vsimultaneous presentation. An experiment tending to settle this point is recorded by Newton (Book i. Part ii., Exp. 10). He used a comb with large teeth to intercept various rays of the spectrum. When it was moved slowly, the various colours could be perceived, but when the speed was increased the result was perfect whiteness. For another form of this experiment, see Newton's Sixth Letter to Oldenburg (Horsley's Edition, Vol. iv., page 335).
In order more fully to satisfy myself on this subject, I took a disc in which were cut a number of sUts, so as to divide it into spokes. In a plane, net-rly passing through the axis of this disc, I placed a blue glass, so that one
* See however Encyc. Metropolitana, Art. "Light," section 502. t lb. sect. 516.
X I have lately seen a passage in Moigno's Cosmos, stating that M. Plateau, in 1819, had obtained jjray by whirling together gamboge and Prussian blue. Correspondance Math, et Phys. de M. Quet«let, Vol. v., p. 221.
EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 147
half of the disc might be seen by transmitted light — blue, and the other by reflected light — white. In the course of the reflected light I placed a yellow glass, and in this way I had two nearly coincident images of the slits, one yellow and one blue. By turning the disc slowly, I observed that in some parts the yellow slits and the blue slits appeared to pass over the field alter- nately, while in others they appeared superimposed, so as to produce alternately their mixture, which was pale pink, and complete darkness. As long as the disc moved slowly I could perceive this, but when the speed became great, the whole field appeared uniformly coloured pink, so that those parts in which the colours were seen successively were indistinguishable from those in which they were presented together to the eye.
Another form in which the experiment may be tried requires only the colour-top above described. The disc should be covered with alternate sectors of any two colours, say red and green, disposed alternately in four quadrants. By placing a piece of glass above the top, in the plane of the axis, we make the image of one half seen by reflection coincide with that of the other seen by transmission. It wiU then be seen that, in the diameters of the top which are parallel and perpendicular to the plane of reflection, the transmitted green coincides with the reflected green, and the transmitted red with the reflected red, so that the result is always either pure red or pure green. But in the diameters intermediate to these, the transmitted red coincides with the reflected green, and vice versa, so that the pure colours are never seen, but only their mixtures. As long as the top is spun slowly, these parts of the disc will appear more steady in colour than those in which the greatest alternations take place ; but when the speed is sufficiently increased, the disc appears per- fectly uniform in colour. From these experiments it appears, that the apparent mixture of colours is not due to a mechanical superposition of vibrations, or to any mutual action of the mixed rays, but to some cause residing in the constitution of the apparatus of vision.
(7) Presentation of the Colours to he mixed one to each Eye.
This method is said not to succeed with some people ; but I have always found that the mixture of colours was perfect, although it was difficult to con- ceive the objects seen by the two eyes as identical. In using the spectacles,
19—2
148 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE.
of which one eye is green and the other red, I have found, when looking at an arrangement of green and red papers, that some looked metallic and others transparent. This arises from the very different relations of brightness of the two colours as seen by each eye through the spectacles, which suggests the false conclusion, that these differences are the result of reflection from a polished surface, or of light transmitted through a clear one.
Note IT.
Results of Experiments with Mr Hay's Papers at Cambridge, November, 1854.
The mean of ten observations made by six observers gave
•449 E+-299 G + -252 B=-224 W+776 Bk (l).
■696 R+-304 G = '181 B + -327 Y + '492 Bk (2).
These two equations served to determine the positions of white and yellow in diagram No. 2. The coeflScient of W is 4*447, and that of yellow 2'506.
From these data we may deduce three other equations, either by calcu- lation, or by measurement on the diagram (No. 2).
Eliminating green from the equations, we find
•565 B + -435 Y = -307 E. + -304 W + -389 Bk (3).
The mean of three observations by three different observers gives
•573 B-f477 Y = ^313 E + ^297 W + -390Bk. Errors of calculation - '008 B + ^008 Y - '006 K + ^007 W - •OOl Bk.
The point on the diagram to which this equation corresponds is the intersec- tion of the lines BY and RW, and the resultant tint is a pinkish-gray.
Eliminating red from the equations, we obtain Calculation "533 B-fl50 G-f317 Y = ^337 W-f -663 Bk"
By 10 observations -537 B-l- '146 G-h ^317 Y= -337 W-f '663 Bk ■ (4).
Errors -'004 -f- -004 — — —
Eliminating blue •660 R-f340 G = -218 Y + -108 W-f '682 Bkl
By 5 observations ^672 R-f '328 G = "224 Y+ '094 W-f672 Bk i (5).
Errors -'012 -f012 -•006 -f014 -f008 I
EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 149
Note III.
On the Tlicory of Compound Colours.
Newton's theorem on the mixture of colours is to be found in his Opticks, Book I., Part ii., Prop. vi.
In a mixtiu'e of primary colours^ the quantity and quality of each being gicen, to know the colour of the compound.
He divides the circumference of a circle into parts proportional to the seven musical intervals, in accordance with